# Higher order parabolic boundary Harnack inequality in $C^{1}$ and
$C^{k,\alpha}$ domains

Teo Kukuljan  Universitat de Barcelona, Departament de Matematiques i
Informàtica, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain.
<EMAIL_ADDRESS>
###### Abstract.

We study the boundary behaviour of solutions to second order parabolic linear
equations in moving domains. Our main result is a higher order boundary
Harnack inequality in $C^{1}$ and $C^{k,\alpha}$ domains, providing that the
quotient of two solutions vanishing on the boundary of the domain is as smooth
as the boundary.

As a consequence of our result, we provide a new proof of higher order
regularity of the free boundary in the parabolic obstacle problem.

###### Key words and phrases:

Parabolic equations, boundary Harnack, free boundary, higher regularity.

###### 2010 Mathematics Subject Classification:

35K10, 35R35.

The author has received funding from the European Research Council (ERC) under
the Grant Agreement No 801867, and from the Swiss National Science Foundation
project 200021_178795. The paper was written under supervision of Xavier Ros-
Oton, we thank him for proposing many ideas and approaches.

##  1\. Introduction

We study the boundary behaviour of solutions of

$\left\\{\begin{array}[]{rcll}\partial_{t}u+Lu&=&f&\text{in }\Omega\cap
Q_{1}\\\ u&=&0&\text{on }\partial\Omega\cap Q_{1},\end{array}\right.$ (1.1)

where $\Omega\subset\mathbb{R}^{n+1}$ is an open set,
$Q_{1}=B_{1}\times(-1,1)$ and $L$ is a second order operator of the form

$Lu(x,t)=-\sum_{i,j=1}^{n}a_{ij}(x,t)\partial_{ij}u(x,t)+\sum_{i=1}^{n}b_{i}(x,t)\partial_{i}u(x,t).$
(1.2)

The matrix $A(x,t):=\left[a_{i,j}(x,t)\right]_{ij}$ is assumed to be uniformly
elliptic and the vector $b(x,t)=\left[b_{i}(x,t)\right]_{i}$ is bounded, that
is, for every $(x,t)$ it holds

$0<\lambda I\leq A(x,t)\leq\Lambda I,\quad\quad||b||_{L^{\infty}}\leq\Lambda.$
(1.3)

In the elliptic version of the problem, classical boundary Schauder estimates
imply that solutions vanishing on the boundary of a $C^{k,\alpha}$ domain are
of class $C^{k,\alpha}$ up to the boundary. This implies that the quotient of
the solution with the distance function to the boundary is $C^{k-1,\alpha}$.
This has been refined by De Silva and Savin [6], who showed that replacing the
distance function with another solution gives that the quotient is
$C^{k,\alpha}$ as well, under positivity assumption on the solution in the
denominator. This is a higher order boundary Harnack estimate.

Such result was extended to the parabolic setting in [3], where Banerjee and
Garofalo established a new parabolic higher order boundary Harnack estimate in
$C^{k,\alpha}$ domains.

The goal of this paper is twofold. On the one hand, we give a new proof of the
result in [3], which is based on blow-up and contradiction arguments. On the
other hand, we extend the result to the case of parabolic $C^{1}$ domains,
which was not covered in [3]. This is important when applying these estimates
to obtain the higher regularity of free boundaries in parabolic obstacle
problems, as we will see below.

We work in domains $\Omega$ which are allowed to be moving in time, and have
to satisfy some parabolic regularity properties.

######  Definition 1.1.

Let $\beta\geq 1$ and let $C^{\beta}_{p}$ be the parabolic Hölder space
defined in Definition 2.2. We say that $\Omega\subset\mathbb{R}^{n+1}$ is
$C^{\beta}_{p}$ in $Q_{1}$, if there is a function $F\colon
Q_{1}^{\prime}\to\mathbb{R}$, with $||F||_{C^{\beta}_{p}(Q_{1}^{\prime})}\leq
C_{0}$ and $F(0,0)=0,$ $\nabla_{x^{\prime}}F(0,0)=0$ so that

$\Omega\cap Q_{1}=\\{(x^{\prime},x_{n},t);\text{ }x_{n}>F(x^{\prime},t)\\}.$

For a domain $\Omega$ as in the above definition we define the parabolic
distance to the boundary as follows

$d(x,t)=\inf_{(z,s)\in\partial\Omega\cap
Q_{1}}\left(|x-z|+|t-s|^{\frac{1}{2}}\right),\quad(x,t)\in\Omega\cap Q_{1}.$

Our first main result says that if we have two solutions of (1.1) in a
$C^{1}_{p}$ domain, with a bounded right-hand side, then their quotient is
nearly as smooth as the boundary. This is a parabolic boundary Harnack
inequality, and reads as follows.

######  Theorem 1.2.

Let $\Omega$ be $C^{1}_{p}$ in $Q_{1}$, and let $L$ be as in (1.2)–(1.3), with
$A\in C(\overline{\Omega})$ and $b\in L^{\infty}(\Omega).$ Let $u_{i}$ be two
solutions of (1.4) with $f_{i}\in L^{\infty}(\Omega\cap Q_{1}).$ Assume that
$|u_{2}|\geq c_{2}d$ with $c_{2}>0$ and $||f_{2}||_{L^{\infty}(\Omega\cap
Q_{1})}+||u_{2}||_{L^{\infty}(\Omega\cap Q_{1})}\leq C_{2}.$

Then for any $\varepsilon\in(0,1)$ we have

$\left|\left|\frac{u_{1}}{u_{2}}\right|\right|_{C^{1-\varepsilon}_{p}(\overline{\Omega}\cap
Q_{1/2})}\leq C\left(||f_{1}||_{L^{\infty}_{p}(\Omega\cap
Q_{1})}+||u_{1}||_{L^{\infty}(\Omega\cap Q_{1})}\right),$

with $C$ depending only on $n,\varepsilon,c_{2},C_{2},\Omega$, the modulus of
continuity of $A$ and ellipticity constants.

It is established through expanding one solution with respect to the other
one, combined with boundary regularity estimates for solutions. With a
similar, but a bit more involved approach, we can also prove the higher order
analogue of the result - the higher order parabolic boundary Harnack
inequality.

######  Theorem 1.3.

Let $\beta>1$, $\beta\not\in\mathbb{N}$. Let $\Omega\subset\mathbb{R}^{n+1}$
be $C^{\beta}_{p}$ in $Q_{1}$ according to Definition 1.1. Let $L$ be an
operator of the form (1.2) satisfying conditions (1.3) with the coefficients
$A,b\in C^{\beta-1}_{p}(\overline{\Omega})$. For $i=1,2$ let $u_{i}$ be a
solution to

$\left\\{\begin{array}[]{rcl l}\partial_{t}u_{i}+Lu_{i}&=&f_{i}&\text{in
}\Omega\cap Q_{1}\\\ u_{i}&=&0&\text{on }\partial\Omega\cap Q_{1},\\\
\end{array}\right.$ (1.4)

with $f_{i}\in C^{\beta-1}_{p}(\Omega\cap Q_{1})$. Assume that $|u_{2}|\geq
c_{2}d$ with $c_{2}>0$ and $||f_{2}||_{C^{\beta-1}_{p}(\Omega\cap
Q_{1})}+||u_{2}||_{L^{\infty}(\Omega\cap Q_{1})}\leq C_{2}.$

Then we have

$\left|\left|\frac{u_{1}}{u_{2}}\right|\right|_{C^{\beta}_{p}(\overline{\Omega}\cap
Q_{1/2})}\leq C\left(||f_{1}||_{C^{\beta-1}_{p}(\Omega\cap
Q_{1})}+||u_{1}||_{L^{\infty}(\Omega\cap Q_{1})}\right),$

with $C$ depending only on $n,\beta,c_{2},C_{2},\Omega$, ellipticity
constants, $||A||_{C^{\beta-1}_{p}(\overline{\Omega})}$ and
$||b||_{C^{\beta-1}_{p}(\overline{\Omega})}$.

Theorem 1.3 is already known under a bit stricter assumptions, see [3]. We
provide a new, different proof, which relaxes the assumption on regularity of
the domain from $C^{1,\alpha}$ in space and time, to $C^{1,\alpha}_{p}$.

One motivation for studying such equations comes from the parabolic obstacle
problem, where we look for a function $v\colon Q_{1}\to\mathbb{R}$ solving

$\partial_{t}v-\Delta v=-f\chi_{\\{v>0\\}},\quad v\geq 0,$ (1.5)

for some function $f$ depending only on $x$, together with some boundary data
on $\partial_{p}Q_{1}=\partial Q_{1}\backslash(B_{1}\times\\{1\\}).$ Of
particular interest is the so called contact set $K=\\{v=0\\},$ and its
boundary $\partial K,$ called the free boundary. If we denote
$\Omega=\\{v>0\\}$, then in particular the solution $v$ solves

$\left\\{\begin{array}[]{rcll}\partial_{t}v-\Delta v&=&-f&\text{in }\Omega\cap
Q_{1}\\\ v&=&0&\text{on }\partial\Omega\cap Q_{1}.\end{array}\right.$

Theorem 1.2 and Theorem 1.3 give a simple proof of the higher regularity of
the free boundary in the parabolic obstacle problem, without making use of a
hodograph transform as in [11, Theorem 3]. Analogous boundary Harnack
inequalities give rise to the bootstrap argument, which yields the higher
order regularity of the free boundaries in similar obstacle problems. Some
examples are [6], where they establish it in the classical case, [10] for the
case when the operator is the fractional Laplacian and [4] for the parabolic
thin obstacle problem.

######  Corollary 1.4.

Let $v\colon Q_{1}\to\mathbb{R}$ solve the parabolic obstacle problem (1.5)
with $f\geq c_{0}>0$, $f\in C^{\theta}(B_{1})$, for some $\theta>1$. Assume
that $(0,0)\in\partial\\{v>0\\}$, that $\partial\\{v>0\\}$ is $C^{1}_{p}$ in
$Q_{1}$ in the sense of Definition 1.1. Then $\partial\\{v>0\\}$ is
$C^{\theta+1}_{p}$ in $Q_{1/2}$.

In particular, if $f\in C^{\infty}(B_{1}),$ then the free boundary is
$C^{\infty}$ near the origin in space and time.

######  Remark 1.5.

In the classical one-phase Stefan problem the function $f$ in the right-hand
side is constantly equal to $1$ and moreover one has $\partial_{t}v\geq 0$.
Then the initial $C^{1,\alpha}$ regularity of the free boundary near regular
free boundary points111Regular points are those at which the blow-up is a
$1$-dimensional profile of the form $\frac{1}{2}(x\cdot e)_{+}^{2}$; see [16]
for more details. is established in [5]. For general right-hand side $f$ in
(1.5), we refer to [16, Theorem 1.7]. There the provided initial regularity is
indeed $C^{1}_{p}$ only.

Let us stress, that one often does not have $C^{1,\alpha}$ initial regularity.
Another example is the free boundary problem for fully non-linear parabolic
equations, studied in [9]. There they establish the $C^{1}$ regularity of the
free boundary in both space and time. This is why it is important to have
Theorem 1.2 in $C^{1}_{p}$ domains in order to establish the higher regularity
of free boundaries.

###  1.1. Strategy of the proofs

The main ingredient for proving Theorem 1.2 and Theorem 1.3 is establishing an
expansion result of the form

$|u_{1}(x,t)-p(x,t)u_{2}(x,t)|\leq
C\left(|x|^{\beta+1}+|t|^{\frac{\beta+1}{2}}\right),\quad(x,t)\in Q_{1},$

for some polynomial $p$ of parabolic degree $\lfloor\beta\rfloor$. To get such
an expansion, we perform a contradiction in combination with a blow-up
argument. The contradiction is reached with Liouville theorem in half-space
for the parabolic setting. We follow the ideas of an analogous result in the
non-local elliptic setting [1]. In particular our proof is different from the
one of De Silva-Savin in [6].

###  1.2. Notation

For a real number, we denote $\lfloor\cdot\rfloor$ its integer part and
$\langle x\rangle=x-\lfloor x\rfloor$.

The ambient space is $\mathbb{R}^{n+1}=\mathbb{R}^{n}\times\mathbb{R},$ where
the first $n$ coordinates we denote with $x=(x_{1},\ldots,x_{n})$ and the last
one with $t$. Sometimes we furthermore split $x=(x^{\prime},x_{n})$, for
$x^{\prime}=(x_{1},\ldots,x_{n-1})$. Accordingly we use the multi-index
notation $\alpha\in\mathbb{N}_{0}^{n+1}$,
$\alpha=(\alpha_{1},\ldots,\alpha_{n},\alpha_{t})$,
$|\alpha|=\alpha_{1}+\ldots+\alpha_{n}+\alpha_{t}$, and furthermore

$\partial^{\alpha}=\left(\frac{\partial}{\partial
x_{1}}\right)^{\alpha_{1}}\circ\ldots\circ\left(\frac{\partial}{\partial
x_{n}}\right)^{\alpha_{n}}\circ\left(\frac{\partial}{\partial
t}\right)^{\alpha_{t}}.$

The gradient operator $\nabla$ and Laplace operator $\Delta$ are taken only in
$x$ variables. When we want to use the full coordinate operators, we denote it
with a subscript $\nabla_{(x,t)}$.

For $\Omega\subset\mathbb{R}^{n+1}$ we denote the time slits with
$\Omega_{t}=\\{x\in\mathbb{R}^{n};\text{ }(x,t)\in\Omega\\}$ and the distance
function to the boundary in space directions only with
$d_{t}(x)=d_{x}(x,t)=\inf_{z\in\partial\Omega_{t}}|z-x|.$ Moreover its
parabolic boundary $\partial_{p}\Omega$ is the set
$\\{(x,t)\in\partial\Omega;\text{ }\forall r>0:\text{
}B_{r}(x)\times(t-r,t)\not\subset\Omega\\}.$

We denote with $Q_{r}(x_{0},t_{0})$ the following parabolic cylinder of radius
$r$ centred at $(x_{0},t_{0})$,

$Q_{r}(x_{0},t_{0})=B_{r}(x_{0})\times(t_{0}-r^{2},t_{0}+r^{2}).$

When $(x_{0},t_{0})=(0,0)$, we denote it simply $Q_{r}$. Sometimes we also
denote

$Q_{r}^{+}=Q_{r}\cap\\{x_{n}>0\\}.$

Finally, $C$ indicates an unspecified constant not depending on any of the
relevant quantities, and whose value is allowed to change from line to line.
We make use of sub-indices whenever we will want to underline the dependencies
of the constant.

###  1.3. Organisation of the paper

In Section 2, we present the definitions of parabolic polynomials, parabolic
derivatives and parabolic Hölder spaces, notion of viscosity solutions, and
cite the results about them which we use. In Section 3 we establish basic
results regarding the behaviour of solutions of (1.1) near the boundary. Well
equipped in Section 4 we prove the boundary Harnack estimate in $C^{1}_{p}$
domains and Theorem 1.2. In the following Section 5 we prove the higher order
boundary Schauder estimates and Theorem 1.3. Finally, the proof of Corollary
1.4 is presented in Section 6. At the end there is an appendix, where we
establish the results regarding the interior regularity of solutions of (1.1)
needed for our specific setting. Moreover we there prove technical auxiliary
results, to lighten the body of the paper.

##  2\. Preliminaries

###  2.1. Parabolic Hölder spaces

We follow the definitions of the parabolic Hölder spaces from [13]. We start
with defining parabolic derivatives and parabolic polynomial spaces.

######  Definition 2.1.

For a multi-index $\alpha\in\mathbb{N}_{0}^{n+1}$, we denote $\alpha_{x}$ the
first $n$ components and $\alpha_{t}$ the last one and define

$|\alpha|_{p}=|\alpha_{x}|+2\alpha_{t}=\sum_{i=1}^{n}\alpha_{i}+2\alpha_{t}.$

Furthermore, we define the parabolic derivatives with

$D^{k}_{p}=\\{\partial^{\alpha};\text{ }|\alpha|_{p}=k\\}.$

Similarly, we define the parabolic polynomial spaces as follows:

$\textbf{P}_{k,p}=\left\\{\sum_{|\alpha|_{p}\leq
k}c_{\alpha}(x,t)^{\alpha};\text{ }c_{\alpha}\in\mathbb{R}\right\\}.$

We say that $k$ is the parabolic degree of polynomial $p$, if $k$ is the least
integer so that $p\in\textbf{P}_{k,p}.$ For a polynomial
$p=\sum_{\alpha}c_{\alpha}(x,t)^{\alpha}$, we denote

$||p||=\sum_{\alpha}|c_{\alpha}|.$

Next we define parabolic Hölder seminorms and spaces.

######  Definition 2.2.

Let $\Omega$ be an open subset of $\mathbb{R}^{n+1}$. For $\alpha\in(0,1]$ we
define the parabolic Hölder seminorm of order $\alpha$ as follows

$\left[u\right]_{C^{\alpha}_{p}(\Omega)}=\sup_{(x,t),(y,s)\in\Omega}\frac{|u(x,t)-u(y,s)|}{|x-y|^{\alpha}+|t-s|^{\frac{\alpha}{2}}},$

and

$\left[u\right]_{C^{\alpha}_{t}(\Omega)}=\sup_{(x,t),(x,s)\in\Omega}\frac{|u(x,t)-u(x,s)|}{|t-s|^{\alpha}}.$

If $\alpha\in(1,2]$, we set

$\displaystyle\left[u\right]_{C^{\alpha}_{p}(\Omega)}$
$\displaystyle=\left[\nabla
u\right]_{C^{\alpha-1}_{p}(\Omega)}+\left[u\right]_{C^{\frac{\alpha}{2}}_{t}(\Omega)}.$

For bigger numbers $\alpha>2$, we set

$\left[u\right]_{C^{\alpha}_{p}(\Omega)}=\left[\nabla
u\right]_{C^{\alpha-1}_{p}(\Omega)}+\left[\partial_{t}u\right]_{C^{\alpha-2}_{p}(\Omega)}.$

When $\alpha\not\in\mathbb{N}$ we say that $u\in C^{\alpha}_{p}(\Omega)$, when
$\left[u\right]_{C^{\alpha}_{p}(\Omega)}<\infty$, and define

$||u||_{C^{\alpha}_{p}(\Omega)}=\sum_{k\leq\lfloor\alpha\rfloor}||D^{k}_{p}u||_{L^{\infty}(\Omega)}+\left[u\right]_{C^{\alpha}_{p}(\Omega)}.$

If $\alpha\in\mathbb{N}$, we say that $u\in C^{\alpha}_{p}(\Omega)$, if there
exists a modulus of continuity
$\omega\colon\left[0,\infty\right)\to\left[0,\infty\right)$ – a continuous,
increasing function with $\omega(0)=0$ – so that for all
$(x,t),(y,s)\in\Omega$

$|D^{\alpha}_{p}u(x,t)-D^{\alpha}_{p}u(y,s)|\leq\omega(|x-y|+|t-s|^{\frac{1}{2}}),$

and

$|D^{\alpha-1}_{p}u(x,t)-D^{\alpha-1}_{p}u(x,s)|\leq|t-s|^{\frac{1}{2}}\omega(|t-s|^{\frac{1}{2}}).$

We set

$||u||_{C^{\alpha}_{p}(\Omega)}=\sum_{k\leq\alpha}||D^{k}_{p}u||_{L^{\infty}(\Omega)}.$

Moreover, we say $u\in C^{\alpha-1,1}_{p}(\Omega)$ if
$\left[u\right]_{C^{\alpha}_{p}(\Omega)}<\infty$ and we set

$||u||_{C^{\alpha-1,1}_{p}(\Omega)}=\sum_{k\leq\alpha-1}||D^{k}_{p}u||_{L^{\infty}(\Omega)}+\left[u\right]_{C^{\alpha}_{p}(\Omega)}.$

######  Remark 2.3.

Breaking down the definition of the parabolic Hölder seminorms, we get that

$\left[u\right]_{C^{\alpha}_{p}(\Omega)}=\left[D^{\lfloor\alpha\rfloor}_{p}u\right]_{C^{\langle\alpha\rangle}_{p}(\Omega)}+\left[D^{\lfloor\alpha\rfloor-1}_{p}u\right]_{C^{\frac{\langle\alpha\rangle+1}{2}}_{t}(\Omega)}.$

Notice that the seminorms are compatible with parabolic scaling. If
$u_{r}(x,t)=u(rx,r^{2}t)$, then

$\left[u_{r}\right]_{C^{\alpha}_{p}(Q_{1})}=r^{\alpha}\left[u\right]_{C^{\alpha}_{p}(Q_{r})}.$

We want to remark as well, that for every power $\beta>0$, we have

$C^{-1}\left(|x|+|t|^{\frac{1}{2}}\right)^{\beta}\leq|x|^{\beta}+|t|^{\frac{\beta}{2}}\leq
C\left(|x|+|t|^{\frac{1}{2}}\right)^{\beta}.$

###  2.2. Viscosity solutions

For the notion of solutions of the equation (1.1) we use the viscosity
solutions.

######  Definition 2.4.

Let $\Omega\subset\mathbb{R}^{n+1}$ be an open set. We say that $u\in
C(\overline{\Omega})$ is a viscosity sub-solution of

$\partial_{t}u+Lu=f$

in $\Omega$, if for any function $\phi\in C^{2}_{p}(\Omega)$ touching $u$ from
below at some point $(x_{0},t_{0})\in\Omega$ it holds

$\partial_{t}\phi(x_{0},t_{0})+L\phi(x_{0},t_{0})\leq f(x_{0},t_{0}).$

Analogously we define viscosity super-solutions; the opposite inequality has
to hold for $C^{2}_{p}(\Omega)$ functions touching $u$ from above.

We say that $u\in C(\overline{\Omega})$ is a viscosity solution of (1.1) if it
is both viscosity sub-solution and viscosity super-solution.

One of the main reasons for using viscosity solutions is, because they are
well behaved under uniform convergence. We often use the stability result [2,
Theorem 1.1]. The existence and uniqueness result is established in [13,
Theorem 5.15], while the interior regularity results are provided in [17,
Theorem 2], when the right-hand side is bounded only, and in [13, Theorem
5.9], when the right-hand side is Hölder continuous.

##  3\. Basic boundary regularity results

Since solutions of (1.1) satisfy the comparison principle (see for example
[13, Lemma 2.1]), a first key to get estimates for solutions near the boundary
is constructing suitable barriers.

Let us start with providing the existence of a so called generalised distance
function. It is comparable to the parabolic distance to the boundary, but
moreover satisfies some additional interior regularity properties, which are
very important later on. The parabolic distance at a point $(x,t)\in\Omega$ is
given as

$\operatorname{dist}_{p}((x,t),\partial\Omega)=\inf_{(z,s)\in\partial\Omega}\left(|x-z|+|t-s|^{\frac{1}{2}}\right).$

######  Lemma 3.1.

Let $\beta\in\left[1,2\right)$ and let $\Omega\subset\mathbb{R}^{n+1}$ be
$C^{\beta}_{p}$ in $Q_{1}$ in the sense of Definition 1.1. Then there exists a
function $d\colon\Omega\cap Q_{1}\to\mathbb{R}$, satisfying

$C^{-1}\operatorname{dist}_{p}(\cdot,\partial\Omega)\leq d\leq
C\operatorname{dist}_{p}(\cdot,\partial\Omega),\quad d\in
C^{\beta}_{p}(Q_{1})\cap C^{2}_{p}(\Omega\cap
Q_{1}),\quad|\partial_{t}d|+|D^{2}d|\leq Cd^{\beta-2},$
$\frac{2}{3}\leq|\nabla d|\leq C.$

Moreover, when $\beta=1$, we have

$|\partial_{t}d|+|D^{2}d|\leq Cd^{-1}\omega(d),$

where $\omega$ is the modolus of continuity of $\nabla F$. The constant $C$
depends only on $\beta$, $||F||_{C^{\beta}_{p}(Q^{\prime}_{1})}$ and $n$.

###### Proof.

The construction is done in [13, Section IV.5]. The second part ($\beta=1$) is
established analogously as [14, Theorem 1.3]. ∎

With aid of the generalised distance we construct a barrier in $C^{1}_{p}$ and
$C^{1,\alpha}_{p}$ domains. The barrier is a non-negative function, vanishing
at the specific point of the boundary with positive right-hand side. When the
boundary is $C^{1}_{p}$ only, the obtained barrier vanishes a bit slower than
linearly.

######  Lemma 3.2.

Let $\Omega\subset\mathbb{R}^{n+1}$ be $C^{1}_{p}$ in $Q_{1}$ in the sense of
Definition 1.1. Let $L$ be an operator of the form (1.2) satisfying conditions
(1.3). Let $\varepsilon\in(0,1)$. Then for every boundary point
$(z,s)\in\partial\Omega\cap Q_{1/2}$ there exists a function $\psi_{(z,s)}$
satisfying the following properties:

$\left\\{\begin{array}[]{rcl
l}(\partial_{t}+L)\phi_{(z,s)}&\geq&d^{-1-\varepsilon}+1&\text{in }\Omega\cap
Q_{1}\cap\\{d<r\\}\\\ \phi_{(z,s)}&\geq&1&\text{in
}\Omega\cap\partial_{p}\left(Q_{1}\cap\\{d<r\\}\right)\\\
\phi_{(z,s)}&\geq&0&\text{in }\Omega\cap Q_{1}\cap\\{d<r\\}\\\
\phi_{(z,s)}(z,s)&=&0&\\\
\phi_{(z,s)}(z+\xi\nu_{x},s)&\leq&Cd^{1-\varepsilon}(z+\xi\nu_{x},s)&\text{for
}\xi<r.\end{array}\right.$

The constants $C$ and $r>0$ depend only on $n,\varepsilon,\lambda,\Lambda$,
$||F||_{C^{1}_{p}}$ and the modulus of continuity of $\nabla F$.

###### Proof.

Set first $\varphi=d^{1-\varepsilon}$. We compute

$\displaystyle(\partial_{t}+L)\varphi$
$\displaystyle=\varepsilon(1-\varepsilon)\nabla d^{T}A\nabla d\cdot
d^{-1-\varepsilon}+(1-\varepsilon)d^{-\varepsilon}(\partial_{t}+L)d$
$\displaystyle\geq\varepsilon(1-\varepsilon)\frac{4\lambda}{9}d^{-1-\varepsilon}-C\Lambda(1-\varepsilon)d^{-1-\varepsilon}\omega(d)-C\Lambda(1-\varepsilon)d^{-\varepsilon},$

thanks to Lemma 3.1. If $(x,t)\in\\{d<r\\},$ for $r$ so that
$\max(\omega(r),r)<\varepsilon\frac{\lambda}{9C\Lambda}$, we get

$(\partial_{t}+L)\varphi\geq\varepsilon(1-\varepsilon)\frac{2\lambda}{9}d^{-1-\varepsilon}.$

The barrier can be taken as
$\phi_{(z,s)}(x,t)=D\left(\varphi(x,t)+E(|x-z|^{2}+|t-s|^{2})\right)$, for
small enough $E>0$ and big enough $D$. ∎

In the case when the domain is $C^{1,\alpha}_{p}$, we can improve the barrier
so that it grows linearly near the boundary.

######  Lemma 3.3.

Let $\beta\in(1,2)$ and let $\Omega\subset\mathbb{R}^{n+1}$ be $C^{\beta}_{p}$
in $Q_{1}$ in the sense of Definition 1.1. Let $L$ be an operator of the form
(1.2) satisfying conditions (1.3). Then for every boundary point
$(z,s)\in\partial\Omega\cap Q_{1/2}$ there exists a function $\psi_{(z,s)}$
satisfying the following properties:

$\left\\{\begin{array}[]{rcl
l}(\partial_{t}+L)\phi_{(z,s)}&\geq&d^{\beta-2}+1&\text{in }\Omega\cap
Q_{1}\cap\\{d<r\\}\\\ \phi_{(z,s)}&\geq&1&\text{in
}\Omega\cap\partial_{p}\left(Q_{1}\cap\\{d<r\\}\right)\\\
\phi_{(z,s)}&\geq&0&\text{in }\Omega\cap Q_{1}\cap\\{d<r\\}\\\
\phi_{(z,s)}(z,s)&=&0&\\\
\phi_{(z,s)}(z+\xi\nu_{x},s)&\leq&Cd(z+\xi\nu_{x},s)&\text{for
}\xi<r.\end{array}\right.$

The constants $C$ and $r>0$ depend only on $n,\beta,\lambda,\Lambda$ and
$||F||_{C^{\beta}_{p}}.$

###### Proof.

Set $\varphi=d-Md^{\beta}$, for some positive constant $M$ specified later. We
compute

$(\partial_{t}+L)\varphi=(\partial_{t}+L)d(1-M\beta
d^{\beta-1})+M\beta(\beta-1)d^{\beta-2}\nabla d^{T}A\nabla d.$

Hence we can estimate

$\displaystyle(\partial_{t}+L)\varphi\geq$
$\displaystyle-|(\partial_{t}+L)d|(1+M\beta
d^{\beta-1})+M\beta(\beta-1)d^{\beta-2}\nabla d^{T}A\nabla d$
$\displaystyle\geq$ $\displaystyle-
Cd^{\beta-2}(1+2Md^{\beta-1})+\frac{4M}{9}\beta(\beta-1)d^{\beta-2}$

Choosing $M$ so that $\frac{4M}{9}\beta(\beta-1)>\max\\{3C,1\\}$, and then $r$
so that $2Md^{\beta-1}<1$, we get that for $x\in\\{d<r\\}\cap\\{d<1\\}$

$(\partial_{t}+L)\varphi\geq d^{\beta-2}\geq\frac{1}{2}(d^{\beta-2}+1).$

We now set
$\phi_{(z,s)}(x,t)=D\left(\varphi(x,t)+E((t-s)^{2}+|z-x|^{2})\right)$.
Constant $E$ has to be small enough so that
$D^{-1}(\partial_{t}+L)\phi_{(z,s)}\geq\frac{1}{4}(d^{\beta-2}+1)$, and then
$D$ is chosen big enough, so that $(\partial_{t}+L)\phi_{(z,s)}\geq
d^{\beta-2}+1$ and that $\phi_{(z,s)}\geq 1$ on
$\Omega\cap\partial_{p}\left(\\{d<r\\}\cap Q_{1}\right)$. Since
$\varphi(z+\xi\nu_{x},s)\leq Cd(z+\xi\nu_{x},s)$, provided that $r$ is small
enough, the same inequality holds for $\phi_{(z,s)}$ as well. ∎

The existence of such barriers straight forward imply that solutions to
parabolic equations grow near the boundary at most linearly (almost linearly
if the boundary is $C^{1}_{p}$).

######  Corollary 3.4.

Let $\beta\in[1,2).$ Let $\Omega\subset\mathbb{R}^{n+1}$ be $C^{\beta}_{p}$ in
$Q_{1}$ in the sense of Definition 1.1 and let $L$ be an operator of the form
(1.2) satisfying conditions (1.3). Suppose $u$ is a solution to

$\left\\{\begin{array}[]{rcl l}\partial_{t}u+Lu&=&f&\text{in }\Omega\cap
Q_{1}\\\ u&=&0&\text{on }\partial\Omega\cap Q_{1}.\\\ \end{array}\right.$

If $\beta=1$,then for every $\varepsilon>0$ have

$|u|\leq C\big{(}||fd||_{L^{\infty}(\Omega\cap
Q_{1})}+||u||_{L^{\infty}(\Omega\cap
Q_{1})}\big{)}d^{1-\varepsilon}\quad\text{in }\Omega\cap Q_{1/2},$

with $C$ depending only on $n,\varepsilon,\lambda,\Lambda,$ and
$||F||_{C^{1}_{p}}.$

When $\beta>1$, we have

$|u|\leq C\big{(}||fd^{2-\beta}||_{L^{\infty}(\Omega\cap
Q_{1})}+||u||_{L^{\infty}(\Omega\cap Q_{1})}\big{)}d\quad\text{in }\Omega\cap
Q_{1/2},$

with $C$ depending only on $n,\beta,\lambda,\Lambda,$ and
$||F||_{C^{\beta}_{p}}.$

###### Proof.

Cases $\beta=1$ and $\beta>1$ are treated in the same way. Let us only prove
the statement when $\beta>1.$

Dividing $u$ with $||fd^{2-\beta}||_{L^{\infty}}+||u||_{L^{\infty}},$ we can
assume that $|u|\leq 1$ and $|f|\leq d^{\beta-2}.$ Take $(x,t)\in\Omega\cap
Q_{1/2}.$ If $d(x,t)<r/2$ for the constant $r$ and function $\phi_{(z,t)}$
from Lemma 3.3, we apply the comparison principle ([13, Lemma 2.1]) on
$\phi_{(z,t)}\pm u$ and get that $|u(x,t)|\leq Cd(x,t).$ Taking $C$ bigger if
necessary, the same holds also if $d_{x}(x,t)\geq r/2$. Note that thanks to
Definition 1.1 the function $d$ is also comparable to $d_{x}$. In case
$\beta=1$ Lemma 3.2 together with comparison principle gives the desired
result. ∎

Having bounds on the growth near the boundary, combined with interior
estimates quickly give Lipschitz bounds up to the boundary (almost Lipschitz
in $C^{1}_{p}$ domains).

######  Corollary 3.5.

For $\beta\in[1,2)$ let $\Omega\subset\mathbb{R}^{n+1}$ be $C^{\beta}_{p}$ in
$Q_{1}$ in the sense of Definition 1.1 and let $L$ be an operator of the form
(1.2) satisfying conditions (1.3), with $A\in C^{0}(\overline{\Omega})$. Let
$u$ be a solution to

$\left\\{\begin{array}[]{rcl l}\partial_{t}u+Lu&=&f&\text{in }\Omega\cap
Q_{1}\\\ u&=&0&\text{on }\partial\Omega\cap Q_{1}.\\\ \end{array}\right.$

If $\beta=1$ we have for every $\varepsilon>0$

$\left[u\right]_{C^{1-\varepsilon}_{p}(\Omega\cap Q_{1/2})}\leq
C\big{(}||fd^{2-\beta}||_{L^{\infty}(\Omega\cap
Q_{1})}+||u||_{L^{\infty}(\Omega\cap Q_{1})}\big{)},$

with $C$ depending only on $n,\varepsilon,\lambda,\Lambda,$
$||F||_{C^{1}_{p}},$ and the modulus of continuity of $A$.

If $\beta>1$ we have

$\left[u\right]_{C^{1}_{p}(\Omega\cap Q_{1/2})}\leq
C\big{(}||fd^{2-\beta}||_{L^{\infty}(\Omega\cap
Q_{1})}+||u||_{L^{\infty}(\Omega\cap Q_{1})}\big{)},$

with $C$ depending only on $n,\beta,\lambda,\Lambda,$ $||F||_{C^{\beta}_{p}},$
and the modulus of continuity of $A$.

###### Proof.

Let first $\beta>1$. Take any $Q_{r}(x_{0},t_{0})\subset
Q_{2r}(x_{0},t_{0})\subset\Omega$ so that $d(x_{0},t_{0})\leq Cr$ with $C$
independent on $u,f,x_{0},t_{0}$. 222The constant $C$ can be taken as
$\frac{\sqrt{S}-1}{S-1}$, where $S=\left[F\right]_{C^{\frac{1}{2}}_{t}}$.
Define $u_{r}(x,t)=u(x_{0}+2rx,t_{0}+4r^{2}t)$. It solves

$\partial_{t}u_{r}+Lu_{r}=4r^{2}f\quad\text{in }Q_{1}.$

Hence by interior regularity results [17, Theorem 2]

$\displaystyle\left[u_{r}\right]_{C_{p}^{1}(Q_{1/2})}$ $\displaystyle\leq
C\big{(}||4r^{2}f||_{L^{\infty}(Q_{2_{r}(x_{0},t_{0})})}+||u_{r}||_{L^{\infty}(Q_{1})}\big{)}$
$\displaystyle\leq Cr\big{(}||fd^{2-\beta}||_{L^{\infty}(\Omega\cap
Q_{1})}+||u||_{L^{\infty}(\Omega\cap Q_{1})}\big{)},$

where we applied the growth control on $u$. Translating it back to $u$, we get

$\left[u\right]_{C_{p}^{1}(Q_{r})}\leq
C\big{(}||fd^{2-\beta}||_{L^{\infty}(\Omega\cap
Q_{1})}+||u||_{L^{\infty}(\Omega\cap Q_{1})}\big{)}.$

Due to Lemma B.2 the claim is proven.

When $\beta=1$, the prove is analogous. The growth control of $u$ is $|u|\leq
Cd^{1-\varepsilon}$, which assures that we can bound
$\left[u\right]_{C_{p}^{1-\varepsilon}(Q_{r})}$ instead. ∎

Having boundary regularity estimates enables us to prove the following
Liouville-type theorem in a half-space. Recall that
$Q_{R}^{+}=Q_{R}\cap\\{x_{n}>0\\}.$

######  Proposition 3.6.

Assume that $v$ solves

$\left\\{\begin{array}[]{rcl
l}\partial_{t}v-\operatorname{tr}(AD^{2}v)&=&P&\text{on }\\{x_{n}>0\\}\\\
v&=&0&\text{in }\\{x_{n}=0\\}\\\
||v||_{L^{\infty}(Q_{R}^{+})}&\leq&C_{0}R^{\gamma}&\forall R>1,\\\
\end{array}\right.$

for some constant, uniformly elliptic matrix $A$, $\gamma>0,$
$\gamma\not\in\mathbb{N}$, and some polynomial
$P\in\textbf{P}_{\lfloor\gamma-2\rfloor,p}$ (if $\gamma<2$, then $P=0$). Then
$v=Qx_{n}$ for some $Q\in\textbf{P}_{\lfloor\gamma-1\rfloor,p}.$

###### Proof.

Again define

$v_{R}(x,t):=R^{-\gamma}v(Rx,R^{2}t).$

Then $||v_{R}||_{L^{\infty}(Q_{1})}\leq C_{0}$,
$(\partial_{t}-\operatorname{tr}(AD^{2}))v_{R}(x,t)=R^{2-\gamma}P(Rx,R^{2}t),$
and $v_{R}=0$ on $\\{x_{n}=0\\}.$ Hence applying Corollary 3.5, we get

$\left[v_{R}\right]_{C_{p}^{1}(Q_{1/2}^{+})}\leq C(||P||+C_{0}),$

which translates to

$\left[v\right]_{C_{p}^{1}(Q_{R/2}^{+})}\leq CR^{\gamma-1}(||P||+C_{0}),$

with $C$ independent of $R$ (same as in the previous proposition). Now we take
arbitrary $h\in\mathbb{R}^{n}$ with $h_{n}=0$ and $\tau\in\mathbb{R}$, and
define

$w_{1}(x,t):=\frac{1}{|h|+|\tau|^{\frac{1}{2}}}(v(x+h,t+\tau)-v(x,t)),$

so that
$||w_{1}||_{L^{\infty}(Q_{R}^{+})}\leq\left[v\right]_{C_{p}^{1}(Q_{R}^{+})}\leq
CR^{\gamma-1}$, as well as

$\left\\{\begin{array}[]{rcl
l}\partial_{t}w_{1}-\operatorname{tr}(AD^{2}w_{1})&=&P_{1}&\text{in
}\\{x_{n}>0\\}\\\ w_{1}&=&0&\text{in }x_{n}=0,\\\ \end{array}\right.$

for some polynomial $P_{1}\in\textbf{P}_{\lfloor\gamma-3\rfloor,p}$. After
$K=\lceil\gamma\rceil$ same steps we conclude that $w_{K}$ is constant which
implies that

$v(x,t)=\sum_{|\alpha|\leq\gamma}v_{\alpha}(x_{n})(x^{\prime},t)^{\alpha},$

for some functions $v_{\alpha}$ depending only on $x_{n}$. Choose now some
maximal multi-index $\alpha$. Then $\partial^{\alpha}v=c_{\alpha}v_{\alpha}$
satisfies

$\left\\{\begin{array}[]{rcl
l}-a_{nn}\frac{d^{2}}{dx_{n}^{2}}v_{\alpha}&=&\partial^{\alpha}P&\text{in
}\\{x_{n}>0\\}\\\ v_{\alpha}&=&0&\text{in }\\{x_{n}=0\\},\\\
\end{array}\right.$

and hence $v_{\alpha}$ is a polynomial. Continuing in a similar manner and
treating higher order coefficients as right-hand side, we conclude that $v$ is
a polynomial itself. Noticing that $v$ satisfies suitable growth control and
vanishes on $\\{x_{n}=0\\}$ we conclude the wanted result. ∎

##  4\. Boundary Harnack estimate in $C^{1}_{p}$ domains

In this section we prove Theorem 1.2. To establish it we first need to show
the following expansion result, saying that solutions of (1.4) can be well
approximated near the boundary by another non-trivial solution.

######  Proposition 4.1.

Let $\Omega\subset\mathbb{R}^{n+1}$ be $C^{1}_{p}$ in $Q_{1}$ in the sense of
Definition 1.1, with $||F||_{C^{1}_{p}(Q_{1}^{\prime})}\leq 1$. Let $L$ be an
operator of the form (1.2) satisfying conditions (1.3), with $A\in
C^{0}(\Omega)$. For $i=1,2$ let $u_{i}$ be a solution to

$\left\\{\begin{array}[]{rcl l}\partial_{t}u_{i}+Lu_{i}&=&f_{i}&\text{in
}\Omega\cap Q_{1}\\\ u_{i}&=&0&\text{on }\partial\Omega\cap Q_{1},\\\
\end{array}\right.$

with $f_{i}\in L^{\infty}(\Omega\cap Q_{1})$. Assume that $|u_{2}|\geq c_{0}d$
with $c_{0}>0$, $||u_{i}||_{L^{\infty}(\Omega)}\leq 1$ and
$||f_{i}||_{L^{\infty}(\Omega)}\leq 1$. Let $\varepsilon\in(0,1)$. Then for
every $(z,s)\in\partial\Omega\cap Q_{1/2}$ there exists a constant
$c_{(z,s)}\in\mathbb{R}$, so that

$|u_{1}(x,t)-c_{(z,s)}u_{2}(x,t)|\leq
C\big{(}|x-z|^{2-\varepsilon}+|t-s|^{\frac{2-\varepsilon}{2}}\big{)}.$

The constant $C$ depends only on $n,\varepsilon,c_{0}$, the modulus of
continuity of $A$ and ellipticity constants.

Moreover for every $(x_{0},t_{0})\in\Omega\cap Q_{1/2}$, such that
$d_{t}(x_{0})=|x_{0}-z|=c_{\Omega}r,$ we have

$\left[u_{1}-c_{(z,t_{0})}u_{2}\right]_{C^{2-\varepsilon}_{p}(Q_{r}(x_{0},t_{0}))}\leq
C.$ (4.1)

###### Proof.

With the same reasoning as in Proposition 5.1 we can assume that
$(z,s)=(0,0)$. We prove the claim by contradiction. Suppose for $i=1,2$ and
$k\in\mathbb{N}$, there exist $\Omega_{k}$ which are $C^{1}_{p}$ in the sense
of Definition 1.1 with $(0,0)\in\partial\Omega_{k}$ and
$||F_{k}||_{C^{1}_{p}(Q_{1}^{\prime})}\leq 1$, functions $u_{i,k}$, $f_{i,k}$
with $||u_{i,k}||_{L^{\infty}}\leq 1$, $u_{2,k}\geq c_{0}d$,
$||f||_{L^{\infty}}\leq 1$ and operators
$L_{k}=-\operatorname{tr}(A_{k}D^{2})+b_{k}\nabla$, with the modulus of
continuity of $A_{k}$ independent of $k$, so that

$\left\\{\begin{array}[]{rcl
l}\partial_{t}u_{i,k}+L_{k}u_{i,k}&=&f_{i,k}&\text{in }\Omega_{k}\cap Q_{1}\\\
u_{i,k}&=&0&\text{on }\partial\Omega_{k}\cap Q_{1},\end{array}\right.$

but

$\sup_{k}\sup_{r>0}r^{-2+\varepsilon}||u_{1,k}-c_{k}u_{2,k}||_{L^{\infty}(Q_{r})}=\infty$

for any choice of constants $c_{k}$. We extend functions $u_{i,k}$ with $0$
outside of $\Omega_{k}$ to functions defined on whole $Q_{1}$. We define

$c_{k,r}=\frac{\int_{Q_{r}}u_{1,k}u_{2,k}}{\int_{Q_{r}}u_{2,k}^{2}},$

so that $\int_{Q_{r}}(u_{1,k}-c_{k,r}u_{2,k})u_{2,k}=0$. Furthermore we define

$\theta(r)=\sup_{k}\sup_{\rho>r}\rho^{-2+\varepsilon}||u_{1,k}-c_{k,r}u_{2,k}||_{L^{\infty}(Q_{\rho})}.$

Thanks to Lemma B.7 $\theta(r)\to\infty$ as $r\downarrow 0.$ Choose a sequence
$r_{m},k_{m}$, with $r_{m}\leq\frac{1}{m}$, so that

$\frac{1}{r_{m}^{2-\varepsilon}\theta(r_{m})}||u_{1,km}-c_{k_{m},r_{m}}u_{2,k_{m}}||_{L^{\infty}(Q_{r_{m}})}\geq\frac{1}{2}.$

We define the blow-up sequence

$v_{m}(x,t):=\frac{1}{r_{m}^{2-\varepsilon}\theta(r_{m})}\left(u_{1,k_{m}}-c_{k_{m},r_{m}}u_{2,k_{m}}\right)(r_{m}x,r_{m}^{2}t).$

Due to the choice of $r_{m},k_{m}$ and $c_{k_{m},r_{m}},$ we have
$1/2\leq||v_{m}||_{L^{\infty}(Q_{1})}\leq 1$ and
$\int_{Q_{1}}v_{m}u_{2,k_{m}}=0$.

We now turn our attention to constants $c_{k,\rho}$. Since
$u_{2,k}>c_{0}d_{k}$, we can estimate

$\displaystyle\rho|c_{k,\rho}-c_{k,2\rho}|\leq$ $\displaystyle
C||c_{k,\rho}u_{2,k}-c_{k,2\rho}u_{2,k}||_{L^{\infty}(Q_{\rho}\cap\\{d_{k}>\rho/2\\})}$
$\displaystyle\leq$ $\displaystyle
C||u_{1,k}-c_{k,\rho}u_{2,k}||_{L^{\infty}(Q_{\rho})}+C||u_{1,k}-c_{k,2\rho}u_{2,k}||_{L^{\infty}(Q_{2\rho})}$
$\displaystyle\leq$ $\displaystyle
C\theta(\rho)\rho^{2-\varepsilon}+C\theta(2\rho)(2\rho)^{2-\varepsilon}\leq
C\theta(\rho)\rho^{2-\varepsilon},$

so that it holds

$|c_{k,\rho}-c_{k,2\rho}|\leq C\theta(\rho)\rho^{1-\varepsilon}.$

Iterating the inequality above we get for any $j\in\mathbb{N}$

$\displaystyle|c_{k,\rho}-c_{k,2^{j}\rho}|\leq\sum_{i=0}^{j-1}|c_{k,2^{i}\rho}-c_{k,2^{i+1}\rho}|\leq$
$\displaystyle C\sum_{i=0}^{j-1}\theta(2^{i}\rho)(2^{i}\rho)^{1-\varepsilon}$
$\displaystyle\leq$ $\displaystyle
C\theta(\rho)\rho^{1-\varepsilon}\sum_{i=0}^{j-1}\frac{\theta(2^{i}\rho)}{\theta(\rho)}2^{i(1-\varepsilon)}$
$\displaystyle\leq$ $\displaystyle C\theta(\rho)(2^{j}\rho)^{1-\varepsilon}.$

It follows that for any $R>1$, we have

$|c_{k,\rho}-c_{k,R\rho}|\leq C\theta(\rho)(R\rho)^{1-\varepsilon},$

and therefore

$||c_{k,\rho}u_{2,k}-c_{k,R\rho}u_{2,k}||_{L^{\infty}(Q_{R\rho})}\leq
C\theta(\rho)(R\rho)^{2-\varepsilon}.$

Hence

$\displaystyle||v_{m}||_{L^{\infty}(Q_{R})}=\frac{1}{r_{m}^{2-\varepsilon}\theta(r_{m})}||u_{1,k_{m}}-c_{k_{m},r_{m}}u_{2,k_{m}}||_{L^{\infty}(Q_{Rr_{m}})}$
$\displaystyle\leq\frac{1}{r_{m}^{2-\varepsilon}\theta(r_{m})}\left(||u_{1,k_{m}}-c_{k_{m},Rr_{m}}u_{2,k_{m}}||_{L^{\infty}(Q_{Rr_{m}})}+||c_{k_{m},Rr_{m}}u_{2,k_{m}}-c_{k_{m},r_{m}}u_{2,k_{m}}||_{L^{\infty}(Q_{Rr_{m}})}\right)$
$\displaystyle\leq\frac{1}{r_{m}^{2-\varepsilon}\theta(r_{m})}\left(\theta(Rr_{m})(Rr_{m})^{2-\varepsilon}+C\theta(r_{m})(Rr_{m})^{2-\varepsilon}\right)\leq
CR^{2-\varepsilon}.$

Moreover for each $\rho>0$ we have

$\frac{|c_{k,\rho}-c_{k,2^{j}\rho}|}{\theta(\rho)}\leq
C\sum_{i=0}^{j}\frac{\theta(2^{j-i}\rho)}{\theta(\rho)}(2^{j-i}\rho)^{1-\varepsilon},$

and choosing $j\in\mathbb{N}$ such that $2^{j}\rho\in\left[1,2\right)$, we
deduce

$\frac{|c_{k,\rho}-c_{k,2^{j}\rho}|}{\theta(\rho)}\leq
C\sum_{i=0}^{j}\frac{\theta(2^{-i}\rho)}{\theta(\rho)}(2^{-i})^{1-\varepsilon}\longrightarrow
0\quad\text{as }\rho\downarrow 0.$

Hence, since $c_{k,\rho}$ is bounded for $\rho\in\left[1,2\right)$, we get

$\frac{|c_{k,\rho}|}{\theta(\rho)}\longrightarrow 0\quad\text{as
}\rho\downarrow 0,$

uniformly in $k$.

We compute

$(\partial_{t}+\tilde{L}_{k_{m}})v_{m}(x,t)=\frac{r_{m}^{\varepsilon}}{\theta(r_{m})}\left(f_{1,k_{m}}-c_{k_{m}r_{m}}f_{2,k_{m}}\right)(r_{m}x,r_{m}^{2}t),$

and hence

$|(\partial_{t}+\tilde{L}_{k_{m}})v_{m}(x,t)|\leq\frac{r_{m}^{\varepsilon}(1+c_{k_{m},r_{m}})}{\theta(r_{m})}.$

The right hand side converges to $0$ uniformly in compact sets in
$\\{x_{n}>0\\}.$ Hence it is also bounded uniformly in $m$, and so by Lemma
3.5 we get uniform Hölder bounds
$\left[v_{m}\right]_{C^{1-\varepsilon}(Q_{R})}\leq C_{R}$. Hence passing to a
subsequence $v_{m}$ converges to $v$ locally uniformly in $\mathbb{R}^{n+1}$,
that by [2, Theorem 1.1] satisfies

$\left\\{\begin{array}[]{rcll}(\partial_{t}+L)v&=&0&\text{in }\\{x_{n}>0\\}\\\
v&=&0&\text{on }\\{x_{n}\leq 0\\},\\\
||v||_{L^{\infty}(Q_{R})}&\leq&CR^{2-\varepsilon}&\text{for all }R\geq 1\\\
||v||_{L^{\infty}(Q_{1})}&\geq&\frac{1}{2}&\\\
\int_{Q_{1}}v\cdot(x_{n})_{+}&=&0,&\end{array}\right.$

where $L$ is some constant coefficient $2$-homogeneous uniformly elliptic
operator. Hence Theorem 3.6 says that $v=c(x_{n})_{+}$ for some
$c\in\mathbb{R}$. Therefore the last two properties of $v$ contradict each
other.

To prove (4.1), we apply interior estimates [17, Theorem 2] on rescaled
function $v_{r}(x,t)=(u_{1}-c_{(z,t_{0})}v_{2})(x_{0}+2rx,t_{0}+4r^{2}t)$, and
take into account the above proven growth control and the uniform boundedness
of $|c_{(z,t_{0})}|$, see Lemma B.5. ∎

Combining the above result with estimates for $u_{2}$ yields Theorem 1.2.

###### Proof of Theorem 1.2.

The proof goes along the same lines as [1, Theorem 1.3].

Choose a cylinder $Q_{r}(x_{0},t_{0})$, so that
$Q_{2_{r}}(x_{0},t_{0})\subset\Omega\cap Q_{3/4},$ and $d(x_{0},t_{0})\leq
Cr$, with $C$ independent of $x_{0},t_{0}$ and $r$. Let $z$ be the closest
point to $x_{0}$ in $\partial\Omega_{t_{0}}$. Take arbitrary $(x,t)$ and
$(y,s)$ in $Q_{r}(x_{0},t_{0})$, denote $c=c_{(z,t_{0})}\in\mathbb{R}$ from
the previous proposition, and compute

$\left|\frac{u_{1}}{u_{2}}(x,t)-\frac{u_{1}}{u_{2}}(y,s)\right|=\left|\left(\frac{u_{1}}{u_{2}}(x,t)-c\right)-\left(\frac{u_{1}}{u_{2}}(y,s)-c\right)\right|\leq$
$\displaystyle\begin{split}\leq&\left|\left(u_{1}-cu_{2}\right)(x,t)-\left(u_{1}-cu_{2}\right)(y,s)\right|\left|u_{2}^{-1}(x,t)\right|+\\\
&+\left|\left(u_{1}-cu_{2}\right)(y,s)\right|\left|u_{2}^{-1}(x,t)-u_{2}^{-1}(y,s)\right|.\\\
\end{split}$ (4.2)

By Proposition 4.1 and Lemma B.2 we have that $u_{1}-cu_{2}$ is
$C^{2-\varepsilon}_{p}(\overline{\mathcal{C}}),$ for a cone
$\mathcal{C}=\cup_{d_{t_{0}}(x_{0})=c_{\Omega}r=|x_{0}-z|}Q_{r}(x_{0},t_{0})$,
with $D^{k}_{p}(u_{1}-pu_{2})(z,t_{0})=0$, for all $k<2$. Hence by Lemma B.1
we can estimate

$|D^{k}_{p}(u_{1}-pu_{2})(x,t)|\leq
C(|x-z|^{2-\varepsilon-k}+|t-t_{0}|^{\frac{2-\varepsilon-k}{2}}),\quad(x,t)\in\mathcal{C},k<2.$

We denote $v=u_{1}-cu_{2}$ and continue estimating (4.2):

$\displaystyle|v(x,t)-v(y,s)|\leq$
$\displaystyle|v(x,t)-v(y,t)|+|v(y,t)-v(y,s)|\leq$ $\displaystyle\leq$
$\displaystyle||D^{1}_{p}v||_{L^{\infty}(Q_{r}(x_{0},t_{0}))}|x-y|+\left[v\right]_{C^{2-\varepsilon}(Q_{r}(x_{0},t_{0}))}|t-s|^{\frac{2-\varepsilon}{2}}$
$\displaystyle\leq$ $\displaystyle
Cr^{1-\varepsilon}r^{\varepsilon}|x-y|^{1-\varepsilon}+Cr^{1}|t-s|^{\frac{1-\varepsilon}{2}}$
$\displaystyle\leq$ $\displaystyle
Cr(|x-y|^{1-\varepsilon}+|t-s|^{\frac{1-\varepsilon}{2}})$

Finally we also estimate

$\displaystyle\left|u_{2}^{-1}(x,t)-u_{2}^{-1}(y,s)\right|$
$\displaystyle\leq\left|u_{2}^{-1}(x,t)u_{2}^{-1}(y,s)\right|\left|u_{2}(x,t)-u_{2}(y,s)\right|$
$\displaystyle\leq Cr^{-2}\left[u_{2}\right]_{C^{1-\varepsilon}(\Omega\cap
Q_{3/4})}(|x-y|^{1-\varepsilon}+|t-s|^{\frac{1-\varepsilon}{2}})$
$\displaystyle\leq
Cr^{-2+\varepsilon}(|x-y|^{1-2\varepsilon}+|t-s|^{\frac{1-2\varepsilon}{2}}),$

where we used Corollary 3.5 for $u_{2}$. Putting it all together we get

$\left|\frac{u_{1}}{u_{2}}(x,t)-\frac{u_{1}}{u_{2}}(y,s)\right|\leq
C(|x-y|^{1-2\varepsilon}+|t-s|^{\frac{1-2\varepsilon}{2}}).$

Up to choosing a different $\varepsilon$ at the beginning, the claim is
proven, thanks to Lemma B.2. ∎

##  5\. Higher order boundary Schauder and boundary Harnack estimates

We proceed with finding a finer, higher order description of solutions near
the boundary. The first goal is to get Schauder type expansions of solutions
at boundary points - approximations with polynomials multiplied with the
distance function. Note that the boundary Schauder estimates are already
known; see for example [13, Section IV.7] or [12, Section 10.3]. Still, the
proof we present here is different and it is useful for the main result: the
higher order parabolic boundary Harnack estimate.

######  Proposition 5.1.

Let $\beta>1$, $\beta\not\in\mathbb{N}$. Let $\Omega\subset\mathbb{R}^{n+1}$
be $C^{\beta}_{p}$ in $Q_{1}$ in the sense of Definition 1.1 with
$||F||_{C^{\beta}_{p}(Q_{1}^{\prime})}\leq 1$. Let $L$ be an operator of the
form (1.2) satisfying conditions (1.3), with $A,b\in
C^{\beta-2}_{p}(\overline{\Omega})$, if $\beta>2$ and $A\in
C^{0}(\overline{\Omega})$, $b\in L^{\infty}(\Omega)$, if $\beta<2$. Let $u$ be
a solution to

$\left\\{\begin{array}[]{rcl l}\partial_{t}u+Lu&=&f&\text{in }\Omega\cap
Q_{1}\\\ u&=&0&\text{on }\partial\Omega\cap Q_{1},\\\ \end{array}\right.$

with $||f||_{C^{\beta-2}_{p}(\Omega)}\leq 1$ if $\beta>2$ and $|f|\leq
d^{\beta-2}$ if $\beta<2.$ Assume that $||u||_{L^{\infty}(\Omega)}\leq 1$.
Then for every $(z,s)\in\partial\Omega\cap Q_{1/2}$ there exists a polynomial
$p_{(z,s)}\in\textbf{P}_{\lfloor\beta-1\rfloor,p}$, so that

$|u(x,t)-p_{(z,s)}(x,t)d(x,t)|\leq C(|x-z|^{\beta}+|t-s|^{\frac{\beta}{2}}).$

The constant $C$ depends only on $n,\beta$, ellipticity constants and
$||A||_{C^{\beta-2}_{p}(\overline{\Omega})},||b||_{C^{\beta-2}_{p}(\overline{\Omega})}$
if $\beta>2$, and the modulus of continuity of $A$, if $\beta<2$..

###### Proof.

Thanks to assumption on $\Omega$ (translations and scaling preserves the
assumptions), we can assume that $(z,s)=(0,0).$ We prove the claim with
contradiction argument. Assume that the claim is false. Then there exist
$\Omega_{k}$ which are $C^{\beta}_{p}$ in $Q_{1}$ in the sense of Definition
1.1, with $||F_{k}||_{C^{\beta}_{p}(Q_{1}^{\prime})}\leq 1$ and
$(0,0)\in\partial\Omega_{k}$, and $u_{k},f_{k},$
$L_{k}=\operatorname{tr}(A_{k}D^{2})+b_{k}\cdot\nabla$, with uniformly bounded
$||A_{k}||_{C^{\beta-2}_{p}(\overline{\Omega}_{k})},||b_{k}||_{C^{\beta-2}_{p}(\overline{\Omega}_{k})}$
if $\beta>2,$ and the modulus of continuity of $A_{k}$ independent of $k$, if
$\beta<2$, satisfying $\partial_{t}u_{k}+L_{k}u_{k}=f_{k}$ in $\Omega_{k}\cap
Q_{1}$. Moreover, $u_{k}=0$ on $\partial\Omega\cap Q_{1},$ $|u_{k}|\leq 1$,
$||f||_{C^{\beta-2}_{p}}\leq 1$ (or $|f_{k}|\leq d_{k}^{\beta-2}$ if
$\beta<2$), $L_{k}$ is $(\lambda,\Lambda)$-uniformly elliptic, but for every
polynomial $p_{k}\in\textbf{P}_{\lfloor\beta-1\rfloor,p}$ we have

$\sup_{r>0}r^{-\beta}||u_{k}-p_{k}d_{k}||_{L^{\infty}(Q_{r})}>k.$

We extend the functions $u_{k}$ and $d_{k}$ with zero in $Q_{1}\cap\Omega^{c}$
so that they are defined in the full cylinder $Q_{1}$, and denote them still
$u_{k},d_{k}$. Let $p_{k,\rho}d_{k}$ be the $L^{2}(Q_{\rho})$ projection of
$u_{k}$ to space $\textbf{P}_{\lfloor\beta-1\rfloor,p}d_{k},$ so that we have

$\int_{Q_{\rho}}(u_{k}-p_{k,\rho}d_{k})pd_{k}=0,$

for every polynomial $p\in\textbf{P}_{\lfloor\beta-1\rfloor,p}.$ Then define
the monotone quantity

$\theta(r)=\sup_{k}\sup_{\rho>r}\rho^{-\beta}||u_{k}-p_{k,\rho}d_{k}||_{L^{\infty}(Q_{\rho})}.$

It follows from Lemma B.4 that $\lim_{r\downarrow 0}\theta(r)=\infty.$ Choose
a sequence $k_{m},r_{m}$ with $r_{m}\leq 1/m$, so that

$\frac{1}{2}\theta(r_{m})\leq
r^{-\beta}||u_{k_{m}}-p_{k_{m},r_{m}}d_{k_{m}}||_{L^{\infty}(Q_{r_{m}})},$

and define the blow-up sequence

$v_{m}(x,t)=\frac{1}{r_{m}^{\beta}\theta(r_{m})}\big{(}u_{k_{m}}(r_{m}x,r_{m}^{2}t)-p_{k_{m},r_{m}}d_{k_{m}}(r_{m}x,r_{m}^{2}t)\big{)}.$

Notice that $||v_{m}||_{L^{\infty}(Q_{1})}\geq 1/2$ and
$\int_{Q_{1}}v_{m}pd_{k_{m}}(r_{m}\cdot,r_{m}^{2}\cdot)=0$ for every
polynomial $p\in\textbf{P}_{\lfloor\beta-1\rfloor,p}.$

Let us now turn our attention to the polynomials $p_{k,\rho}$. We write

$p_{k,\rho}(x,t)=\sum_{|\alpha|_{p}\leq\lfloor\beta-1\rfloor}p_{k,\rho}^{(\alpha)}(x,t)^{\alpha},\quad
p_{k,\rho}^{(\alpha)}\in\mathbb{R}.$

Using a rescaled version of [1, Lemma A.10] and that $d_{k}\geq c\rho$ in
$Q_{\rho}\cap\\{d_{k}>\rho/2\\}$, we estimate for any $\alpha$ such that
$|\alpha|_{p}\leq\lfloor\beta-1\rfloor$

$\displaystyle\rho^{|\alpha|_{p}+1}|p_{k,\rho}^{(\alpha)}-p_{k,2\rho}^{(\alpha)}|\leq$
$\displaystyle
C\rho||p_{k,\rho}-p_{k,2\rho}||_{L^{\infty}(Q_{\rho}\cap\\{d_{k}>\rho/2\\})}$
$\displaystyle\leq$ $\displaystyle
C||p_{k,\rho}d_{k}-p_{k,2\rho}d_{k}||_{L^{\infty}(Q_{\rho}\cap\\{d_{k}>\rho/2\\})}$
$\displaystyle\leq$ $\displaystyle
C||u_{k}-p_{k,\rho}d_{k}||_{L^{\infty}(Q_{\rho})}+C||u_{k}-p_{k,2\rho}d_{k}||_{L^{\infty}(Q_{2\rho})}$
$\displaystyle\leq$ $\displaystyle
C\theta(\rho)\rho^{\beta}+C\theta(2\rho)(2\rho)^{\beta}\leq
C\theta(\rho)\rho^{\beta},$

so that it holds

$|p_{k,\rho}^{(\alpha)}-p_{k,2\rho}^{(\alpha)}|\leq
C\theta(\rho)\rho^{\beta-1-|\alpha|_{p}}.$

Iterating the inequality above we get for any $j\in\mathbb{N}$

$\displaystyle|p_{k,\rho}^{(\alpha)}-p_{k,2^{j}\rho}^{(\alpha)}|\leq\sum_{i=0}^{j-1}|p_{k,2^{i}\rho}^{(\alpha)}-p_{k,2^{i+1}\rho}^{(\alpha)}|\leq$
$\displaystyle
C\sum_{i=0}^{j-1}\theta(2^{i}\rho)(2^{i}\rho)^{\beta-1-|\alpha|_{p}}$
$\displaystyle\leq$ $\displaystyle
C\theta(\rho)\rho^{\beta-1-|\alpha|_{p}}\sum_{i=0}^{j-1}\frac{\theta(2^{i}\rho)}{\theta(\rho)}2^{i(\beta-1-|\alpha|_{p})}$
$\displaystyle\leq$ $\displaystyle
C\theta(\rho)(2^{j}\rho)^{\beta-1-|\alpha|_{p}}.$

It follows that for any $R>1$, we have

$|p_{k,\rho}^{(\alpha)}-p_{k,R\rho}^{(\alpha)}|\leq
C\theta(\rho)(R\rho)^{\beta-1-|\alpha|_{p}},$

and therefore

$||p_{k,\rho}d_{k}-p_{k,R\rho}d_{k}||_{L^{\infty}(Q_{R\rho})}\leq
C\theta(\rho)(R\rho)^{\beta}.$

Hence

$\displaystyle||v_{m}||_{L^{\infty}(Q_{R})}=\frac{1}{r_{m}^{\beta}\theta(r_{m})}||u_{k_{m}}-p_{k_{m},r_{m}}d_{k_{m}}||_{L^{\infty}(Q_{Rr_{m}})}$
$\displaystyle\leq\frac{1}{r_{m}^{\beta}\theta(r_{m})}\left(||u_{k_{m}}-p_{k_{m},Rr_{m}}d_{k_{m}}||_{L^{\infty}(Q_{Rr_{m}})}+||p_{k_{m},Rr_{m}}d_{k_{m}}-p_{k_{m},r_{m}}d_{k_{m}}||_{L^{\infty}(Q_{Rr_{m}})}\right)$
$\displaystyle\leq\frac{1}{r_{m}^{\beta}\theta(r_{m})}\left(\theta(Rr_{m})(Rr_{m})^{\beta}+C\theta(r_{m})(Rr_{m})^{\beta}\right)\leq
CR^{\beta}.$

Moreover for each $\rho>0$ we have

$\frac{|p_{k,\rho}^{(\alpha)}-p_{k,2^{j}\rho}^{(\alpha)}|}{\theta(\rho)}\leq
C\sum_{i=0}^{j}\frac{\theta(2^{j-i}\rho)}{\theta(\rho)}(2^{j-i}\rho)^{\beta-1-|\alpha|_{p}},$

and choosing $j\in\mathbb{N}$ such that $2^{j}\rho\in\left[1,2\right)$, we
deduce

$\frac{|p_{k,\rho}^{(\alpha)}-p_{k,2^{j}\rho}^{(\alpha)}|}{\theta(\rho)}\leq
C\sum_{i=0}^{j}\frac{\theta(2^{-i}\rho)}{\theta(\rho)}(2^{-i})^{\beta-1-|\alpha|_{p}}\longrightarrow
0\quad\text{as }\rho\downarrow 0.$

Hence, since $p_{k,\rho}^{(\alpha)}$ is bounded for $\rho\in\left[1,2\right)$,
we get

$\frac{|p_{k,\rho}^{(\alpha)}|}{\theta(\rho)}\longrightarrow 0\quad\text{as
}\rho\downarrow 0.$

We compute

$\displaystyle(\partial_{t}+\tilde{L}_{k_{m}})v_{m}(x,t)$
$\displaystyle=\frac{r_{m}^{2-\beta}}{\theta(r_{m})}\big{(}f_{k_{m}}-(\partial_{t}-\tilde{L}_{k_{m}})(p_{k_{m},r_{m}}d_{k_{m}})\big{)}(r_{m}x,r_{m}^{2}t)$
$\displaystyle=\left\\{\begin{array}[]{cl}\frac{r_{m}^{2-\beta}}{\theta(r_{m})}o_{m,\beta}&\text{
if }\beta<2\\\ P_{m}+\frac{1}{\theta(r_{m})}o_{m,\beta}&\text{ if
}\beta>2,\end{array}\right.$

where $P_{m}\in\textbf{P}_{\lfloor\beta-2\rfloor,p}$ is a suitable Taylor
polynomial of quantities on the right-hand side. Note that $f_{k_{m}}$ and
$(\partial_{t}-\tilde{L}_{k_{m}})(p_{k_{m},r_{m}}d_{k_{m}})$ are both
$C^{\beta-2}_{p}$ functions, and they can be approximated with a polynomial up
to order $|x|^{\beta-2}+|t|^{\frac{\beta-2}{2}}$ (see Lemma B.1), if $\beta>2$
and otherwise we bound the $L^{\infty}$ norm. The reminder we denote with
$o_{m,\beta}$, so

$o_{m,\beta}(x,t)=\left\\{\begin{array}[]{cl}\big{(}f_{k_{m}}-(\partial_{t}-\tilde{L}_{k_{m}})(p_{k_{m},r_{m}}d_{k_{m}})\big{)}(r_{m}x,r_{m}^{2}t)&\text{
if }\beta<2\\\
\big{(}f_{k_{m}}-(\partial_{t}-\tilde{L}_{k_{m}})(p_{k_{m},r_{m}}d_{k_{m}})-P_{m}\big{)}(r_{m}x,r_{m}^{2}t)&\text{
if }\beta>2,\end{array}\right.$

In the case $\beta<2$ we can estimate
$\frac{r_{m}^{2-\beta}}{\theta(r_{m})}|o_{m,\beta}|\leq
C\frac{|p_{k_{m},r_{m}}|}{\theta(r_{m})}d_{m}^{\beta-2}(x,0)$, with $C$
independent of $m$, using the assumption on $f_{m}$, estimates on
$|\partial_{t}d|$ and $|D^{2}d_{m}|$ (see Lemma 3.1), while
$\frac{p_{k_{m},r_{m}}}{\theta(r_{m})}$ converges to zero. In the case
$\beta>2$, we have $|\theta(r_{m})^{-1}o_{m,\beta}|\leq
C\frac{||p_{k_{m},r_{m}}||}{\theta(r_{m})}(|x|^{\beta-2}+|t|^{\frac{\beta-2}{2}})$.
Since $||v_{m}||_{L^{\infty}(Q_{R})}\leq CR^{\beta}$ independently of $m$ and
since $\frac{1}{\theta(r_{m})}o_{m,\beta}$ are bounded uniformly in $m$, we
get that $P_{m}$ are uniformly bounded as well (see the end of the proof of
Proposition 5.4). Therefore in both cases the term with $o_{m,\beta}$
converges to zero in $L^{\infty}_{\mathrm{loc}}(\\{x_{n}>0\\}).$ Thanks to
Corollary 3.5 we get uniform Lipschitz bounds for $v_{m}$ on every $Q_{R}$.
Passing to a subsequence, the convergence result [2, Theorem 1.1] assures the
local uniform convergence of $v_{m}$ to some function $v$ defined in
$\mathbb{R}^{n+1}\cap\\{x_{n}>0\\},$ satisfying

$\left\\{\begin{array}[]{rcll}(\partial_{t}+L)v&=&P&\text{in }x_{n}>0\\\
v&=&0&\text{in }x_{n}=0,\\\ ||v||_{L^{\infty}(Q_{R}^{+})}&\leq&CR^{\beta}&\\\
||v||_{L^{\infty}(Q_{1}^{+})}&\geq&\frac{1}{2}&\\\
\int_{Q_{1}}vpx_{n}&=&0&\text{for every
}p\in\textbf{P}_{\lfloor\beta-1\rfloor,p},\end{array}\right.$

where $L$ is a constant coefficient, second order $(\lambda,\Lambda)$-elliptic
operator and $P\in\textbf{P}_{\lfloor\beta-2\rfloor,p}$ if $\beta>2$ and $P=0$
if $\beta<2.$ Hence by Liouville theorem (Proposition 3.6) $v$ has to be equal
to $qx_{n}$ for some polynomial $q$ in $\textbf{P}_{\lfloor\beta-1\rfloor,p}$,
which gives the contradiction. ∎

######  Remark 5.2.

When $\beta\in(1,2)$ the function $d$ is taken from Lemma 3.1. When $\beta>2$,
we take the composition of the boundary flattening map
($\phi(x^{\prime},x_{n},t)=(x^{\prime},x_{n}-F(x^{\prime},t),t),$ with the
$n$-th projection. Concretely, $d(x,t)=x_{n}-F(x^{\prime},t)$, which is
$C^{\beta}_{p}(Q_{1}).$ In both cases we get that $r^{-1}d(rx,r^{2}t)$
converges to $x_{n}$ locally uniformly.

Additionally, the estimate $d(rx,r^{2}t)\leq Crd(x,rt)\leq C_{1}rd(x,0)$,
works because the boundary is flat enough.

The obtained description of solutions near the boundary implies the regularity
up to the boundary, as follows.

######  Corollary 5.3.

Let $\beta>1$, $\beta\not\in\mathbb{N}$. Let $\Omega\subset\mathbb{R}^{n+1}$
be $C^{\beta}_{p}$ in $Q_{1}$ in the sense of Definition 1.1. Let $L$ be an
operator of the form (1.2) satisfying conditions (1.3), with $A,b\in
C^{\beta-2}_{p}(\overline{\Omega})$, if $\beta>2$ and $A\in
C^{0}(\overline{\Omega})$ if $\beta<2$. Let $u$ be a solution to

$\left\\{\begin{array}[]{rcl l}\partial_{t}u+Lu&=&f&\text{in }\Omega\cap
Q_{1}\\\ u&=&0&\text{on }\partial\Omega\cap Q_{1},\\\ \end{array}\right.$

with $||f||_{C^{\beta-2}_{p}(\overline{\Omega}\cap Q_{1})}\leq C_{0}$ if
$\beta>2$ and $|f|\leq C_{0}d^{\beta-2}$ if $\beta<2.$

Then

$\left|\left|u\right|\right|_{C^{\beta}_{p}(\Omega\cap Q_{1/2})}\leq CC_{0}.$

The constant $C$ depends only on $n,\beta,\Omega$, ellipticity constants and
$||A||_{C^{\beta-2}_{p}(\overline{\Omega})},||b||_{C^{\beta-2}_{p}(\overline{\Omega})}$
if $\beta>2$, and the modulus of continuity of $A$, if $\beta<2$.

###### Proof.

Dividing the equation with $C_{0}$ if necessary, we may assume that $C_{0}=1$.
Thanks to Lemma B.2, it is enough to prove

$\left[u\right]_{C^{\beta}_{p}(Q_{r}(x_{0},t_{0}))}\leq C,$

whenever $d_{t_{0}}(x_{0})=|x_{0}-z|\leq Cr$, with $C$ independent of
$x_{0},t_{0},r$, and $Q_{2r}(x_{0},t_{0})\subset\Omega\cap Q_{1/2}$. To prove
that, take $p_{(z,t_{0})}$ from Proposition 5.1, and define

$u_{r}(x,t):=(u-p_{(z,t_{0})}d)(x_{0}+2rx,t_{0}+(2r)^{2}t),\quad(x,t)\in
Q_{1}.$

Then in the case $\beta>2$ use interior regularity estimates from Proposition
A.4, to get

$\left[u_{r}\right]_{C^{\beta}_{p}(Q_{1/2})}\leq
C\left(r^{\beta}||f+(\partial_{t}+L)(p_{(z,t_{0})}d)||_{C^{\beta-2}_{p}(\Omega\cap
Q_{1})}+||u-p_{(z,t_{0})}d||_{L^{\infty}(Q_{2r}(x_{0},t_{0}))}\right)$

and hence

$\left[u-p_{(z,t_{0})}d\right]_{C^{\beta}_{p}(Q_{r}(x_{0},t_{0}))}\leq C,$

thanks to Proposition 5.1, the fact that $p_{(z,t_{0})}$ are uniformly bounded
(see Lemma B.5) and the fact that $d\in C^{\beta}_{p}(\overline{\Omega}\cap
Q_{1}).$

In case $\beta<2$, use [17, Theorem 2] to get

$\left[u_{r}\right]_{C^{\beta}_{p}(Q_{1/2})}\leq
C\left(r^{2}||f+(\partial_{t}+L)(p_{(z,t_{0})}d)||_{L^{\infty}(Q_{2r}(x_{0},t_{0}))}+||u-p_{(z,t_{0})}d||_{L^{\infty}(Q_{2r}(x_{0},t_{0}))}\right)$

and hence due to similar reasons as before

$\left[u-p_{(z,t_{0})}d\right]_{C^{\beta}_{p}(Q_{r}(x_{0},t_{0}))}\leq C.$

This implies that

$\left[u\right]_{C^{\beta}_{p}(Q_{r}(x_{0},t_{0}))}\leq C,$

thanks to regularity of $d$ and boundedness of $p_{(z,t_{0})},$ as wanted. ∎

We can now establish expansions of one solution with respect to the other.
This result is the key ingredient to prove the boundary Harnack estimate. It
is the higher order version of Proposition 4.1 Since the order exceeds $2$,
the constant in the expansion is replaced with a polynomial, which causes some
difficulties in the proof.

######  Proposition 5.4.

Let $\beta>1$, $\beta\not\in\mathbb{N}$. Let $\Omega\subset\mathbb{R}^{n+1}$
be $C^{\beta}_{p}$ in $Q_{1}$ in the sense of Definition 1.1, with
$||F||_{C^{\beta}_{p}(Q_{1}^{\prime})}\leq 1$. Let $L$ be an operator of the
form (1.2) satisfying conditions (1.3), with $A,b\in
C^{\beta-1}_{p}(\overline{\Omega})$. For $i=1,2$ let $u_{i}$ be a solution to

$\left\\{\begin{array}[]{rcl l}\partial_{t}u_{i}+Lu_{i}&=&f_{i}&\text{in
}\Omega\cap Q_{1}\\\ u_{i}&=&0&\text{on }\partial\Omega\cap Q_{1},\\\
\end{array}\right.$

with $f_{i}\in C^{\beta-1}_{p}(\overline{\Omega}\cap Q_{1})$. Assume that
$|u_{2}|\geq c_{0}d$ with $c_{0}>0$, $||u_{i}||_{L^{\infty}(\Omega)}\leq 1$
and $||f_{i}||_{C^{\beta-1}_{p}(\overline{\Omega})}\leq 1$. Then for every
$(z,s)\in\partial\Omega\cap Q_{1/2}$ exists a polynomial
$p_{(z,s)}\in\textbf{P}_{\lfloor\beta\rfloor,p}$, so that

$|u_{1}(x,t)-p_{(z,s)}(x,t)u_{2}(x,t)|\leq
C\big{(}|x-z|^{\beta+1}+|t-s|^{\frac{\beta+1}{2}}\big{)}.$

The constant $C$ depends only on $n,\beta,c_{0}$,
$||A||_{C^{\beta-1}_{p}(\overline{\Omega})},||b||_{C^{\beta-1}_{p}(\overline{\Omega})}$
and ellipticity constants.

Moreover for every $(x_{0},t_{0})\in\Omega\cap Q_{1/2}$, such that
$d_{t}(x_{0})=|x_{0}-z|=c_{\Omega}r,$ we have

$\left[u_{1}-p_{(z,t_{0})}u_{2}\right]_{C^{\beta+1}_{p}(Q_{r}(x_{0},t_{0}))}\leq
C.$ (5.1)

###### Proof.

With the same reasoning as in Proposition 5.1 we can assume $(z,s)=(0,0)$.

We write

$p(x,t)=\sum_{|\alpha|_{p}\leq\beta}p_{\alpha}\cdot(x,t)^{\alpha}=p^{(0)}+\sum_{1\leq|\alpha|_{p}\leq\beta}p_{\alpha}(x,t)^{\alpha}=p^{(0)}+p^{(1)}(x,t),$

for some constants $p_{\alpha}$. In view of Proposition 5.1

$u_{2}(x,t)=p_{2}(x,t)d(x,t)+v_{2}(x,t),\quad\quad
p_{2}\in\textbf{P}_{\lfloor\beta-1\rfloor,p},\text{ }|v_{2}(x,t)|\leq
C(|x|^{\beta}+|t|^{\frac{\beta}{2}}),$

and then the claim is equivalent to

$\left|u_{1}(x,t)-p^{(0)}u_{2}(x,t)-p^{(1)}(x,t)p_{2}(x,t)d(x,t)-p^{(1)}(x,t)v_{2}(x,t)\right|\leq
C\big{(}|x-z|^{\beta+1}+|t-s|^{\frac{\beta+1}{2}}\big{)},$

which is furthermore equivalent to

$\left|u_{1}(x,t)-\tilde{p}^{(0)}u_{2}(x,t)-\tilde{p}^{(1)}(x,t)d(x,t)\right|\leq
C\big{(}|x-z|^{\beta+1}+|t-s|^{\frac{\beta+1}{2}}\big{)},$

for a suitable polynomial $\tilde{p}\in\textbf{P}_{\lfloor\beta\rfloor,p}$.

We prove that by contradiction. So suppose for $i=1,2$, and $k\in\mathbb{N}$
there exist $\Omega_{k},$ which are $C^{\beta}_{p}$ in $Q_{1}$ in the sense of
Definition 1.1, with $0\in\partial\Omega_{k}$ and
$||F_{k}||_{C^{\beta}_{p}(Q_{1}^{\prime})}\leq 1$, $u_{i,k},$ $f_{i,k}$ and
$(\lambda,\Lambda)$-elliptic operator
$L_{k}=-\operatorname{tr}(A_{k}D^{2})+b_{k}\cdot\nabla$, with uniformly
bounded norms
$||A_{k}||_{C^{\beta-1}_{p}(\overline{\Omega}_{k})},||b_{k}||_{C^{\beta-1}_{p}(\overline{\Omega}_{k})}$,
so that $||u_{i,k}||_{L^{\infty}}\leq 1,$ $||f_{i,k}||_{C^{\beta-1}_{p}}\leq
1,$ $u_{2,k}\geq c_{0}d$ and

$\left\\{\begin{array}[]{rcl
l}\partial_{t}u_{i,k}+L_{k}u_{i,k}&=&f_{i,k}&\text{in }\Omega_{k}\cap Q_{1}\\\
u_{i,k}&=&0&\text{on }\partial\Omega_{k}\cap Q_{1},\end{array}\right.$

but

$\sup_{k}\sup_{r>0}r^{-\beta-1}||u_{1,k}-p^{(0)}_{k}u_{2,k}-p^{(1)}_{k}d||_{L^{\infty}(Q_{r})}=\infty$

for every choice of polynomials $p_{k}\in\textbf{P}_{\lfloor\beta\rfloor,p}$.
We extend $u_{i,k}$ and $d_{k}$ to functions defined on whole $Q_{1}$ with
zero outside of $\Omega_{k}$. Define $p_{k,r}$ as the $L^{2}(Q_{r})$
projection of $u_{1,k}$ on the space $\\{q^{(0)}u_{2,k}+q^{(1)}d_{k};\text{
}q\in\textbf{P}_{\lfloor\beta\rfloor,p}\\}$, so that

$\int_{Q_{r}}(u_{1,k}-p_{k,r}^{(0)}u_{2,k}-p_{k,r}^{(1)}d_{k})(q^{(0)}u_{2,k}+q^{(1)}d_{k})=0$

for every polynomial $q\in\textbf{P}_{\lfloor\beta\rfloor,p}.$ Furthermore, we
define

$\theta(r)=\sup_{k}\sup_{\rho>r}\rho^{-\beta-1}||u_{1,k}-p_{k,\rho}^{(0)}u_{2,k}-p_{k,\rho}^{(1)}d_{k}||_{L^{\infty}(Q_{\rho})}.$

Lemma B.7, ensures that $\theta(r)\to\infty$ as $r\downarrow 0$. Pick a
sequence $r_{m},k_{m}$, with $r_{m}\leq 1/m$, so that

$\frac{1}{r_{m}^{\beta+1}\theta(r_{m})}||u_{1,k_{m}}-p_{r_{m},k_{m}}^{(0)}u_{2,k_{m}}-p_{r_{m},k_{m}}^{(1)}d_{k_{m}}||_{L^{\infty}(Q_{r_{m}})}\geq\frac{1}{2}.$

We define the blow-up sequence

$v_{m}(x,t):=\frac{1}{r_{m}^{\beta+1}\theta(r_{m})}\left(u_{1,k_{m}}-p_{r_{m},k_{m}}^{(0)}u_{2,k_{m}}-r_{r_{m},k_{m}}^{(1)}d_{k_{m}}\right)(r_{m}x,r_{m}^{2}t).$

Note that $||v_{m}||_{L^{\infty}(Q_{1})}\leq 1/2$, as well as
$\int_{Q_{1}}v_{m}(q^{(0)}u_{2,k_{m}}+q^{(1)}d_{k_{m}})=0$ for every
polynomial $q\in\textbf{P}_{\lfloor\beta\rfloor,p}.$ With same arguments as in
Proposition 5.1, we get $||v_{m}||_{L^{\infty}(Q_{R})}\leq CR^{\beta+1}$, and
$\frac{|p_{r,k}^{(\alpha)}|}{\theta(r)}\to 0$ uniformly in $k$ as $r\downarrow
0.$

Now we turn our attention to

$(\partial_{t}+\tilde{L}_{k_{m}})v_{m}(x,t)=\frac{1}{r_{m}^{\beta-1}\theta(r_{m})}\left(f_{1,k_{m}}-p_{k_{m},r_{m}}^{(0)}f_{2,k_{m}}-(\partial_{t}+\tilde{L}_{k_{m}})(p_{k_{m},r_{m}}^{(1)}d_{k_{m}})\right)(r_{m}x,r_{m}^{2}t).$

We want to approximate the right-hand side as well as possible with a
polynomial. By assumption $f_{i,k_{m}}\in C^{\beta-1}_{p}$ and hence we can
approximate it up to order $|x|^{\beta-1}+|t|^{\frac{\beta-1}{2}}$, that is

$|f_{i,m}(x,t)-P_{i,m}(x,t)|\leq
C\left(|x|^{\beta-1}+|t|^{\frac{\beta-1}{2}}\right),$

for suitable polynomials $P_{i,m}$. Let us turn now to the term with the
distance function

$\displaystyle(\partial_{t}+\tilde{L}_{k_{m}})(p_{k_{m},r_{m}}^{(1)}d_{k_{m}})=$
$\displaystyle(\partial_{t}+\tilde{L}_{k_{m}})p_{k_{m},r_{m}}^{(1)}d_{k_{m}}+p_{k_{m},r_{m}}^{(1)}(\partial_{t}+\tilde{L}_{k_{m}})d_{k_{m}}$
$\displaystyle-2\nabla p_{k_{m},r_{m}}^{(1)}A_{k_{m}}\nabla d_{k_{m}}$

Let us first consider the case $\beta>2$. Then $d_{k_{m}}\in C^{\beta}_{p}$,
and hence $\nabla d_{k_{m}}\in C^{\beta-1}_{p}$, and
$(\partial_{t}+\tilde{L}_{k_{m}})d_{k_{m}}\in C^{\beta-2}_{p}$. Therefore
$d_{k_{m}}$ and $\nabla d_{k_{m}}$ can be approximated up to order
$|x|^{\beta-1}+|t|^{\frac{\beta-1}{2}}$, while the remaining term can only be
approximated up to order $|x|^{\beta-2}+|t|^{\frac{\beta-2}{2}}$. But this
term is multiplied with $p_{k_{m},r_{m}}^{(1)}$, which can be estimated with
$|x|+|t|$, and hence we get

$\left|(\partial_{t}+\tilde{L}_{k_{m}})(p_{k_{m},r_{m}}^{(1)}d_{k_{m}})(x,t)-P^{\prime}_{m}(x,t)\right|\leq
C\left(|x|^{\beta-1}+|t|^{\frac{\beta-1}{2}}\right).$

Thus we deduce

$|(\partial_{t}+\tilde{L}_{k_{m}})v_{m}(x,t)-P_{m}(x,t)|\leq\frac{C||p_{k_{m},r_{m}}||}{\theta(r_{m})}\left(|x|^{\beta-1}+|t|^{\frac{\beta-1}{2}}\right),$

for a suitable polynomial $P_{m}\in\textbf{P}_{\lfloor\beta-1\rfloor,p}.$ Note
that $\frac{||p_{k_{m},r_{m}}||}{\theta(r_{m})}\to 0$, as $m\to\infty.$

In the case $\beta\in(1,2)$, we have $d_{k_{m}}\in C^{\beta}$, and
$|D^{2}d_{k_{m}}|\leq Cd^{\beta-2}.$ Hence we get

$|(\partial_{t}+\tilde{L}_{k_{m}})v_{m}(x,t)-P_{m}(x,t)|\leq\frac{C||p_{k_{m},r_{m}}||}{\theta(r_{m})}\left(|x|^{\beta-1}+|t|^{\frac{\beta-1}{2}}+(|x|+|t|)d_{k_{m}}^{\beta-2}\right).$

In both cases we get local uniform convergence of the residue
$o_{m}:=(\partial_{t}+\tilde{L}_{k_{m}})v_{m}(x,t)-P_{m}(x,t)$ to zero. Denote
$\Omega_{m}=\\{(x,t);\text{ }(r_{m}x,r_{m}^{2}t)\in\Omega_{k_{m}}\\}$ and
split $v_{m}=v_{m,1}+v_{m,2}$, with

$\left\\{\begin{array}[]{rcl
l}(\partial_{t}+\tilde{L}_{k_{m}})v_{m,1}&=&P_{m}&\text{in }\Omega_{m}\cap
Q_{1}\\\ v_{m,1}&=&0&\text{in }\partial_{p}(\Omega_{m}\cap
Q_{1}),\end{array}\right.$

and

$\left\\{\begin{array}[]{rcl
l}(\partial_{t}+\tilde{L}_{k_{m}})v_{m,2}&=&o_{m}&\text{in }\Omega_{m}\cap
Q_{1}\\\ v_{m,2}&=&v_{m}&\text{in }\partial_{p}(\Omega_{m}\cap
Q_{1}).\end{array}\right.$

Note that the existence of $v_{m,1},v_{m,2}$ is given by [13, Theorem 5.15].
Since $v_{m}$ are bounded independently of $m$, we can get a bound
$|v_{m,2}|\leq Cd_{m},$ with $C$ independent of $m$, using a barrier
constructed in Lemma 3.3. Since $\Omega_{m}$ are closer and closer to
$\\{x_{n}>0\\},$ we can estimate $|v_{m,2}|\leq C$, independently of $m$.
Hence we have uniform bounds for $||v_{m,1}||_{L^{\infty}(Q_{1})}$ as well,
which by Lemma B.6 implies also the uniform boundedness of $||P_{m}||.$ Then
we apply Corollary 3.5 rescaled to any $Q_{R}$, to get uniform Lipschitz
bounds for $v_{m}$. Together with the convergence result [2, Theorem 1.1] this
ensures that a subsequence of the blow-up sequence converges to some function
$v$ satisfying

$\left\\{\begin{array}[]{rcll}(\partial_{t}+L)v&=&P&\text{in }\\{x_{n}>0\\}\\\
v&=&0&\text{on }\\{x_{n}=0\\},\\\
||v||_{L^{\infty}(Q_{R})}&\leq&CR^{\beta+1}&\text{for all }R\geq 1\\\
||v||_{L^{\infty}(Q_{1})}&\geq&\frac{1}{2}&\\\
\int_{Q_{1}}vp(x_{n})_{+}&=&0&\text{for every
}p\in\textbf{P}_{\lfloor\beta\rfloor,p},\end{array}\right.$

where $P\in\textbf{P}_{\lfloor\beta-1\rfloor,p}.$ Proposition 3.6 then says
that $v$ equals $qx_{n}$, $q\in\textbf{P}_{\lfloor\beta\rfloor,p}$, which
gives the contradiction with the last two properties of $v$, as wanted.

Finally we prove also (5.1). If $c_{\Omega}$ is big enough, then for every
point $(x_{0},t_{0})$ with $d(x_{0},t_{0})=c_{\Omega}r$ we have
$Q_{2r}(x_{0},t_{0})\subset\Omega$. It suffices applying Proposition A.4 on
the rescaled function
$v_{r}(x,t)=(u_{1}-p_{z,t_{0}}u_{2})(x_{0}+2rx,t_{0}+4r^{2}t)$, take into
account the just proven bound and noticing that $||p_{(z,t_{0})}||$ can be
uniformly bounded thanks to Lemma B.5. ∎

We now have all ingredients to prove Theorem 1.3.

###### Proof of Theorem 1.3.

The proof goes along the same lines as Theorem 1.2, just that the estimates
with higher order parabolic seminorms are a bit more complicated.

Note that

$\left[\frac{u_{1}}{u_{2}}\right]_{C^{\beta}_{p}(\Omega\cap
Q_{1/2})}=\left[D^{\lfloor\beta\rfloor}_{p}\frac{u_{1}}{u_{2}}\right]_{C^{\langle\beta\rangle}_{p}(\Omega\cap
Q_{1/2})}+\left[D^{\lfloor\beta\rfloor-1}_{p}\frac{u_{1}}{u_{2}}\right]_{C^{\frac{1+\beta}{2}}_{t}(\Omega\cap
Q_{1/2})}.$

We start with estimating the first term.

Choose $\gamma\in\mathbb{N}^{n+1}$, with $|\gamma|_{p}=\lfloor\beta\rfloor$
and a cylinder $Q_{r}(x_{0},t_{0})$, so that
$Q_{2_{r}}(x_{0},t_{0})\subset\Omega\cap Q_{3/4},$ and $d(x_{0},t_{0})\leq
Cr$, with $C$ independent of $x_{0},t_{0}$ and $r$. Let $z$ be the closest
point to $x_{0}$ in $\partial\Omega_{t_{0}}$. Take arbitrary $(x,t)$ and
$(y,s)$ in $Q_{r}(x_{0},t_{0})$, denote
$p=p_{(z,t_{0})}\in\textbf{P}_{\lfloor\beta\rfloor,p}$ and compute

$\left|\partial^{\gamma}\frac{u_{1}}{u_{2}}(x,t)-\partial^{\gamma}\frac{u_{1}}{u_{2}}(y,s)\right|=\left|\partial^{\gamma}\left(\frac{u_{1}}{u_{2}}(x,t)-p(x,t)\right)-\partial^{\gamma}\left(\frac{u_{1}}{u_{2}}(y,s)-p(y,s)\right)\right|\leq$
$\displaystyle\begin{split}\leq&\sum_{\alpha\leq\gamma}\left|\partial^{\alpha}\left(u_{1}-pu_{2}\right)(x,t)-\partial^{\alpha}\left(u_{1}-pu_{2}\right)(y,s)\right|\left|\partial^{\gamma-\alpha}u_{2}^{-1}(x,t)\right|+\\\
&+\sum_{\alpha\leq\gamma}\left|\partial^{\alpha}\left(u_{1}-pu_{2}\right)(y,s)\right|\left|\partial^{\gamma-\alpha}u_{2}^{-1}(x,t)-\partial^{\gamma-\alpha}u_{2}^{-1}(y,s)\right|.\end{split}$
(5.2)

By Proposition 5.4 and Lemma B.2 we have that $u_{1}-pu_{2}$ is
$C^{\beta+1}_{p}(\overline{\mathcal{C}}),$ for a suitable cone
$\mathcal{C}=\cup_{d_{t_{0}}(x_{0})=c_{\Omega}r=|x_{0}-z|}Q_{r}(x_{0},t_{0})$,
with $D^{k}_{p}(u_{1}-pu_{2})(z,t_{0})=0$, for all $k<\beta+1$. Hence by Lemma
B.1 we can estimate

$|D^{k}_{p}(u_{1}-pu_{2})(x,t)|\leq
C(|x-z|^{\beta+1-k}+|t-t_{0}|^{\frac{\beta+1-k}{2}}),\quad(x,t)\in\mathcal{C},k<\beta+1.$

Next take $\alpha\in\mathbb{N}^{n+1}$ and compute

$\partial^{\alpha}u_{2}^{-1}=\sum_{l\leq|\alpha|}\frac{1}{u_{2}^{l+1}}\sum_{\alpha_{1}+\ldots+\alpha_{l}=\alpha}c_{\alpha_{1},\ldots,\alpha_{l}}\partial^{\alpha_{1}}u_{2}\cdots\partial^{\alpha_{l}}u_{2}.$

Since by Corollary 5.3 $u_{2}\in C^{\beta}_{p}(\Omega\cap Q_{1})$ this implies

$||\partial^{\alpha}u_{2}^{-1}||_{L^{\infty}(Q_{r}(x_{0},t_{0}))}\leq
Cr^{-(|\alpha|+1)},$

when $|\alpha|_{p}<\beta,$ as well as

$\left[\partial^{\alpha}u_{2}^{-1}\right]_{C^{\langle\beta\rangle}_{p}(Q_{r}(x_{0},t_{0}))}\leq
Cr^{-(|\alpha|+1)},$ (5.3)

when $|\alpha|_{p}=\lfloor\beta\rfloor,$ and

$\left[\partial^{\alpha}u_{2}^{-1}\right]_{C^{\frac{\langle\beta\rangle+1}{2}}_{t}(Q_{r}(x_{0},t_{0}))}\leq
Cr^{-(|\alpha|+1)},$ (5.4)

when $|\alpha|_{p}=\lfloor\beta\rfloor-1.$

We are now equipped to estimate (5.2). Take $\alpha\leq\gamma$, with
$|\alpha|_{p}<\beta$ and estimate

$\displaystyle|\partial^{\alpha}v(x,t)-\partial^{\alpha}v(y,s)|\leq$
$\displaystyle|\partial^{\alpha}v(x,t)-\partial^{\alpha}v(y,t)|+|\partial^{\alpha}v(y,t)-\partial^{\alpha}v(y,s)|\leq$
$\displaystyle\leq$
$\displaystyle||D^{|\alpha|_{p}+1}_{p}v|||x-y|+||D^{|\alpha|_{p}+2}_{p}v|||t-s|$
$\displaystyle\leq$ $\displaystyle
Cr^{\beta+1-|\alpha|_{p}-1}r^{1-\langle\beta\rangle}|x-y|^{\langle\beta\rangle}+r^{\beta+1-|\alpha|_{p}-2}r^{2(1-\frac{\langle\beta\rangle}{2})}|t-s|^{\frac{\langle\beta\rangle}{2}}$
$\displaystyle\leq$ $\displaystyle
Cr^{\lfloor\beta\rfloor+1-|\alpha|_{p}}(|x-y|^{\langle\alpha\rangle}+|t-s|^{\frac{\langle\beta\rangle}{2}}).$

If $|\alpha|_{p}=\lfloor\beta\rfloor$, then we need to use the definition of
$\left[\partial^{\alpha}v\right]_{C^{\frac{1+\langle\beta\rangle}{2}}_{t}}$,
to get the same estimate. Similarly, if $|\gamma-\alpha|_{p}<\beta-2$,

$\displaystyle|\partial^{\gamma-\alpha}u_{2}^{-1}(x,t)-\partial^{\gamma-\alpha}u_{2}^{-1}(y,s)|\leq$
$\displaystyle||D^{|\gamma-\alpha|+1}u_{2}^{-1}||(|x-y|+|t-s|)$
$\displaystyle\leq$ $\displaystyle
Cr^{-|\gamma-\alpha|-2}(r^{1-\langle\beta\rangle}|x-y|^{\langle\beta\rangle}+r^{2(1-\frac{\langle\beta\rangle}{2})}|t-s|^{\frac{\langle\beta\rangle}{s}})$
$\displaystyle\leq$ $\displaystyle
Cr^{-|\gamma-\alpha|-1-\langle\beta\rangle}(|x-y|^{\langle\beta\rangle}+|t-s|^{\frac{\langle\beta\rangle}{2}}).$

When $|\gamma-\alpha|_{p}=\lfloor\beta\rfloor-1$ we have to split the
difference into the spatial part and time part and on the time part use (5.4).
When $|\gamma-\alpha|_{p}=\lfloor\beta\rfloor$, we directly use (5.3) to get
the same estimate in other cases. Plugging it all in expression (5.2), we get

$\eqref{termWithExpanisons}\leq\sum_{\alpha\leq\gamma}Cr^{\lfloor\beta\rfloor+1-|\alpha|_{p}}Cr^{-|\gamma-\alpha|-1}(|x-y|^{\langle\beta\rangle}+|t-s|^{\frac{\langle\beta\rangle}{2}})+$
$+\sum_{\alpha\leq\gamma}Cr^{\beta+1-|\alpha|_{p}}r^{-|\gamma-\alpha|-1-\langle\beta\rangle}(|x-y|^{\langle\beta\rangle}+|t-s|^{\frac{\langle\beta\rangle}{2}}).$

It remains to notice that $|\gamma-\alpha|\leq|\gamma-\alpha|_{p}$, and hence
$\eqref{termWithExpanisons}\leq
C(|x-y|^{\langle\beta\rangle}+|t-s|^{\frac{\langle\beta\rangle}{2}})$.

We argue similarly that
$\left[D^{\lfloor\beta\rfloor-1}_{p}\frac{u_{1}}{u_{2}}\right]_{C^{\frac{1+\beta}{2}}_{t}(\Omega\cap
Q_{1/2})}\leq C$. Choose $|\gamma|_{p}=\lfloor\beta\rfloor-1$ remember the
degree of $p$ and compute

$\left|\partial^{\gamma}\frac{u_{1}}{u_{2}}(x,t)-\partial^{\gamma}\frac{u_{1}}{u_{2}}(x,s)\right|=\left|\partial^{\gamma}\left(\frac{u_{1}}{u_{2}}(x,t)-p(x,t)\right)-\partial^{\gamma}\left(\frac{u_{1}}{u_{2}}(x,s)-p(x,s)\right)\right|\leq$
$\displaystyle\begin{split}\leq&\sum_{\alpha\leq\gamma}\left|\partial^{\alpha}\left(u_{1}-pu_{2}\right)(x,t)-\partial^{\alpha}\left(u_{1}-pu_{2}\right)(x,s)\right|\left|\partial^{\gamma-\alpha}u_{2}^{-1}(x,t)\right|+\\\
&+\sum_{\alpha\leq\gamma}\left|\partial^{\alpha}\left(u_{1}-pu_{2}\right)(x,s)\right|\left|\partial^{\gamma-\alpha}u_{2}^{-1}(x,t)-\partial^{\gamma-\alpha}u_{2}^{-1}(x,s)\right|\\\
\leq&\sum_{\alpha\leq\gamma}||D^{|\alpha|_{p}+2}_{p}v||\cdot|t-s|\cdot||D^{|\gamma-\alpha|}u_{2}^{-1}||+\\\
&+\sum_{1\leq|\alpha|\leq|\gamma|}||D^{|\alpha|_{p}}_{p}v||\cdot||D^{|\gamma-\alpha|+1}u_{2}^{-1}||\cdot|t-s|\\\
&+||v||\cdot\left[\partial^{\gamma}u_{2}^{-1}\right]_{C^{\frac{1+\langle\beta\rangle}{2}}_{t}}\cdot|t-s|^{\frac{1+\langle\beta\rangle}{2}}\\\
\leq&\sum_{\alpha\leq\gamma}Cr^{\beta+1-|\alpha|_{p}-2}r^{2(\frac{1-\langle\beta\rangle}{2})}|t-s|^{\frac{1+\langle\beta\rangle}{2}}Cr^{-|\gamma-\alpha|-1}\\\
&+\sum_{1\leq|\alpha|\leq|\gamma|}Cr^{\beta+1-|\alpha|_{p}}Cr^{-|\gamma-\alpha|-2}r^{2(\frac{1-\langle\beta\rangle}{2})}|t-s|^{\frac{1+\langle\beta\rangle}{2}}\\\
&+Cr^{\beta+1}Cr^{-|\gamma|-1}r^{2(\frac{1-\langle\beta\rangle}{2})}|t-s|^{\frac{1+\langle\beta\rangle}{2}}\\\
\leq&C|t-s|^{\frac{1+\langle\beta\rangle}{2}}.\end{split}$

Therefore the claim is proven, thanks to Lemma B.2.

The same steps prove the claim in case $\beta=1$, but we use Proposition 4.1
instead of Proposition 5.4, and the $C^{\beta}_{p}$ estimate for $u_{2}$ up to
the boundary is replaced by $C^{1-\varepsilon}_{p}$ from Corollary 3.5. ∎

##  6\. Higher regularity of free boundary in obstacle problem

Our parabolic higher order boundary Harnack inequality gives a simple way to
prove that once the free boundary in the parabolic obstacle problems is
$C^{1}_{p}$, it is in fact $C^{\infty}$. The reason is that the normal to the
free boundary can be expressed with the quotients of partial derivatives of
the solution to obstacle problem, which are as smooth as the boundary. This
gives the bootstrap argument, and does not require any use of a hodograph
transform as in [11].

Let us start with preliminary results that we need in the proof of Corollary
1.4.

######  Lemma 6.1.

Let $v\colon Q_{1}\to\mathbb{R}$ solve (1.5) with $f\in C^{\theta}(B_{1})$,
for some $\theta>0$, with $f(0)=1$. Assume that $(0,0)\in\partial\\{v>0\\}$ is
a regular free boundary point.

Then $v\in C^{1,1}_{p}(Q_{1})$, up to rotation of coordinates the free
boundary is $C^{1}_{p}$ in $Q_{r}$ for some $r>0$ in the sense of Definition
1.1, $v_{t}\in C(Q_{r})$ and $\partial_{\nu}v\geq c_{1}d$, with $c_{1}>0$,
where $\nu$ is the unit spatial normal vector of the free boundary at $(0,0)$.

###### Proof.

The $C^{1,1}_{p}(Q_{1})$ regularity of solutions of the obstacle problem
follows from [15, Theorem 1.6]. The $C^{1}_{p}$ regularity of the free
boundary is provided in [16, Theorem 1.7]. Denote the boundary defining map
with $F$.

Let us now prove the continuity of the time derivative. This is well known
when $f\equiv 1$ (see [5]). For completeness we prove it next in our setting.
Since inside $\Omega$ the solution $v$ is $C^{\theta+2}_{p}$ and $v\equiv 0$
in $\Omega^{c}$, we have to show that

$\lim_{(x,t)\to(z,s)}\partial_{t}v(x,t)=0,$

for every free boundary point in $Q_{r}$. Choose a point $(x,t)\in\Omega\cap
Q_{r}$, such that $(x,\tau)\in\Omega^{c}$ for $\tau\leq s$. Let
$(x,s),(x^{\prime},F(x^{\prime},t),t)$ be free boundary points. Therefore by
[15, Theorem 1.6] we have $v(x,t)\leq C|x_{n}-F(x^{\prime},t)|^{2}.$ But since
$F\in C^{1}_{p}$, and $x_{n}=F(x^{\prime},s)$, we have $v(x,t)\leq
C|F(x^{\prime}t)-F(x^{\prime},s)|^{2}\leq
C(|t-s|^{\frac{1}{2}}\omega(|t-s|^{\frac{1}{2}}))^{2}$, where $\omega$ is the
modulus of continuity of $\nabla F$. This gives
$|u(x,t)-u(x,s)|=u(x,t)\leq|t-s|\eta(|t-s|)$, for some modulus of continuity
$\eta$.

Finally, let us show that $\partial_{\nu}v\geq c_{1}d$ where $c_{1}>0$, $d$ is
the regularised distance to the $\partial\Omega$ and $\nu$ is its spatial
normal vector at $(0,0)$. Without loss of generality assume that $\nu=e_{n}$.
Having the bound $||u||_{C^{1,1}_{p}(Q_{1})}\leq C$, together with [15,
Theorem 1.7] and Arzela-Ascoli theorem imply that $v$ converges to its blow up
at any free boundary point $(x_{0},t_{0})\in\partial\Omega\cap Q_{r}$
uniformly (independently of the free boundary point) in $C^{0,1}_{p}$, namely

$\left|\left|\frac{1}{r^{2}}v(x_{0}+rx,t_{0}+r^{2}t)-\frac{f(x_{0})}{2}(x\nu_{x_{0}})_{+}^{2}\right|\right|_{C^{0,1}_{p}(Q_{1})}\longrightarrow
0$

as $r\to 0,$ uniformly in $(x_{0},t_{0})$. Hence reading it for the $n-$th
derivative, this means that there exists a number $r_{0}>0$ so that for all
$r<r_{0}$ we have

$\left|\left|\frac{1}{r}v_{n}(x_{0}+rx,t_{0}+r^{2}t)-f(x_{0})(\nu_{x_{0}}e_{n})(x\nu_{x_{0}})_{+}\right|\right|_{L^{\infty}(Q_{1})}\leq\frac{1}{4}.$

Restricting ourselves to a potentially smaller neighbourhood of $(0,0)$, so
that $\nu_{x_{0}}e_{n}>\frac{3}{4}$, $|\nu_{x_{0}}-e_{n}|\leq\frac{1}{5},$
$|f(x_{0})-1|>\frac{1}{4}$ with a triangle inequality we deduce that for
$(x,t_{0})\in Q_{r}((x_{0},t_{0}))\cap\\{(x-x_{0})e_{n}>\frac{r}{2}\\}$ we
have

$\partial_{n}v(x,t_{0})\geq
f(x_{0})(\nu_{x_{0}}e_{n})((x-x_{0})\nu_{x_{0}})-\frac{r}{4}\geq\frac{r}{4}.$

But since $d_{x}(x,t_{0})\leq|x-x_{0}|=r,$ and the estimate is independent of
$(x_{0},t_{0})$, we get that
$\partial_{v}\geq\frac{1}{4}d_{x}\geq\frac{1}{4C}d$, since the two distances
are comparable. ∎

We are now equipped enough to prove Corollary 1.4.

###### Proof of Corollary 1.4.

Denote $\Omega=\\{v>0\\}.$ By Lemma 6.1, rescaling and rotation if necessary
$\Omega$ is $C^{1}_{p}$ in $Q_{1}$, the solution $v$ is $C^{1,1}$ in space and
$C^{1}$ in time. This implies that in $\Omega^{c}$ both
$v_{t}:=\partial_{t}v=0$ and for $i=1,\ldots,n$ also $v_{i}:=\partial_{i}v=0$.
Without loss of generality assume that the normal vector to $\Omega_{0}$ at
$0$ is $e_{n}$. Hence all the partial derivatives of $v$ solve

$\left\\{\begin{array}[]{rcll}(\partial_{t}-\Delta)w&=&\partial_{e}f&\text{in
}\Omega\cap Q_{1}\\\ w&=&0&\text{on }\partial\Omega\cap
Q_{1}.\end{array}\right.$

Remember that $f$ is independent of time and by assumption $f\in
C^{\theta}(B_{1})$ and hence $\partial_{e}f\in C^{\theta-1}_{p}(Q_{1})$.
Moreover Lemma 6.1 says that $v_{n}\geq c_{1}d$ with $c_{1}>0$ in a
neighbourhood of $0$. Hence we can apply Theorem 1.3, which gives that all
quotients $v_{i}/v_{n}$ and $v_{t}/v_{n}$ are
$C^{1-\varepsilon}_{p}(\overline{\Omega}\cap Q_{r_{2}}),$ with bounds on the
norms.

Now notice that every component the normal vector $\nu(x,t)$ to the level set
$\\{v=t\\}$, $t>0$ can be expressed as

$\nu^{i}(x,t)=\frac{\partial_{i}v}{|\nabla_{(x,t)}v|}(x,t)=\frac{\partial_{i}v/\partial_{n}v}{\left(\sum_{j=1}^{n-1}(\partial_{j}v/\partial_{n}v)^{2}+1+(\partial_{t}v/\partial_{n}v)^{2}\right)^{1/2}}.$

Letting $t\downarrow 0$, we get that the normal vector is
$C^{1-\varepsilon}_{p}(\partial\Omega\cap Q_{r/2}).$ Hence the free boundary
is $C^{2-\varepsilon}_{p}$.

The same reasoning gives that if the boundary is $C^{\beta}_{p}$ it is
$C^{\beta+1}_{p}$, as long as $\beta\leq\theta$ and hence the claim is proven.
∎

##  Appendix A Interior regularity results

In this section we state the interior regularity results used in the body of
the paper. Even though the results are not new, we provide the proofs since
the claims are adapted to our specific setting. For overview of the theory of
parabolic second order equations, we refer to [12, 13].

Our techniques rely on contradiction and blow up arguments. The contradiction
at the end is usually provided by Liouville type results. We start with
establishing such kind of result, saying that if a function solves an equation
in the full space $\mathbb{R}^{n+1}$ with a polynomial in the right-hand side,
then the function is a polynomial as well. The proof is based on using
interior estimates iteratively on arbitrarily big cylinders.

######  Proposition A.1.

Assume that $v$ solves

$\left\\{\begin{array}[]{rcl
l}\partial_{t}v-\operatorname{tr}(AD^{2}v)&=&P&\text{in }\mathbb{R}^{n+1}\\\
||v||_{L^{\infty}(Q_{R})}&\leq&C_{0}R^{\gamma}&\forall R>1,\\\
\end{array}\right.$

for some constant, uniformly elliptic matrix $A$, $\gamma>0,$
$\gamma\not\in\mathbb{N}$, and some polynomial $P$ of parabolic order less
than $\gamma-2$ (if $\gamma<2$, then $P=0$). Then $v$ is a polynomial of
parabolic order less than $\gamma$.

###### Proof.

Take $R>1$ and define

$v_{R}(x,t):=R^{-\gamma}v(Rx,R^{2}t),$

so that
$(\partial_{t}v_{R}-\operatorname{tr}(AD^{2}v_{R}))(x,t)=R^{-\gamma+2}P(Rx,R^{2}t)=P_{R}(x,t).$
Note that $P_{R}$ is again a polynomial of the same order and the coefficients
reduce as $R$ increases. Moreover, by assumption
$||v_{R}||_{L^{\infty}(Q_{1})}\leq C_{0}$. Interior regularity result [17,
Theorem 2] gives

$\left[v_{R}\right]_{C_{p}^{0,1}(Q_{1/2})}\leq
C(||P_{R}||+||v_{R}||_{L^{\infty}(Q_{1})})\leq C(||P||+C_{0}),$

and hence

$\left[v\right]_{C_{p}^{0,1}(Q_{R/2})}\leq CR^{\gamma-1}(||P||+C_{0}).$

Choose arbitrary $h\in\mathbb{R}^{n},$ $\tau\in\mathbb{R}$, and define

$w_{1}(x,t)=\frac{v(x+h,t+\tau)-v(x,t)}{|h|+|\tau|}.$

Then $w_{1}$ solves

$\partial_{t}w_{1}-\operatorname{tr}(AD^{2}w_{1})=P_{1},$

for some polynomial $P_{1}\in\textbf{P}_{\lfloor\gamma-1\rfloor,p}$ and it
satisfies the growth control

$||w_{1}||_{L^{\infty}(Q_{R})}\leq C\left[v\right]_{C^{0,1}(Q_{R})}\leq
CR^{\gamma-1}.$

Repeating the same procedure we conclude that

$\left[w_{\lceil\gamma\rceil}\right]_{C^{0,1}(Q_{R/2})}\leq
CR^{\gamma-\lceil\gamma\rceil},$

which gives that $w_{\lceil\gamma\rceil}$ is constant. This implies that $v$
is a polynomial, and the assumed growth assures that it is of parabolic order
less than $\gamma$. ∎

Since we define the parabolic Hölder seminorms a bit differently than for
example in [12], we prove the a priori Schauder estimates in our setting as
well.

######  Proposition A.2.

Assume $u\in C^{2+\alpha}_{p}(Q_{1})$ solves the equation

$\partial_{t}u+Lu=f,\quad\quad\text{in }Q_{1},$

with $f\in C^{\alpha}_{p}(Q_{1}),$ $A\in C^{\alpha}_{p}(Q_{1})$ and $b\in
C^{\alpha}_{p}(Q_{1})$. Then we have

$\left|\left|u\right|\right|_{C^{2+\alpha}_{p}(Q_{1/2})}\leq
C(\left[f\right]_{C^{\alpha}_{p}(Q_{1})}+||u||_{L^{\infty}(Q_{1})}),$

where $C$ depends only on $n,\alpha$, ellipticity constants and Hölder norms
of the coefficients.

###### Proof.

Thanks to [7, Lemma 2.23] and interpolation inequality, it suffices to prove
that for every $\delta>0$ there exists $C$ so that

$\left[u\right]_{C^{2+\alpha}_{p}(Q_{1/2})}\leq\delta\left[u\right]_{C^{2+\alpha}_{p}(Q_{1})}+C(\left[f\right]_{C^{\alpha}_{p}(Q_{1})}+||u||_{C^{2}_{p}(Q_{1})}).$

Dividing the equation with a constant, we can assume that
$\left[f\right]_{C^{\alpha}_{p}(Q_{1})}+||u||_{C^{2}_{p}(Q_{1})}=1$.

We argue with contradiction. Assume that there is $\delta>0$ so that for every
$k\in\mathbb{N}$ there exist $u_{k},f_{k}$ and $L_{k}$
$(\lambda,\Lambda)$-uniformly elliptic operators so that
$\partial_{t}u_{k}+L_{k}u_{k}=f_{k}$ in $Q_{1}$, but

$\left[u_{k}\right]_{C^{2+\alpha}_{p}(Q_{1/2})}>\delta\left[u_{k}\right]_{C^{2+\alpha}_{p}(Q_{1})}+k.$

Choose points $(x_{k},t_{k}),$ and $(y_{k},s_{k})$, so that

$\displaystyle c(n)\left[u_{k}\right]_{C^{2+\alpha}_{p}(Q_{1/2})}\leq$
$\displaystyle\frac{|D^{2}u_{k}(x_{k},t_{k})-D^{2}u_{k}(y_{k},s_{k})|}{|x_{k}-y_{k}|^{\alpha}+|t_{k}-s_{k}|^{\frac{\alpha}{2}}}+\frac{|\partial_{t}u_{k}(x_{k},t_{k})-\partial_{t}u_{k}(y_{k},s_{k})|}{|x_{k}-y_{k}|^{\alpha}+|t_{k}-s_{k}|^{\frac{\alpha}{2}}}$
$\displaystyle+\frac{|Du_{k}(x_{k},t_{k})-Du_{k}(x_{k},s_{k})|}{|t_{k}-s_{k}|^{\frac{1+\alpha}{2}}}.$

Define $\rho_{k}=|x_{k}-y_{k}|+|t_{k}-s_{k}|^{\frac{1}{2}}.$ Then the above
inequality implies

$\displaystyle c(n)\left[u_{k}\right]_{C^{2+\alpha}_{p}(Q_{1/2})}\leq$
$\displaystyle\frac{|D^{2}u_{k}(x_{k},t_{k})-D^{2}u_{k}(y_{k},s_{k})|}{\rho_{k}^{\alpha}}+\frac{|\partial_{t}u_{k}(x_{k},t_{k})-\partial_{t}u_{k}(y_{k},s_{k})|}{\rho_{k}^{\alpha}}$
$\displaystyle+\frac{|Du_{k}(x_{k},t_{k})-Du_{k}(x_{k},s_{k})|}{\rho_{k}^{1+\alpha}}.$

Applying triangle inequality and bounding the terms with
$||u_{k}||_{C^{2}_{p}(Q_{1})}$, we get

$c(n)\left[u_{k}\right]_{C^{2+\alpha}_{p}(Q_{1/2})}\leq||u_{k}||_{C^{2}_{p}(Q_{1})}(\rho_{k}^{-\alpha}+\rho_{k}^{-1+\alpha}),$

but thanks to the contradiction assumption,

$c(n)\left[u_{k}\right]_{C^{2+\alpha}_{p}(Q_{1/2})}\leq\frac{\left[u_{k}\right]_{C^{2+\alpha}_{p}(Q_{1/2})}}{k}(\rho_{k}^{-\alpha}+\rho_{k}^{-1+\alpha}),$

which assures that $\rho_{k}\to 0$ as $k\to\infty.$

Now we define the blow-up sequence

$v_{k}(x,t)=\frac{1}{\rho_{k}^{2+\alpha}\left[u_{k}\right]_{C^{2+\alpha}_{p}(Q_{1})}}\left(u_{k}(x_{k}+\rho_{k}x,t_{k}+\rho_{k}^{2}t)-p_{k}(x,t)\right),$

where $p_{k}$ is a polynomial of parabolic order $2$, so that
$v_{k}(0,0)=Dv_{k}(0,0)=D^{2}v_{k}(0,0)=\partial_{t}v_{k}(0,0)=0$. Now take
any $R<\frac{1}{2\rho_{k}}.$ Since $p_{k}$ is of lower order, it holds

$\left[v_{k}\right]_{C^{2+\alpha}_{p}(Q_{R})}=\frac{1}{\left[u_{k}\right]_{C^{2+\alpha}_{p}(Q_{1})}}\left[u_{k}\right]_{C^{2+\alpha}_{p}(Q_{R\rho_{k}}(x_{k},t_{k}))}\leq
1,$

and hence also $||v_{k}(x,t)||_{L^{\infty}(Q_{R})}\leq R^{2+\alpha}$ and in
particular $\left[v_{k}\right]_{C^{2+\alpha}_{p}(Q_{1})}\leq 1$. Moreover, by
definition of $(x_{k},t_{k})$ and $\rho_{k}$ we have

$\left[v_{k}\right]_{C^{2+\alpha}_{p}(Q_{1})}\geq\delta c(n).$

Furthermore

$\displaystyle(\partial_{t}+\tilde{L}_{k})v_{k}(x,t)=$
$\displaystyle\frac{1}{\rho_{k}^{\alpha}\left[u_{k}\right]_{C^{2+\alpha}_{p}(Q_{1})}}\Bigg{(}f_{k}(x_{k}+\rho_{k}x,t_{k}+\rho_{k}^{2}t)-$
$\displaystyle-\partial_{t}u_{k}(x_{k},t_{k})-\sum_{i,j}a^{k}_{i,j}(x_{k}+\rho_{k}x,t_{k}+\rho_{k}^{2}t)\partial_{ij}u_{k}(x_{k},t_{k})+$
$\displaystyle+\sum_{i}b^{k}_{i}(x_{k}+\rho_{k}x,t_{k}+\rho_{k}^{2}t)\partial_{i}u_{k}(x_{k},t_{k})+$
$\displaystyle+\rho_{k}\sum_{i}b^{k}_{i}(x_{k}+\rho_{k}x,t_{k}+\rho_{k}^{2}t)\sum_{j}\partial_{ij}u_{k}(x_{k},t_{k})x_{j}\Bigg{)}.$

Taking into account the regularity of the coefficients, with adding and
subtracting suitable terms, we get

$\displaystyle(\partial_{t}+\tilde{L}_{k})v_{k}(x,t)=$
$\displaystyle\frac{1}{\rho_{k}^{\alpha}\left[u_{k}\right]_{C^{2+\alpha}_{p}(Q_{1})}}\Bigg{(}f_{k}(x_{k}+\rho_{k}x,t_{k}+\rho_{k}^{2}t)-f_{k}(x_{k},t_{k})+$
$\displaystyle+\sum_{i,j}o_{A,k}(x,t)\partial_{ij}u_{k}(x_{k},t_{k})+\sum_{i}o_{b,k}(x,t)\partial_{i}u_{k}(x_{k},t_{k})+$
$\displaystyle+\rho_{k}\sum_{i}b_{i}^{k}(x_{k}+\rho_{k}x,t_{k}+\rho_{k}^{2}t).\Bigg{)}$

and hence by contradiction assumption

$|(\partial_{t}+\tilde{L}_{k})v_{k}(x,t)|\leq\frac{C}{k}\left(\left[f_{k}\right]_{C^{\alpha}_{p}}+||A_{k}||_{C^{\alpha}_{p}}+||b_{k}||_{C^{\alpha}_{p}}+\rho_{k}^{1-\alpha}\Lambda\right),$

on any compact set inside $Q_{\frac{1}{2\rho_{k}}}$.

Passing to a subsequence, Arzela-Ascoli theorem provides that $v_{k}$ converge
in $C^{2}_{p}(\mathbb{R}^{n+1})$ on compact sets to some function $v$, which
by [2, Theorem 1.1] solves

$\left\\{\begin{array}[]{rcll}(\partial_{t}-\operatorname{tr}(AD^{2}))v&=&0&\text{in
}\mathbb{R}^{n+1}\\\ ||v||_{L^{\infty}(Q_{R})}&\leq&R^{2+\alpha}&\text{for all
}R>0\\\ v(0)=Dv(0)=D^{2}v(0)=\partial_{t}v(0)&=&0&\\\
\left[v\right]_{C^{2+\alpha}_{p}(Q_{1})}&\geq&\delta c(n),&\end{array}\right.$

for some uniformly elliptic, constant matrix $A$. Hence by Liouville theorem
(Proposition A.1) $v$ is a polynomial of parabolic order $2$, which is a
contradiction with last two properties of $v$. ∎

Next we establish the higher order a priori estimates.

######  Corollary A.3.

Let $k\in\mathbb{N}$, $k\geq 2$, and $\alpha\in(0,1)$. Assume $u\in
C^{k,\alpha}_{p}(Q_{1})$ solves

$(\partial_{t}+L)u=f\quad\quad\text{in }Q_{1},$

for some $f\in C^{k-2,\alpha}_{p}(Q_{1})$, $A,b\in C^{k-2,\alpha}_{p}(Q_{1})$.
Then

$\left|\left|u\right|\right|_{C^{k+\alpha}_{p}(Q_{1/2})}\leq
C(\left[f\right]_{C^{k-2,\alpha}_{p}(Q_{1})}+||u||_{L^{\infty}(Q_{1})}),$

where $C$ depends only on $n,\alpha$, ellipticity constants and Hölder norms
of the coefficients.

###### Proof.

We prove the claim by induction. Proposition A.2 proves the case $k=2$. So let
$k>2$ and assume the claim is true for all $l<k$. Choose now
$e\in\mathbb{S}^{n-1}.$ Then by induction hypothesis

$\left[\partial_{e}u\right]_{C^{k-1+\alpha}_{p}(Q_{1/2})}\leq
C(\left[\partial_{e}f\right]_{C^{k-3,\alpha}_{p}(Q_{1})}+||\partial_{e}u||_{L^{\infty}(Q_{1})}),$

which after interpolation inequality implies

$\left[Du\right]_{C^{k-1+\alpha}_{p}(Q_{1/2})}\leq
C(\left[f\right]_{C^{k-2,\alpha}_{p}(Q_{1})}+||u||_{L^{\infty}(Q_{1})}).$

Hence also

$\left[D^{2}u\right]_{C^{k-2+\alpha}_{p}(Q_{1/2})}\leq
C(\left[f\right]_{C^{k-2,\alpha}_{p}(Q_{1})}+||u||_{L^{\infty}(Q_{1})}),$

and so using that $\partial_{t}u=f-Lu$ we also get

$\left[\partial_{t}u\right]_{C^{k-2+\alpha}_{p}(Q_{1/2})}\leq
C(\left[f\right]_{C^{k-2,\alpha}_{p}(Q_{1})}+||u||_{L^{\infty}(Q_{1})}).$

Combined, the inequalities render

$\left[u\right]_{C^{k+\alpha}_{p}(Q_{1/2})}\leq
C(\left[f\right]_{C^{k-2,\alpha}_{p}(Q_{1})}+||u||_{L^{\infty}(Q_{1})}),$

and the claim is proven. ∎

We conclude the section by showing that solutions to equations with regular
right-hand side are indeed as regular as the a priori estimates indicate.

######  Proposition A.4.

Let $k\in\mathbb{N}$, $k\geq 2$, and $\alpha\in(0,1)$. Let $A,b,f\in
C^{k-2,\alpha}_{p}(Q_{1})$, and $u$ a solution to

$(\partial_{t}+L)u=f,\quad\text{in }Q_{1}.$

Then $u\in C^{k,\alpha}_{p}(Q_{1}),$ with

$\left|\left|u\right|\right|_{C^{k+\alpha}_{p}(Q_{1/2})}\leq
C(\left[f\right]_{C^{k-2,\alpha}_{p}(Q_{1})}+||u||_{L^{\infty}(Q_{1})}),$

where $C$ depends only on $n,\alpha,k$, ellipticity constants and Hölder norms
of the coefficients.

###### Proof.

We only need to prove the regularity of $u$, since we have already established
the estimates.

If $k=2$, then the regularity is justified by [13, Theorem 5.9]. For $k>2$,
with the same argument as in [12, Theorem 8.12.1] we establish that $\nabla
u\in C^{k-1,\alpha}_{p}(Q_{1})$. Using the equation, this also gives that
$\partial_{t}u\in C^{k-2,\alpha}_{p}(Q_{1})$. ∎

##  Appendix B Technical tools and lemmas

######  Lemma B.1.

Let $\Omega\subset\mathbb{R}^{n+1}$ be a domain, such that $\Omega_{0}$ is
convex and $\\{t;(t,x)\in\Omega\\}$ is connected for every $x$. Let
$\beta\not\in\mathbb{N}$ and assume $u\in C^{\beta}_{p}(\overline{\Omega})$,
with $D^{k}_{p}u(0,0)=0$, for every $k\leq\lfloor\beta\rfloor$. Then for every
$k\leq\beta$ and $r>0$ we have

$||D^{k}_{p}u||_{L^{\infty}(\Omega\cap Q_{r})}\leq Cr^{\beta-k}.$

###### Proof.

We prove the claim with induction on $\beta$ and $k$. If
$k=\lfloor\beta\rfloor$, then it follows from the definition of
$C^{\beta}_{p}$.

If $k=\lfloor\beta\rfloor-1,$ we estimate

$|D^{k}_{p}u(x,t)|\leq|D^{k}_{p}u(x,t)-D^{k}_{p}u(x,0)|+|D^{k}u(x,0)|\leq
C|t|^{\frac{1+\langle\beta\rangle}{2}}+\int_{0}^{|x|}||D^{k+1}_{p}u||_{L^{\infty}(Q_{s})}ds,$

and hence by induction hypothesis

$|D^{k}_{p}u(x,t)|\leq Cr^{1+\langle\beta\rangle}.$

If $k<\lfloor\beta\rfloor-1,$ we use $\partial_{t}D^{k}_{p}u\subset
D^{k+2}_{p}u$ and estimate

$|D^{k}_{p}u(x,t)-D^{k}_{p}u(x,0)|\leq\int_{0}^{t}||D^{k+2}_{p}u||_{L^{\infty}(Q_{s})}ds,$

to obtain the same result. ∎

######  Lemma B.2.

Let $\beta>0$ and let $\Omega\subset\mathbb{R}^{n+1}$ be
$C^{\max(\beta,1)}_{p}$ in $Q_{1}$ in the sense of Definition 1.1. Assume that
a function $u\colon\Omega\cap Q_{1}\to\mathbb{R}$ satisfies

$\left[u\right]_{C^{\beta}_{p}(Q_{r}(x_{0},t_{0}))}\leq C_{0},$

whenever $Q_{2r}(x_{0},t_{0})\subset\Omega\cap Q_{1}$ and
$d_{t_{0}}(x_{0})\leq C_{1}r$, with $C_{1}$ depending only on $\Omega$.

Then

$\left[u\right]_{C^{\beta}_{p}(\Omega\cap Q_{1/2})}\leq CC_{0}.$

The constant $C$ depends only on $n,\beta$ and $\Omega$.

###### Proof.

Thanks to parabolic scaling and covering argument, we can assume $C_{1}=3$,
and that $\left[u\right]_{C^{\beta}_{p}(Q_{\frac{3r}{2}}(x_{0},t_{0}))}\leq
C_{0}$, or in other words, we have

$\left[u\right]_{C^{\beta}_{p}(Q_{r}(x_{0},t_{0}))}\leq C_{0},$

whenever $d_{t_{0}}(x_{0})=2r.$

Let us first prove, that
$\left[u\right]_{C^{\beta}_{p}(\mathcal{C}_{(z,t_{0})})}\leq CC_{0},$ where

$\mathcal{C}_{(z,t_{0})}=\cup_{d_{t_{0}}(x_{0})=2r=|x_{0}-z|}Q_{r}(x_{0},t_{0}),$

is a ”parabolic cone” starting at a boundary point
$(z,t_{0})\in\partial\Omega.$ First, notice that

$\left[D^{\lfloor\beta\rfloor-1}_{p}u\right]_{C^{\frac{1+\langle\beta\rangle}{2}}_{t}(\mathcal{C}_{(z,t_{0})}}\leq
C_{0},$

since the points we choose for comparing the function values have to lie in
the same cylinder $Q_{r}(x_{0},t_{0})$, because the space coordinate coincide.

Now choose arbitrary $(x,t)$ and $(y,s)$ in $\mathcal{C}_{(z,t_{0})}$. Without
loss of generality assume $d(x,t)\leq d(y,s)$. Denote $x^{\prime}$ and
$y^{\prime}$ the central points of cylinders from $\mathcal{C}_{(z,t_{0})}$,
containing $(x,t)$ and $(y,s)$ respectively. We want to estimate

$|D^{\lfloor\beta\rfloor}_{p}u(x,t)-D^{\lfloor\beta\rfloor}_{p}u(y,s)|\leq
CC_{0}(|x-y|^{\langle\beta\rangle}+|t-s|^{\frac{\langle\beta\rangle}{2}}).$

If $2d_{t_{0}}(x)>d_{t_{0}}(y)$, both points are contained in
$Q_{\frac{3}{4}d_{t_{0}}(y)}(y,t_{0})$, and hence another covering argument
assures the above estimate. Otherwise we have $d_{t_{0}}(y)\leq 2|x-y|$ as
well as $|x-x^{\prime}|\leq\frac{d_{t_{0}}(x)}{2}$ and
$|y-y^{\prime}|\leq\frac{d_{t_{0}}(y)}{2}.$ For $i=0,1,\ldots$ denote
$x_{i}=x^{\prime}+2^{-i}(y^{\prime}-x^{\prime})$ and $r_{i}=d_{t_{0}}(x_{i})$,
so that $(x_{i+1},t),(x_{i},t)\in Q_{\frac{r_{i}}{2}}(x_{i},t_{0})$. Hence we
have

$\displaystyle|D^{\lfloor\beta\rfloor}_{p}u(x,t)-D^{\lfloor\beta\rfloor}_{p}u(y,s)|\leq$
$\displaystyle|D^{\lfloor\beta\rfloor}_{p}u(y,s)-D^{\lfloor\beta\rfloor}_{p}u(y^{\prime},t)|+$
$\displaystyle+\sum_{i=0}^{K}|D^{\lfloor\beta\rfloor}_{p}u(x_{i},t)-D^{\lfloor\beta\rfloor}_{p}u(x_{i+1},t)|+$
$\displaystyle+|D^{\lfloor\beta\rfloor}_{p}u(x,t)-D^{\lfloor\beta\rfloor}_{p}u(x^{\prime},t)|$
$\displaystyle\leq$ $\displaystyle
C_{0}(|y-y^{\prime}|^{\langle\beta\rangle}+|t-s|^{\frac{\langle\beta\rangle}{2}})+\sum_{i=0}^{K}C_{0}|x_{i}-x_{i+1}|^{\langle\beta\rangle}$
$\displaystyle+C_{0}|x-x^{\prime}|^{\langle\beta\rangle}$ $\displaystyle\leq$
$\displaystyle
C_{0}\left(2|x-y|^{\langle\beta\rangle}+|t-s|^{\frac{\langle\beta\rangle}{2}}+|x-y|^{\langle\beta\rangle}\sum_{i=0}^{\infty}2^{-i\langle\beta\rangle}\right)$
$\displaystyle=$ $\displaystyle
C_{\beta}C_{0}\left(|x-y|^{\langle\beta\rangle}+|t-s|^{\frac{\langle\beta\rangle}{2}}\right).$

Let us now turn to the general case. Pick $(x,t),(y,s)\in\Omega\cap Q_{1/2}$.
Denote $z_{x}\in\partial\Omega_{t}$, $z_{y}\in\partial\Omega_{s}$ so that
$d_{t}(x)=|x-z_{x}|$ and $d_{s}(y)=|y-z_{y}|$. Now choose
$x_{1}\in\Omega_{t}$, a central point of the cylinder from
$\mathcal{C}_{(z_{x},t)}$, so that $|x-x_{1}|\leq 2|t-s|^{\frac{1}{2}}$, and
that $(x_{1},s)\in\mathcal{C}_{(z_{x},t)}$. If the boundary is flat enough,
this is possible, meaning that we have to restrict ourselves to a small
neighbourhood of the boundary before if necessary. Now take
$z_{xy}\in\partial\Omega_{s}$, so that $d_{s}(x_{1})=|x_{1}-z_{x,y}|,$ and
choose a point $y_{1}\in\Omega_{s}$, so that
$(y_{1},s)\in\mathcal{C}_{(z_{xy},s)}\cap\mathcal{C}_{(z_{y},s)},$ but
$|x_{1}-y_{1}|\leq 2\left(|x-y|+|t-s|^{\frac{1}{2}}\right)$. Note that
$(x,t)\in\mathcal{C}_{(z_{x},t)},$ as well as
$(y,s)\in\mathcal{C}_{(z_{y},s)},$ and hence

$\displaystyle|D^{\lfloor\beta\rfloor}_{p}u(x,t)-D^{\lfloor\beta\rfloor}_{p}u(y,s)|\leq$
$\displaystyle|D^{\lfloor\beta\rfloor}_{p}u(x,t)-D^{\lfloor\beta\rfloor}_{p}u(x_{1},s)|+|D^{\lfloor\beta\rfloor}_{p}u(x_{1},s)-D^{\lfloor\beta\rfloor}_{p}u(y_{1},s)|$
$\displaystyle+|D^{\lfloor\beta\rfloor}_{p}u(y_{1},s)-D^{\lfloor\beta\rfloor}_{p}u(y,s)|$
$\displaystyle\leq$ $\displaystyle
C_{\beta}C_{0}\left(|x-x_{1}|^{\langle\beta\rangle}+|t-s|^{\frac{\langle\beta\rangle}{2}}\right)+C_{\beta}C_{0}|x_{1}-y_{1}|^{\langle\beta\rangle}$
$\displaystyle+C_{\beta}C_{0}|y_{1}-y|^{\langle\beta\rangle}$
$\displaystyle\leq$ $\displaystyle
CC_{0}\left(|x-y|^{\langle\beta\rangle}+|t-s|^{\frac{\langle\beta\rangle}{2}}\right).$

Finally, choose two points $(x,t),(x,s)\in\Omega\cap Q_{1/2}$. Let
$d_{t}(x)=|x-z_{t}|$ and $d_{s}(x)=|x-z_{s}|$. Pick $x_{1}$ and $x_{2}$ in
$\Omega_{t}\cap\Omega_{s}$, so that $x,x_{1},x_{2}$ lie on the same line,
$|x-x_{1}=|x_{1}-x_{2}|$, $|x-x_{1}|\leq 2|t-s|^{\frac{1}{2}}$, and that
$(x,t),(x_{1},t),(x_{2},t),(x_{1},s),(x_{2},s)\in\mathcal{C}_{(z_{t},t)}$. If
$\Omega$ is flat enough, this is possible, otherwise we restrict ourselves to
the smaller neighbourhood of the boundary. Therefore we have

$\displaystyle|D^{\lfloor\beta\rfloor-1}_{p}u(x,t)-D^{\lfloor\beta\rfloor-1}_{p}u(x,s)|\leq$
$\displaystyle|D^{\lfloor\beta\rfloor-1}_{p}u(x,t)-2D^{\lfloor\beta\rfloor-1}_{p}u(x_{1},t)+D^{\lfloor\beta\rfloor-1}_{p}u(x_{2},t)|$
$\displaystyle+|D^{\lfloor\beta\rfloor-1}_{p}u(x,s)-2D^{\lfloor\beta\rfloor-1}_{p}u(x_{1},s)+D^{\lfloor\beta\rfloor-1}_{p}u(x_{2},s)|$
$\displaystyle+2|D^{\lfloor\beta\rfloor-1}_{p}u(x_{1},t)-D^{\lfloor\beta\rfloor-1}_{p}u(x_{1},s)|$
$\displaystyle+|D^{\lfloor\beta\rfloor-1}_{p}u(x_{2},t)-D^{\lfloor\beta\rfloor-1}_{p}u(x_{2},s)|$
$\displaystyle\leq$ $\displaystyle
2CC_{0}|x-x_{1}|^{1+\langle\beta\rangle}+3CC_{0}|t-s|^{\frac{1+\langle\beta\rangle}{2}}$
$\displaystyle\leq$ $\displaystyle
CC_{0}|t-s|^{\frac{1+\langle\beta\rangle}{2}},$

and so the claim is proven. ∎

######  Remark B.3.

We need the boundary to be at least $C^{0,1}_{p}$, so that the cones
$\mathcal{C}_{(z,s)}$ come up to the boundary.

######  Lemma B.4.

Let $\beta>0$ $\beta\not\in\mathbb{N}$, $\Omega\subset\mathbb{R}^{n+1}$, with
$0\in\partial\Omega$, and let $u\in C(\overline{Q}_{1}).$ Assume that for
every $r\in(0,1)$ we have a polynomial
$p_{r}\in\textbf{P}_{\lfloor\beta\rfloor,p}$ so that

$||u-p_{r}d||_{L^{\infty}(Q_{r})}\leq C_{0}r^{\beta+1}.$

Then there exists a polynomial $p_{0}\in\textbf{P}_{\lfloor\beta\rfloor,p}$
which satisfies

$||u-p_{0}d||_{L^{\infty}(Q_{r})}\leq CC_{0}r^{\beta+1},\quad r\in(0,1),$

where $C$ depends only on $n,$ and $\beta$.

###### Proof.

The proof is the same as the one of [1, Lemma 4.3], just that every $B_{r}$ is
replaced with $Q_{r}$, and polynomials are of bounded parabolic degree. ∎

######  Lemma B.5.

Let $\Omega\subset\mathbb{R}^{n+1}$, and assume that for every point
$z\in\partial\Omega\cap Q_{1}$ there exists a polynomial
$p_{z}\in\textbf{P}_{K}$, so that

$||u(x)-p_{z}(x)v(x)||_{L^{\infty}(B_{1}(z)\cap\Omega)}\leq C_{0},$

for some functions $u,v\in L^{\infty}(\Omega)$, with $C_{0}$ independent of
$z$. Assume also that $v\geq d$.

Then we have $||p_{z}||\leq CC_{0},$ for some $C$ independent of $z$.

###### Proof.

Find a non-empty, open set $B\subset\cap_{z\in\partial\Omega\cap
B_{1}}B_{1}(z),$ that is away from the boundary $B\subset\\{x;\text{
}d(x)>r\\}$, for some $r>0$. Then we can estimate

$\displaystyle||p_{z}||_{L^{\infty}(B)}\leq\frac{1}{r}||p_{z}v||_{L^{\infty}(B_{1}(z))}$
$\displaystyle\leq\frac{1}{r}\left(||u-p_{z}v||_{L^{\infty}(B_{1}(z))}+||u||_{L^{\infty}(B_{1}(z))}\right)$
$\displaystyle\leq C(C_{0}+||u||_{L^{\infty}(\Omega)}).$

The claim follows from [1, Lemma A.10]. ∎

######  Lemma B.6.

Let $D\subset\mathbb{R}^{n+1}$ be a bounded domain and let $v$ be a solution
to

$\left\\{\begin{array}[]{rcll}(\partial_{t}+L)v&=&P&\text{in }D\\\
v&=&0&\text{in }\partial_{p}D,\end{array}\right.$

For some polynomial $P$. Then

$||P||_{L^{\infty}(D)}\leq C||v||_{L^{\infty}(D)},$

where $C$ depends only on $n,D,$ ellipticity constants and the degree of $P$.

###### Proof.

If $P=0$ we have nothing to prove. Otherwise, dividing the equation with a
constant we can assume that $||P||_{L^{\infty}(D)}=1$. Assume by
contradiction, that there are sequences of $(\lambda,\Lambda)$-uniformly
elliptic operators $L_{k}$, polynomials $P_{k}$ of fixed degree, with
$||P_{k}||_{L^{\infty}(D)}=1$, and solutions $v_{k}$ of

$\left\\{\begin{array}[]{rcll}(\partial_{t}+L_{k})v_{k}&=&P_{k}&\text{in }D\\\
v_{k}&=&0&\text{in }\partial_{p}D,\end{array}\right.$

and that

$\frac{1}{k}>||v_{k}||_{L^{\infty}(D)}.$

Since $P_{k}$ are bounded in a finite dimensional vector space, we can get a
subsequence converging uniformly in $D$ to some polynomial $P_{0}$. Hence by
convergence result [2, Theorem 1.1] the limit function $v_{0}$ should satisfy

$\left\\{\begin{array}[]{rcll}(\partial_{t}+L_{0})v_{0}&=&P_{0}&\text{in }D\\\
v_{0}&=&0&\text{in }\partial_{p}D.\end{array}\right.$

But $v_{0}\equiv 0$, while $||P_{0}||_{L^{\infty}(D)}=1$, which gives a
contradiction. ∎

######  Lemma B.7.

Let $\beta>0$ $\beta\not\in\mathbb{N}$, $\Omega\subset\mathbb{R}^{n+1}$, with
$0\in\partial\Omega$, and let $u_{1},u_{2}\in C(\overline{Q}_{1}).$ Assume
that for every $r\in(0,1)$ we have a polynomial
$p_{r}\in\textbf{P}_{\lfloor\beta\rfloor,p}$ so that

$||u_{1}-p_{r}^{(0)}u_{2}-p_{r}^{(1)}d||_{L^{\infty}(Q_{r})}\leq
C_{0}r^{\beta+1}.$

Furthermore assume $c_{0}d\leq u_{2}\leq C_{0}d$, for some $c_{0}>0$. Then
there exists a polynomial $p_{0}\in\textbf{P}_{\lfloor\beta\rfloor,p}$ which
satisfies

$||u_{1}-p_{0}^{(0)}u_{2}-p_{0}^{(1)}d||_{L^{\infty}(Q_{r})}\leq
CC_{0}r^{\beta+1},\quad r\in(0,1),$

where $C$ depends only on $n,c_{0}$ and $\beta$.

###### Proof.

The proof is the same as the one of [1, Lemma 4.5], with $Q_{r}$ instead of
$B_{r}$. ∎

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