# Existence for the supercooled Stefan problem in general dimensions

Sunhi Choi, Inwon C. Kim, & Young-Heon KIm  Department of Mathematics
University of Arizona
Tucson, AZ, USA<EMAIL_ADDRESS>Department of Mathematics
University of California at Los Angeles
Los Angeles, CA, USA<EMAIL_ADDRESS>Department of Mathematics
University of British Columbia
Vancouver, V6T 1Z2 Canada<EMAIL_ADDRESS>
###### Abstract.

In this paper we prove global-time existence of weak solutions to the
supercooled Stefan problem in general space dimensions, starting from a
general class of initial domain and data. In addition, our solution is
maximal, in the sense that its support is maximal among all comparable weak
solutions starting from the same initial data.

Our approach is based on a free target optimization problem for Brownian
stopping times. While this approach is taken previously in [22],the novel
feature of our optimization problem here is that it creates particle dynamics
with a prescribed initial domain of general geometry. To achieve this, our
main idea is to introduce a superharmonic cost function in the optimization
problem, which encourages the target measure to accumulate near the prescribed
domain boundary as much as possible. A central ingredient in our proof lies in
the usage of dual problem: we prove the dual attainment and use the dual
optimal solution to characterize the primal optimal solution. It follows in
turn that the underlying particle dynamics yields a solution to the
supercooled Stefan problem.

IK is partially supported by NSF (DMS 2153254). YHK is partially supported by
the Natural Sciences and Engineering Research Council of Canada (NSERC), with
Discovery Grant RGPIN-2019-03926, as well as Exploration Grant
(NFRFE-2019-00944) from the New Frontiers in Research Fund (NFRF). YHK is also
a member of the Kantorovich Initiative (KI) that is supported by PIMS Research
Network (PRN) program. We thank PIMS for their generous support. Part of this
work is completed during YHK’s visit at Korea Advanced Institute of Science
and Technology (KAIST), and he thanks their hospitality and excellent
environment. ©2023 by the author. All rights reserved.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Assumptions
    2. 2.2 Randomized stopping times
    3. 2.3 Superharmonic functions.
    4. 2.4 Strictly subharmonic order
  3. 3 Primal problem
  4. 4 Duality
    1. 4.1 Proof of Theorem 4.2
  5. 5 Attainment of the dual problem
  6. 6 Saturation property for the primal optimal solution
  7. 7 Existence for the supercooled Stefan problem
    1. 7.1 Irregular evolution is generic for $(St)$
  8. 8 Examples of initial data for existence and non-existence results.

##  1\. Introduction

The main objective of this paper is to prove global existence results for the
supercooled Stefan problem $(St)$ in general dimensions, for a class of
general initial data.

The problem $(St)$ describes freezing of supercooled water into ice, and can
be written as

(1.1) $(\eta-\chi_{\\{\eta>0\\}})_{t}-\frac{1}{2}\Delta\eta=0\hbox{ in
}\mathbb{R}^{d}\times(0,\infty).$

Here $-\eta$ denotes the temperature of the supercooled water [28, 29]. The
term $\chi_{\\{\eta>0\\}}$ represents the unit amount of change in latent heat
when water turns into ice. Heuristically, the problem corresponds to an
interacting particle system where active particles generate Brownian motion in
the supercooled water region, stop at the boundary of the region and turn into
the ice. The dynamics of the free boundary $\partial\\{\eta>0\\}$ is the
central interest in this problem.

The problem is famously ill-posed with general initial data: most known
examples and results are in one space dimension. The available existence
results in multiple space dimensions are limited to either self-similar ones
[21] or the ones with specially chosen initial data by an elliptic variational
inequality [9, 22]. This is in contrast to the stable Stefan problem that
usually describes ice melting into water, which has been extensively studied
in the literature; see e.g. [14], [2], [6], [12] and the book [24].

In one space dimension, it is well-known that solutions of (1.1) develop
finite-time discontinuities of their free boundary [26, 20, 5, 3, 4], when the
initial data $\eta(x,0)$ is larger than $1$ near the initial free boundary
$\partial\\{\eta(\cdot,0)>0\\}$. Intuitively speaking, such discontinuity is
due to the excess of active particles present near the water-ice interface,
which generates infinite rate of freezing. The jump discontinuity of the free
boundaries is the main source of non-existence and non-uniqueness of weak
solutions in one space dimension (see for instance the discussion in [3, 8]).
On the other hand, still in one space dimension, for initial data that is
small and smooth in its support, it is known that the free boundary does not
jump: indeed there exists a unique smooth solution, see for instance [10, 11,
4]. See also the discussion of this dichotomy with the critical density of
size $1$ near the initial free boundary, in terms of one-dimensional particle
systems in [7].

These results motivate our investigation on the global existence of weak
solutions for $(St)$ in general space dimensions. Roughly speaking, our result
shows that such existence result holds if there is enough room for the initial
data to diffuse within its initial support, to form a unit-density ice at the
end of the freezing process. More precisely, we present a result that
addresses all initial data that are compactly supported and less than $1$, and
at the same time gives a general condition that allows initial data that are
larger than $1$ away from its support boundary.

To our best knowledge, ours is the first global-time existence result for
$(St)$ that starts from initial data with such generality, even in one space
dimension; in general dimensions, the initial domain is only required to be
bounded with locally Lipschitz boundary.

Before stating the main theorem, we first explain our condition:

######  Definition 1.1.

We say that $\eta_{0}\in L^{\infty}(\mathbb{R}^{d})$ satisfies the condition
$(C_{0})$ if the following holds:

  * $(C_{0})$:

There exists a constant $0<\delta<1$, such that

$\displaystyle\eta_{0}\hbox{ is subharmonically ordered with }\nu,\hbox{ for
some $\nu$ with }\nu\leq\delta\chi_{\\{\eta_{0}>0\\}}.$

Here the subharmonic order means that there is a Brownian motion starting from
the distribution $\eta_{0}$ and ending with $\nu$, with a strictly positive
stopping time; see Definition 2.7 in Section 2.

The condition $(C_{0})$ is natural in terms of the particle dynamics
associated with the freezing process in $(St)$, where the (negative) heat
particles follow the Brownian motion in the supercooled water region, hit the
ice front, and become frozen. In this context $\nu$ represents the final
distribution of frozen ice that are generated by those Brownian particles. The
inequality $\nu\leq\delta\chi_{\\{\eta_{0}>0\\}}$, expressing the upper bound
for the capacity of the latent heat in ice, is inspired by the critical
threshold $1$ for the constant density observed in the literature. It is worth
pointing out that any initial data which is equal or larger than $1$ near the
boundary of its positive set cannot satisfy $(C_{0})$, since then there is no
room for diffusing such a mass by the Brownian motion to another measure with
less than unit density supported in the same positive set. On the other hand,
we mention that

if $\eta_{0}\leq\delta$ for some constant $\delta<1$ then it automatically
satisfies $(C_{0})$.

As we will see later, the condition $(C_{0})$ presents a plethora of initial
data that are larger than $1$ in the interior of its positive set.

Now we are ready to state our main existence theorem.

######  Theorem A.

Let $\eta_{0}$ satisfy $(C_{0})$, with its positive set $\\{\eta_{0}>0\\}$
open, bounded, and with Lipschitz boundary. In addition suppose that
$|\\{\eta_{0}=1\\}|=0$.

Then there exists a weak solution $\eta\in
L^{1}(\mathbb{R}^{d}\times[0,\infty))$ to (1.1) with initial data $\eta_{0}$
that satisfies the following additional properties:

  * •

$\\{\eta>0\\}$ is open and its initial domain $\limsup_{t\to
0^{+}}\\{\eta(\cdot,t)>0\\}$ equals $\\{\eta_{0}>0\\}$, possibly minus a
measure zero set (see Section 7.1).

  * •

$\eta$ is maximal in the sense of Definition 1.2 below.

Our notion of maximality bears similarity to the ‘minimal jump’ solution
discussed in [8] for one space dimension:

######  Definition 1.2 (Maximal solution).

A weak solution (see Definition 7.1) $\eta$ to (1.1) is maximal if for any
other weak solution $\tilde{\eta}$ with the same initial data and initial
domain,

$\displaystyle\\{\eta>0\\}\subseteq\\{\tilde{\eta}>0\\}\hbox{ implies
}\eta=\tilde{\eta}.$

The proof of Theorem A will be presented in Section 7.

The additional condition $|\\{\eta_{0}=1\\}|=0$ in the above theorem, or the
constant $\delta<1$, is necessary for our approach (see Remark 7.4 and Remark
5.4), but it appears not essential to the existence result. Let us indeed
propose what we expect to be a sharp condition for the global existence
result:

$\displaystyle(C):\quad\quad\eta_{0}\hbox{ is strictly subharmonically ordered
with }\nu,\hbox{ for some $\nu$ with }\nu\leq\chi_{\\{\eta_{0}>0\\}}.$

It is indeed a necessary condition as a consequence of the analysis in [22],
see Theorem 7.2. Our conjecture is that the condition is also sufficient:

######  Conjecture.

There exists a global-time weak solution of $(St)$ with initial data
$\eta_{0}$ and initial support $\\{\eta_{0}>0\\}$ if and only if $\eta_{0}$
satisfies condition $(C)$.

While Theorem A doesn’t fully answer our conjecture, it almost does. Moreover
there is a broad class of initial data that meets the condition $(C_{0})$.
Remark 2.8 discusses a simple example of $\eta_{0}$ satisfying $(C_{0})$.
Furthermore in Section 8 we discuss concrete conditions for $\eta_{0}$ that
fails the sharp condition $(C)$, and conditions for $\eta_{0}$ that implies
$(C_{0})$. These conditions generate many nontrivial examples for the initial
data our result holds, beyond those that are smaller than $1$: see Corollarys
8.1 and 8.3 and remarks therein.

We expect that the evolution of the free boundary in Theorem A will generate
many intriguing irregularities that goes much beyond topological ones. Even
with smooth and small initial data $\eta_{0}$, the evolution of the positive
set $\\{\eta>0\\}$ appears to be irregular for most choices of the initial
domain $U:=\\{\eta_{0}>0\\}$, for space dimension larger than $1$. For
instance, one may expect that the set $\\{\eta(x,t)>0\\}$ continuously
decreases from $U$ initially to a compact subset of $U$ eventually.
Interestingly, however, one can check that this is not the case if $U$ has a
non-smooth ($C^{\infty}$) boundary point: see Section 7.1.

Let us briefly describe our analysis in the paper. We consider a novel
optimization problem (3.1), where we minimize the expected value of a given
superharmonic function $u$ against a measure supported inside a given domain
$U$. The admissible class of measures are those generated by the Brownian
stopping times from a given initial distribution $\eta_{0}$ supported in $U$.
The optimal measure $\nu^{*}$ corresponds to the final distribution of ice in
$(St)$ after the evolution of $\eta$ is complete, starting from the initial
data $\eta_{0}$ and the initial domain $U$.

Ours is an improved version of the optimization problem recently considered in
[22], where a limited global existence is obtained for the supercooled Stefan
problem. More precisely, the result in [22] requires the initial positive set
to be given by an elliptic variational inequality, while in the present paper
we incorporate arbitrary initial support $U$ in our problem. Importantly,
superharmonicity of the cost $u$ makes the Brownian particles to move as much
as possible, as the longer the stopping time, the smaller the corresponding
expected value of $u$; this principle is manifested in such a way that the
final optimal distribution $\nu^{*}$ is to be as close to the boundary of the
initial domain as possible. See Section 3, where we present the connection
between our optimization problem and the Stefan problem $(St)$, which leads to
the discussion of condition $(C)$.

The key step in our method is to consider the dual problem (4.1), for which we
prove no-duality-gap (Theorem 4.2) and dual attainment (Theorem 5.3). The dual
formulation is crucial in showing the saturation property, Theorem 6.5, of the
optimal measure, where the saturated region is determined by the dual optimal
solution in an explicit way. We then use the results of [22] to connect the
optimal measure with a solution to the Stefan problem to prove our main
theorems in Section 7.

There are important questions to be answered in future work. For example we
suspect that our class of weak solutions given in Theorem 7.3 feature
continuous evolution of the free boundary (see Remark 7.5), though this
remains to be confirmed. Moreover, we also conjecture that uniqueness holds
for maximal solutions given in Definition 1.2, for a given pair of initial
data and initial domain. With our approach, such uniqueness would be
equivalent to the statement that the optimization problem (3.1) yields the
same optimal solution independent of the choice of the cost function $u$; in
the present paper we only have uniqueness of the solution for a given
superharmonic cost $u$ (see Corollary 6.6).

Acknowledgements. We thank Mathav Murugan and Zhen-Qing Chen for helpful
discussions on probabilistic aspects.

##  2\. Preliminaries

###  2.1. Assumptions

Throughout the paper we use the following assumptions.

  * •

$U$ is a bounded open set in $\mathbb{R}^{d}$, with locally Lipschitz
boundary.

  * •

For each $r>0$, we denote $U_{r}:=\\{x\in U\ |\ d(x,\partial U)>r\\}$.

  * •

$u:\bar{U}\to\mathbb{R}_{\geq 0}$ is a bounded, continuous function on
$\bar{U}$ such that it is smooth in $U$ and

$\Delta u<0\hbox{ in }U.$

  * •

$f:\mathbb{R}^{d}\to\mathbb{R}_{>0}$ is a bounded, positive measurable
function.

  * •

$\mu$ has a bounded density supported in $\bar{U}$ and

$\mu[U]\leq\int_{U}fdx.$

  * •

The expression $\tau\oplus\sigma$ is the concatenation of the two stopping
times, namely, the Brownian particles first move until $\tau$, and then they
further move for the additional time $\sigma$.

The Lipschitz assumption on $\partial U$ is used to ensure that the stopping
times associated with $U$ are well approximated with those associated with
$U_{r}$: see Remark 2.5. While we expect that it can be generalized to a
broader class of domains, some very irregular domains may be problematic with
our choice of approximation, such as those with positive Lebesgue measure
boundaries.

###  2.2. Randomized stopping times

We now define a few standard notions for Brownian motion, borrowing the
notation from [22]. We assume the same assumptions as in [22] on the
probability space. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},\mathbb{P})$ be a filtered probability space that incorporates the brownian
motion $W_{t}:\Omega\to\mathbb{R}^{d}$.

A random variable $\tau:\Omega\to\mathbb{R}^{+}$ is called a stopping time if
$\tau^{-1}([0,t])\in\mathcal{F}_{t}$ for any $t\geq 0$. Stopping times can be
regarded as a rule that prescribes when each particle of the process stops
moving. An important example of stopping time, to be used frequently in this
paper, is the exit time associated with a domain. Let $W_{t}$ denote a
Brownian motion, and For a given domain $D$ in $\mathbb{R}^{d}$, let
$\tau^{D}$ denote the exit time for $D$, that is,

$\displaystyle\tau^{D}:=\inf\\{t>0\ |\ W_{t}\in D^{c}\\}$

(Here, we use the convention that $\tau^{D}=0$ for those Brownian paths
starting already from $D^{c}$.) Then this is a stopping time.

For our analysis we introduce a relaxed notion of the stopping time, called a
randomized stopping time, which assigns a probability distribution to each
particle instructing when to stop.

######  Definition 2.1.

Let $\mathcal{M}(\mathbb{R}^{+})$ be the collection of probability measures on
$\mathbb{R}^{+}$. A randomized stopping time $\tau$ is a map from $\Omega$ to
$\mathcal{M}(\mathbb{R}^{+})$ such that

$\hbox{ For each }t\geq 0,\hbox{the map }w\in\Omega\to\tau(w)([0,t])\hbox{ is
}\mathcal{F}_{t}-\hbox{measurable. }$

For a randomized stopping time $\tau$ and
$\varphi:\mathbb{R}^{+}\to\mathbb{R}$, we denote

$\mathbb{E}[\varphi(\tau)]=\mathbb{E}\left[\int_{\mathbb{R}^{+}}\varphi(t)\tau(dt)\right].$

We say that $\nu$ is the distribution of the stopped Brownian motion $W_{t}$
at the (randomized) stopping time $\tau$, that is $W_{\tau}\sim\nu$, if

$\mathbb{E}[g(W_{\tau})]=\int g(z)d\nu(z)\quad\hbox{ for any }g\in
C(\mathbb{R}^{d}).$

For more detailed discussions on randomized stopping time, we refer to [22].

###  2.3. Superharmonic functions.

In this section we will collect different notions of superharmonic functions
and their relations. Let us first begin with the standard notion.

######  Definition 2.2.

A lower semi-continuous function
$\phi:\mathcal{O}\to\mathbb{R}\cup\\{-\infty\\}$ is superharmonic if for every
$x\in\mathcal{O}$ and for every ball $B$ in $\mathcal{O}$ with center at $x$,
it satisfies

$\phi(x)\leq\frac{1}{|B|}\int_{B}\phi(y)dy.$

Now, we consider a maximum principle of superharmonic function via stopping
times; the use of the stopping time interpretation of superharmonicity will be
crucial in the dual formulation. The following is an application of a well-
known lemma (see. e.g. [19, Lemma 2.3 ]):

######  Lemma 2.3.

Let $\phi$ be a superharmonic function in $U$. For each $x\in\mathbb{R}^{d}$,
let $W^{x}_{t}$ denote a Brownian motion at time $t$ started from $x$ at
$t=0$. Then, for each $r>0$ and for a.e. $x\in U$, and for each (randomized)
stopping time $\tau$ that satisfies $\tau\leq\tau^{U_{r}}$,

(2.1) $\displaystyle\phi(x)\geq\mathbb{E}\left[\phi(W^{x}_{\tau})\right].$

###### Proof.

Observe that (2.1) is trivial for $x\in U\setminus U_{r}$ as $\tau^{D}=0$ by
our convention for those Brownian paths staring from such $x$. For $x\in
U_{r}$, we can apply [19, Lemma 2.3] as $\phi$ is in $LSC(\overline{U_{r}})$.
∎

Lemma 2.3 induces a simple but important result for us (in Sections 4 and 6)
where we crucially uses the condition $\nu\leq f$:

######  Lemma 2.4.

Let $\tau\leq\tau^{D}$ be a (randomized) stopping time with $W_{0}\sim\mu$ and
$W_{\tau}\leq\nu$ for some $\nu$. Assume that $\nu\leq f$. For each $r>0$,
define $\tau_{r}:=\tau\wedge\tau^{U_{r}}$ and let $\nu_{r}$ be the
distribution $W_{\tau_{r}}\sim\nu_{r}$ with $W_{0}\sim\mu$. Then,

(2.2) $\lim_{r\to 0}\nu_{r}(E)=\nu(E)\hbox{ for any measurable subset }E\hbox{
of }U.$

Moreover, if $\phi$ is superharmonic then

(2.3) $\displaystyle\int\phi(x)d\mu(x)\geq\int\phi(y)d\nu(y).$

###### Proof.

Observe that $\nu_{r}\leq\nu$ as measures on $U_{r}$, because by definition
$\tau_{r}=\tau\wedge\tau^{U_{r}}$ it follows that the only possibility for
$\nu_{r}$ is bigger than or equal to $\nu$ is along $\partial U_{r}$, where
$\nu_{r}$ has singular part. Since $\nu\leq f$ and $f$ is supported on $U$,
$\lim_{r\to 0}\nu(U_{r}^{c})=0$. We show that the singular part of $\nu_{r}$
converges to zero.

For sufficiently small $r>0$, the probability of the Brownian motion started
from a point $x\in\partial U_{r}$ to hit $\partial U_{\sqrt{r}}$ before it
hits $\partial U$ is bounded above by $C(r/\sqrt{r})^{\beta}$, for a
dimensional constant $\beta>0$ depending on the Lipschitz constant of $U$.
This implies

$\lim_{r\to 0}\nu_{r}(\partial U_{r})\leq\lim_{r\to
0}(\nu(U_{\sqrt{r}}^{c})+Cr^{\beta/2})=0.$

We thus conclude that $\nu_{r}$ converges to $\nu$ as a measure on $U$, namely
(2.2). In fact the density of the absolutely continuous part of $\nu_{r}$
monotonically converges to the density $\nu$.

Now note that (2.3) follows by applying (2.2) to Lemma 2.3 because

$\displaystyle\int\phi(x)d\mu(x)\geq\int\mathbb{E}\left[\phi(W^{x}_{\tau_{r}})\right]d\mu(x)=\int\phi(y)d\nu_{r}(y)\to\int\phi(y)d\nu(y)\quad\hbox{
as }r\to 0^{+}.$

(For the last limit one can use the monotone convergence theorem applied to
the absolutely continuous part of $\nu_{r}$.) This completes the proof. ∎

######  Remark 2.5.

Lemma 2.4 can be proved with more general domains $U$ beyond those with
locally Lipschitz boundaries, though we choose not to pursue this for
simplicity of presentation. In particular our main results in Sections 4 and 6
holds with domains $U$ for which the stopping times
$\tau^{*}_{r}:=\tau^{*}\wedge\tau^{U_{r}}$ converge to $\tau^{*}$ in terms of
their associated distributions when the target distribution has a bounded
density; this property is satisfied for instance, by $U$ whose boundary is
Lipschitz away from a finite number of inward or outward cusps.

Lemma 2.4 also holds when $U$ satisfies the exterior cone condition; for any
$x\in\partial U$, there is a cone $V\subset U^{c}$ with vertex at $x$,
constant opening and height, and axis $e(x)$ depending on $x$. More generally,
the lemma holds under a weaker condition that for any $x\in\partial U$ and
small $r>0$, there is a ball $B(y,s)\subset U^{c}\cap B(x,r)$ with $s\approx
r$. This condition allows Koch snowflakes as an initial domain $U$.

Furthermore in two dimension, Lemma 2.4 holds for any simply connected bounded
open set $U$; when $d=2$, Beurling’s projection theorem [16, Theorem 9.3]
implies that the probability of the Brownian motion started from a point
$x\in\partial U_{r}$ to hit $\partial U_{\sqrt{r}}$ before it hits $\partial
U$ is bounded above by $C(r)$ with $\lim_{r\to 0}C(r)=0$, and hence the proof
of the lemma works.

We now turn to another important definition relevant to the notion of
superharmonicity with stopping times. Given a measurable function
$\phi:\bar{U}\to\mathbb{R}$, we define its superharmonic envelope
$\phi^{sp}:\bar{U}\to\mathbb{R}$ as

(2.4)
$\displaystyle\phi^{sp}(x):=\sup_{\tau\leq\tau^{U}}\mathbb{E}\left[\phi(W^{x}_{\tau})\right].$

where $\tau$ is any (randomized) stopping time such that $\tau\leq\tau^{U}$;
see e.g. [19].

######  Lemma 2.6.

For $\phi\in C(\bar{U})$, the function $\phi^{sp}:U\to\mathbb{R}$ is lower
semi-continuous and satisfies (2.1).

###### Proof.

We first show that $\phi^{sp}$ is a measurable function on $U$. To see this
for each given $\delta>0$, use the measurable selection theorem to get a
measurable function $x\in U\to\tau^{\delta,x}$, from $U$ to the space of
randomized stopping times endowed with the $\hbox{weak}^{*}$-topology over the
space of paths, such that the function
$\psi_{\delta}(x):=\mathbb{E}\left[\phi(W^{x}_{\tau^{\delta,x}})\right]$
satisfies

$\psi_{\delta}(x)\geq\phi^{sp}(x)-\delta$ for each $x\in U$.

Since $\psi_{\delta}\leq\phi^{sp}$ from the definition of $\phi^{sp}$, we see
that as $\delta\to 0$, $\psi_{\delta}\to\phi^{sp}$ for each $x$. Therefore,
the measurability of $\psi_{\delta}$ will prove the measurability of
$\phi^{sp}$, as a pointwise limit of measurable functions. Now since $\phi\in
C(\bar{U})$, the function
$\tau\mapsto\mathbb{E}\left[\phi(W^{x}_{\tau})\right]$ is continuous function,
therefore the composition
$x\mapsto\tau^{\delta,x}\mapsto\mathbb{E}\left[\phi(W^{x}_{\tau^{\delta,x}})\right]$
is measurable. This proves the measurability of $\psi_{\delta}$.

Next, the superharmonic mean value property of $\phi^{sp}$ follows from the
dynamic programming principle. For each stopping time $\tau\leq\tau^{U}$, note
that
$\phi^{sp}(W^{x}_{\tau})=\sup_{\tau\oplus\sigma\leq\tau^{U}}\mathbb{E}\left[\phi(W^{x}_{\tau\oplus\sigma})\right]$
by the definition (2.4) of the superharmonic envelope. Then

$\displaystyle\mathbb{E}\left[\phi^{sp}(W^{x}_{\tau})\right]=\mathbb{E}\left[\sup_{\tau\oplus\sigma\leq\tau^{U}}\mathbb{E}\left[\phi(W^{x}_{\tau\oplus\sigma})\right]\right]\leq\mathbb{E}\left[\phi^{sp}(x)\right]=\phi^{sp}(x)$

it follows that $\phi^{sp}$ satisfies (2.1) and it is lower semi-continuous
(see [30, Proposition 2.8]). ∎

###  2.4. Strictly subharmonic order

Two measures $\mu,\nu$ on $\mathbb{R}^{d}$ with equal volume is said to be in
subharmonic order $\mu\leq_{SH}\nu$, if for each open set $\mathcal{O}$
containing the support of $\mu+\nu$,

$\int\varphi(x)d\mu(x)\leq\int\varphi(x)d\nu(x)\quad\hbox{ for each smooth
subharmonic function in }\mathcal{O}.$

It is known that for compactly supported $\mu$ and $\nu$, $\mu\leq_{SH}\nu$ if
and only if there is a randomized stopping time $\tau$ for the Brownian motion
such that

(2.5) $W_{0}\sim\mu\hbox{ and }W_{\tau}\sim\nu\hbox{ with
}\mathbb{E}(\tau)<\infty.$

Moreover, for many types of cost functions, for instance,
$\mathcal{C}(\tau)=\mathbb{E}(\tau^{2})$, there exists an optimal (randomized)
stopping time that satisfies (2.5) that minimizes the cost, see [1, 18, 19];
in fact, the optimal stopping time $\tau^{*}$ is a non-randomized stopping
time. In our problem we will first optimize the target measure in the primal
problem (3.1) for the given measure $\mu$, and then use the optimal stopping
time for the optimal target measure $\nu^{*}$ to establish the connection to
the Stefan problem.

For our analysis we need a stronger ordering that prevents any instant
stopping of the Brownian particles when following the optimal stopping time,
to ensure that the corresponding solution of the Stefan problem evolves with
the initial distribution $\mu$.

######  Definition 2.7.

A pair ($\mu$, $\nu$) is strictly subharmonically ordered if $\mu\leq_{SH}\nu$
and with its optimal stopping time $\tau^{*}$ strictly positive.

######  Remark 2.8.

We will make use of the fact that, due to [22, Theorem 8.3], any compactly
supported measure $\mu$ whose density is smaller than $1$ is strictly
subharmonically ordered to $\nu^{*}$, which optimizes our primal problem
(3.1). It is also not hard to construct $\mu$ whose density is larger than 1
away from the boundary of their support, which are still strictly
subharmonically ordered to $\nu$ for some $\nu<1$. For a simple example, given
$0<\delta<1$ and $M>1$, let $\mu=\delta\chi_{B(0,1)}+M\chi_{B(0,r)}$ for some
$r>0$. If $r$ is sufficiently small (i.e., $r\leq r_{0}(n,\delta,M)$), then
$\mu\leq_{SH}\nu:=\delta^{\prime}\chi_{B(0,1)}$ for some
$\delta<\delta^{\prime}<1$ and its optimal stopping time $\tau^{*}>0$. For
more general examples, see Corollary 8.3 and Remark 8.4.

##  3\. Primal problem

For $\mu$, $f$ and $u$ as given in Section 2, let us consider the following
optimization problem about stopping times $\tau$ and their target
distributions $W_{\tau}\sim\nu$:

(3.1) $\displaystyle P(\mu,f,u)=\inf\left\\{\int u(x)d\nu(x)\ \Big{|}\
\tau\leq\tau^{U},W_{\tau}\sim\nu\leq f\chi_{U}\ \&\ W_{0}\sim\mu\right\\}.$

The constraint $\nu\leq f\chi_{U}$ means that $\nu$ has a density which is
less than or equal to $f\chi_{U}$ a.e. It is easy to see that $P(\mu,f,u)$ has
optimal solutions.

######  Theorem 3.1 (Existence of a primal solution).

The problem $P(\mu,f,u)$ has an optimal solution if the eligible set of $\nu$
is non-empty.

###### Proof.

We consider the stopping times $\tau$, and its corresponding distributions
$\nu$, that are admissible for the problem $P(\mu,f,u)$. Let
$(\tau_{i},\nu_{i})$ be a minimizing sequence. We can view $\tau_{i}$ as
randomized stopping times (see e.g. [1] or [22]), which converge via the
weak∗-topology to a randomized stopping time $\tau_{\infty}$, with
$\tau_{\infty}$ and its distribution giving the pair
$(\tau_{\infty},\nu_{\infty})$. Notice that $\nu_{i}\to\nu_{\infty}$ in the
weak∗-topology, therefore we can keep $\nu_{\infty}\leq f\chi_{U}$ and also
that

$\displaystyle P(\mu,f,u)=\lim_{i\to\infty}\int ud\nu_{i}=\int
ud\nu_{\infty}.$

From the result of [1, 18], there is a corresponding (non-randomized) stopping
time $\tau^{*}$ satisfying the conditions of $P(\mu,f,u)$, therefore
$\nu_{\infty}$ is admissible. This completes the proof. ∎

######  Remark 3.2.

In Corollary 6.6, we prove the uniqueness of an optimal solution to the primal
problem $P(\mu,f,u)$, assuming

(3.2) $\mu$ is subharmonically ordered with some $\nu$ with $\nu\leq\delta
f\chi_{U}$ for some $\delta\in(0,1)$,

which is a slightly weaker condition than $(C_{0})$ in Definition 1.1, as
$\mu$ may not be positive everywhere in $U$. Note that (3.2) is automatically
satisfied if $\mu\leq\delta f$.

One of the motivations for (3.1) is that its optimal target measure will be
supported near the boundary of $U$ as much as possible, due to the super-
averaging property (2.1) of the superharmonic function $u$: the strict
superharmonicity $\Delta u<0$ makes the stopping time $\tau$ to be as large as
possible. To see this, observe that for $W_{0}\sim\mu$,
$W_{\tau_{i}}\sim\nu_{i}$, $i=1,2$, if $\tau_{1}\leq\tau_{2}$ then

$\displaystyle\int u(x)d\nu_{1}\geq\int u(x)d\nu_{2},\hbox{ and ``$>$'' if
$Prob[\tau_{1}<\tau_{2}]>0$}$

as we see from Ito’s formula

$\displaystyle\int u(x)d\nu_{i}=\int
u(x)d\mu(x)+\mathbb{E}\left[\int_{0}^{\tau_{i}}\frac{1}{2}\Delta
u(W_{t})dt\right].$

For the same reason, one can expect that the optimal strategy for the Brownian
particles is to move as much as possible so that the target measure is
saturated in its support: see Proposition 6.1 where this argument is presented
in the interior of the target measure support. A rigorous proof of saturation
will be given in Section 6 (Theorem 6.5), where we present a delicate argument
using the dual variable to avoid the difficulty arising from the unknown
topological nature of the support for the optimal target measure.

If the optimal stopping time were strictly positive, together with the
saturation property, we can relate $P(\mu,f,u)$ to the supercooled Stefan
problem, as we will see in Section 7. Due to the fact that the target measure
is supported up to the boundary of $U$, it will follow that the optimal
stopping time for $P(\mu,f,u)$ provides a global solution to $(St)$ with the
initial domain $U$, yielding our Theorem Theorem A. The strict positivity of
the stopping time can be ensured for instance if $\mu$ satisfies
$|\\{\mu=f\\}|=0$, due to Theorem 8.3 (a) in [22]. Later in the paper we will
impose the assumption (3.2) which is also used to obtain a technical
compactness estimate for the dual attainment (Theorem 5.3).

Of course to consider $P(\mu,f,u)$, we also need to ensure that the eligible
set of $\nu$ is non-empty. This is certainly the case with the condition (3.2)
since then $\mu$ is in the eligible set. Our conjecture says that in principle
our approach should hold under the assumption $(C)$ which says (a) the
admissible set of $\nu$ is non-empty and (b) when the optimal stopping time is
strictly positive.

##  4\. Duality

We consider the dual problem:

(4.1) $\displaystyle
D(\mu,f,u)=\sup_{(\phi,\,\psi)\in\mathcal{D}}\left[\int\phi(y)df(y)-\int\psi(x)d\mu(x)\right]$

where

(4.2) $\displaystyle\mathcal{D}:=\\{(\phi,\psi)\in L^{1}(\bar{U})\ |\ \phi\leq
0,\,\,\ \psi\geq\phi-u\hbox{ in $U$ and $\psi$ is superharmonic }\\}.$

We first establish the weak duality:

######  Theorem 4.1 (Weak duality).

For $\mu$, $f$, $u$ as given in section 2,

$\displaystyle D(\mu,f,u)\leq P(\mu,f,u).$

###### Proof.

Consider an admissible primal/dual pair $(\nu,(\phi,\psi))$ for $P(\mu,f,u)$
and $D(\mu,f,u)$. Namely, for a stopping time $\tau$
$\tau\leq\tau^{U},W_{\tau}\sim\nu\leq f\chi_{U}\ \&\ W_{0}\sim\mu$ and
$(\phi,\psi)\in\mathcal{D}$ as in (4.2). It suffices to show that

$\displaystyle\int\phi df-\int\psi d\mu\leq\int ud\nu.$

The subharmonic ordering (2.3) and the fact $\psi\geq\phi-u$ yield

(4.3) $\displaystyle\int\psi d\mu\geq\int\psi d\nu\geq\int(\phi-u)d\nu.$

Now let us consider the following inequalities:

$\displaystyle\int\phi df-\int\psi d\mu$ $\displaystyle\leq\int\phi
df-\int(\phi-u)d\nu\quad\hbox{ (due to \eqref{observation_1})}$
$\displaystyle=\int\phi(df-d\nu)+\int ud\nu$ $\displaystyle\leq\int
ud\nu\quad\hbox{( since $\phi\leq 0$, $\nu\leq f$ on $U$)}.$

Thus proves the weak duality. ∎

We in fact have a no-duality-gap result which is crucial for the present
paper. For its proof, we introduce a modified dual problem:

(4.4) $\displaystyle
D^{\prime}(\mu,f,u)=\sup_{(\phi,\,\psi)\in\mathcal{D}^{\prime}}\left[\int\phi(y)df(y)-\int\psi(x)d\mu(x)\right]$

where

$\displaystyle\mathcal{D}^{\prime}:=\\{(\phi,\psi)\in L^{1}(\bar{U})\ |\
\phi\leq 0\,\ \&\
\psi(x)\geq\mathbb{E}\left[(\phi-u)(W^{x}_{\tau})\right]\hbox{ for any
$\tau\leq\tau^{U}$}\\}.$

Then since $\mathcal{D}^{\prime}\subset\mathcal{D}$, we observe

(4.5) $\displaystyle D^{\prime}(\mu,f,u)\leq D(\mu,f,u).$

######  Theorem 4.2 (No Duality Gap).

For $\mu$, $f$, $u$ as given in section 2,

(4.6) $\displaystyle P(\mu,f,u)=D^{\prime}(\mu,f,u).$

and hence $P(\mu,f,u)=D^{\prime}(\mu,f,u)=D(\mu,f,u)$.

Note that the last equality of Theorem 4.2 follows from Theorem 4.1 with
(4.5), (4.6) since

$P(\mu,f,u)=D^{\prime}(\mu,f,u)\leq D(\mu,f,u)\leq P(\mu,f,u).$

Before proving this theorem in Section 4.1, let us give heuristics that
utilize the ‘min-max’ principle, where “$=$” is a nonrigorous part:

$\displaystyle P(\mu,f,u)$
$\displaystyle=\inf_{\tau\leq\tau^{U}\&\,W_{0}\sim\pi_{1}\&W_{\tau}\sim\pi_{2}}\left\\{\int
ud\pi_{2}+\sup_{\phi\leq
0,\,\psi}\left[\int\phi(df-d\pi_{2})+\int\psi(d\pi_{1}-d\mu)\right]\right\\}$
$\displaystyle\hbox{``}=\hbox{''}\sup_{\phi\leq
0,\,\psi}\left\\{\inf_{\tau\leq\tau^{U}\&\,W_{0}\sim\pi_{1}\&W_{\tau}\sim\pi_{2}}\left[\int
ud\pi_{2}+\int\phi(df-d\pi_{2})+\int\psi(d\pi_{1}-d\mu)\right]\right\\}.$

Then after rearranging, the last expression becomes

$\displaystyle=\sup_{\phi\leq
0,\,\psi}\left\\{\inf_{\tau\leq\tau^{U}\&\,W_{0}\sim\pi_{1}\&W_{\tau}\sim\pi_{2}}\left[\int\phi
df-\int\psi d\mu+\int(u-\phi)d\pi_{2}+\int\psi d\pi_{1}\right]\right\\}$
$\displaystyle=\sup_{\phi\leq 0,\,\psi}\left\\{\int\phi df-\int\psi
d\mu+\inf_{\tau\leq\tau^{U}\&\,W_{0}\sim\pi_{1}\&W_{\tau}\sim\pi_{2}}\mathbb{E}[u(W_{\tau})-\phi(W_{\tau})+\psi(W_{0})]\right\\}.$

Here the infimum in the last expression goes to $-\infty$ by choosing
$\pi_{1}$ and $\pi_{2}$ which are not necessarily probability measures, unless
$\psi(W_{0})\geq\phi(W_{\tau})-u(W_{\tau})\hbox{ a.s. for any
$\tau\leq\tau^{U}$}$. Therefore, the last expression is the same as

$\displaystyle\sup\left\\{\int\phi df-\int\psi d\mu\ \Big{|}\ \phi\leq 0\ \&\
\psi(W_{0})\geq\phi(W_{\tau})-u(W_{\tau})\hbox{ a.s. for any
$\tau\leq\tau^{U}$}\right\\}.$

This gives an informal proof that $P(\mu,f,u)=D^{\prime}(\mu,f,u)$. Below we
give a rigorous proof.

###  4.1. Proof of Theorem 4.2

Define

$\displaystyle\tilde{D}(\mu,f,u)=\sup_{\phi\in C(\bar{U})\ \&\ \phi\leq
0}\left[\int\phi(y)df(y)-\int(\phi-u)^{sp}(x)d\mu(x)\right].$

Note that obviously $\tilde{D}(\mu,f,u)\leq D^{\prime}(\mu,f,u)$ as the pair
$(\phi,(\phi-u)^{sp})$, with $0\geq\phi\in C(\bar{U})$, satisfies the
admissibility for $D^{\prime}(\mu,f,u)$. Therefore it suffices to prove
$\tilde{D}(\mu,f,u)=P(\mu,f,u)$ because of the weak duality (Theorem 4.1).

We apply Fenchel-Rockafellar duality (see [31, Theorem 1.9]) and Sion’s
minimax theorem, namely,

######  Theorem 4.3 (Fenchel-Rockafellar duality, Theorem 1.9 of [31]).

Let $E$ be a normed vector space, $E^{*}$ its topological dual space, and
$\Theta$, $\Xi$ two convex functions on $E$ with values in
$\mathbb{R}\cup\\{+\infty\\}$. Let $\Theta^{*}$, $\Xi^{*}$ be the Legendre-
Fenchel transforms of $\Theta$, $\Xi$ respectively. Assume that there exists
$z_{0}\in E$ such that

$\displaystyle\Theta(z_{0})<+\infty,\quad\Xi(z_{0})<+\infty,\hbox{$\Theta$ is
continuous at $z_{0}$.}$

Then,

(4.7) $\displaystyle\inf_{E}[\Theta+\Xi]=\max_{z^{*}\in
E^{*}}\left[-\Theta^{*}(-z^{*})-\Xi^{*}(z^{*})\right].$

(Here $\max$ on the righthand side is a part of the result.)

######  Theorem 4.4 (Sion’s minimax theorem [27]).

Let $X$ be a compact convex subset of a linear topological space, and $Y$ a
convex subset of a linear topological space. Let $f:X\times Y\to\mathbb{R}$ be
a function satisfying

  * (a)

$f(x,\cdot)$ is upper semicontinuous and quasi-concave on $Y$ for each $x\in
X$;

  * (b)

$f(\cdot,y)$ is lower semicontinuous and quasi-convex on $X$ for each $y\in
Y$. Then

$\sup_{y\in Y}\min_{x\in X}f(x,y)=\min_{x\in X}\sup_{y\in Y}f(x,y).$

The rest of the proof follows a similar strategy of the proof of Kantorovich
duality as in [31]. Here, our assumption that $U$ is bounded, thus its closure
$\bar{U}$ is compact, gives a simplification; it can be relaxed but for
simplicity of the paper we do not follow it.

Let $E=C(\bar{U})$ be the space of all continuous functions on $\bar{U}$,
equipped with the sup-norm $\|\cdot\|_{\infty}$. The topological dual of $E$
is the space of Radon measures on $\bar{U}$, that is, $E^{*}=M(\bar{U})$,
normed by the total variation. We now define

$\displaystyle\Theta:\phi\in
C(\bar{U})\mapsto-\int\phi(y)df(y)+\int(\phi-u)^{sp}(x)d\mu(x).$

and

$\displaystyle\Xi:\phi\in C(\bar{U})\mapsto\begin{cases}0&\text{if $\phi\leq
0$ },\\\ +\infty&\text{otherwise}.\end{cases}$

Let us verify the conditions of $\Theta$ and $\Xi$ for the Fenchel-Rockafellar
theorem. The function $\Xi$ is obviously convex. For $\Theta$, notice that the
function
$\phi\mapsto(\phi-u)^{sp}=\sup_{\tau\leq\tau^{U}}\mathbb{E}\left[(\phi-u)(W^{x}_{\tau})\right]$
is convex, which implies $\Theta$ is convex on $C(\bar{U})$. On the other
hand, consider $z_{0}\equiv 0$, then, $\Theta$ is continuous at $z_{0}$ (with
respect to sup-norm $\|\cdot\|_{\infty}$) and
$\Theta(z_{0}),\Xi(z_{0})<\infty$. Therefore, we can apply (4.7). Notice that

$\displaystyle\inf_{\psi\in C(\bar{U})}\left[\Theta(\psi)+\Xi(\psi)\right]$
$\displaystyle=-\left\\{\sup_{\phi\in C(\bar{U})\ \&\ \phi\leq
0}\left[\int\phi(y)df(y)-\int(\phi-u)^{sp}(x)d\mu(x)\right]\right\\}$
$\displaystyle=-\tilde{D}(\mu,f,u).$

Next, let us consider the Legendre-Fenchel transforms $\Theta^{*}$, $\Xi^{*}$
of $\Theta$ and $\Xi$, respectively.

For each $\nu\in M(\bar{U})$, we have

$\displaystyle\Xi^{*}(\nu)$ $\displaystyle=\sup_{\phi\in
C(\bar{U})}\left\\{\int\phi(x)d\nu(x)-\Xi(\phi)\right\\}$
$\displaystyle=\sup_{\phi\in C(\bar{U})\ \&\ \phi\leq
0}\left\\{\int\phi(x)d\nu(x)\right\\}$ $\displaystyle=\begin{cases}0&\text{if
$\nu\geq 0$},\\\ +\infty&\text{otherwise}.\end{cases}$

On the other hand,

$\displaystyle\Theta^{*}(-\nu)$ $\displaystyle=\sup_{\phi\in
C(\bar{U})}\left\\{-\int\phi(x)d\nu(x)-\Theta(\phi)\right\\}$
$\displaystyle=\sup_{\phi\in
C(\bar{U})}\left\\{-\int\phi(x)d\nu(x)+\int\phi(y)df(y)-\int(\phi-u)^{sp}(x)d\mu(x)\right\\}.$

Notice that

$\displaystyle\int(\phi-u)^{sp}(x)d\mu(x)=\sup_{\tau\leq\tau^{U}\ \&\
W_{0}\sim\mu\ \&\
W_{\tau}\sim\tilde{\nu}}\left[\int(\phi(y)-u(y))d\tilde{\nu}(y)\right].$

Therefore,

(4.8) $\displaystyle\Theta^{*}(-\nu)$ $\displaystyle=\sup_{\phi\in
C(\bar{U})}\left\\{\inf_{\tau\leq\tau^{U}\ \&\ W_{0}\sim\mu\ \&\
W_{\tau}\sim\tilde{\nu}}\left[-\int\phi(y)(d\nu+d\tilde{\nu}-df)+\int
u(y)d\tilde{\nu}(y)\right]\right\\}$

Here, the function
$(\phi,\tilde{\nu})\mapsto-\int\phi(y)(d\nu+d\tilde{\nu}-df)+\int
u(y)d\tilde{\nu}(y)$ is linear in $\phi$ and $\tilde{\nu}$, and the set
$\\{\tilde{\nu}\ |\ \tau\leq\tau^{U}\ \&\ W_{0}\sim\mu\ \&\
W_{\tau}\sim\tilde{\nu}\\}$ is compact with respect to the weak*-topology as a
closed set of probability measures on the compact set $\bar{U}$. (Notice that
the set of randomized stopping times, as probability measures, is endowed with
the weak*-topology, that is, the weak* topology is its original topology.)

Hence the infimum is attained at some $\tilde{\nu}$ for a given $\phi$, and we
can apply Theorem 4.4 to get

$\displaystyle\Theta^{*}(-\nu)$ $\displaystyle=\sup_{\phi\in
C(\bar{U})}\left\\{\inf_{\tau\leq\tau^{U}\ \&\ W_{0}\sim\mu\ \&\
W_{\tau}\sim\tilde{\nu}}\left[-\int\phi(y)(d\nu+d\tilde{\nu}-df)+\int
u(y)d\tilde{\nu}(y)\right]\right\\}$ $\displaystyle=\sup_{\phi\in
C(\bar{U})}\left\\{\min_{\tau\leq\tau^{U}\ \&\ W_{0}\sim\mu\ \&\
W_{\tau}\sim\tilde{\nu}}\left[-\int\phi(y)(d\nu+d\tilde{\nu}-df)+\int
u(y)d\tilde{\nu}(y)\right]\right\\}.$ $\displaystyle=\min_{\tau\leq\tau^{U}\
\&\ W_{0}\sim\mu\ \&\ W_{\tau}\sim\tilde{\nu}}\left\\{\sup_{\phi\in
C(\bar{U})}\left[-\int\phi(y)(d\nu+d\tilde{\nu}-df)+\int
u(y)d\tilde{\nu}(y)\right]\right\\}.$

Observe that $\int u(y)d\tilde{\nu}$ is bounded by
$\|u\|_{\infty}\tilde{\nu}(\bar{U})=\|u\|_{\infty}\mu(U)$. Therefore,

$\displaystyle\Theta^{*}(-\nu)\geq\min_{\tau\leq\tau^{U}\ \&\ W_{0}\sim\mu\
\&\ W_{\tau}\sim\tilde{\nu}}\left\\{\sup_{\phi\in
C(\bar{U})}\left[-\int\phi(y)(d\nu+d\tilde{\nu}-df)\right]\right\\}-\|u\|_{\infty}\mu(U).$

Given $\nu$, suppose that $\tilde{\nu}+\nu\neq f$ on a set with positive
measure. Then for some $\phi\in C(\bar{U})$ we have

$-\int\phi(y)(d\nu+d\tilde{\nu}-df)\not=0.$

Then, by multiplying $\phi$ by a constant $\lambda$ (with the opposite sign if
necessary), we can make

$-\int\lambda\phi(y)(d\nu+d\tilde{\nu}-df)\longrightarrow\infty\quad\hbox{ as
$|\lambda|\to\infty$.}$

Therefore, the above supremum in (4.8) becomes

$\displaystyle\Theta^{*}(-\nu)=\min_{\tau\leq\tau^{U}\ \&\,W_{0}\sim\mu\
\&\,W_{\tau}\sim\tilde{\nu}\hbox{ and }\tilde{\nu}+\nu=f}\int
u(y)d\tilde{\nu}(y)$

where it is understood that this minimum is $+\infty$ if there is no such
$\tilde{\nu}$ with the condition “$\tau\leq\tau^{U}\ \&\,W_{0}\sim\mu\
\&\,W_{\tau}\sim\tilde{\nu}\hbox{ and }\tilde{\nu}+\nu=f$.” Then together with
$\Xi^{*}$, this implies

$\displaystyle\max_{z^{*}\in
E^{*}}\left[-\Theta^{*}(-z^{*})-\Xi^{*}(z^{*})\right]$
$\displaystyle=-\min\left\\{\int u(y)d\tilde{\nu}(y);\hbox{ there exist
$\tilde{\nu}\leq f$, $\tau$ such that $\tau\leq\tau^{U}\ \&\ W_{0}\sim\mu\ \&\
W_{\tau}\sim\tilde{\nu}$}\right\\}$ $\displaystyle=-P(\mu,f,u).$

This completes the proof of Theorem 4.2.

##  5\. Attainment of the dual problem

In this section, we prove the dual attainment using the following lemma and
proposition for superharnomic functions.

######  Lemma 5.1 (Local Maximum Principle (c.f. Theorem 9.20 in [17])).

Let $U\subset\mathbb{R}^{d}$ be a bounded open set. Let $\psi:U\to\mathbb{R}$
satisfy $\Delta\psi\leq 0$ and $\psi\leq 0$. Then for any ball $B_{r}$ such
that $2B_{r}\subset U$ and for any $p>0$, there exists a constant $C=C(p,r,U)$
such that

$\displaystyle\sup_{B_{r}}|\psi|\leq
C\frac{1}{|2B_{r}|}\left(\int_{2B_{r}}|\psi|^{p}\right)^{1/p}$

###### Proof.

This is a special case of Theorem 9.20 of [17]. ∎

Lemma 5.1 also holds for LSC nonsmooth superharmonic functions due to the
standard mollifier process: We can approximate $\phi$ by smooth superharmonic
functions, using a convolution with a smooth mollifier. This lemma then
provides the following proposition.

######  Proposition 5.2.

Let $U\subset\mathbb{R}^{d}$ be a bounded open set. Let $\psi$ and $\psi_{j}$
be measurable LSC functions on $U$ such that

  * •

$\psi_{j}\leq 0$ in $U$;

  * •

$\psi_{j}$’s are superharmonic in $U$;

  * •

$\int_{U}|\psi_{i}|dx\leq C$ for some universal constant $C$ (independent of
$j$);

  * •

$\psi_{j}\to\psi$ as $j\to\infty,$ for a.e. $x\in U$.

Then, $\psi$ is superharmonic in $U$.

###### Proof.

From Lemma 5.1 applied with $p=1$ we have that in every compact ball in $U$
there is a uniform lower bound on $\psi_{j}$’s; the lower bound may depend on
the compact ball, but not on $j$. Therefore we can apply Fatou’s lemma to show
the desired property of Definiton 2.2 for $\psi$ for a.e. $x\in U$. ∎

We now prove the dual attainment, using the above results.

######  Theorem 5.3 (Dual attainment).

For $\mu$, $f$, $u$ given as in section 2, let $\mu$ satisfy (3.2). Then, the
dual problem $D(\mu,f,u)$ is attained with an optimal solution
$(\phi^{*},\psi^{*})\in\mathcal{D}$.

###### Proof.

We prove the theorem in five steps. Recall
$D(\mu,f,u)=D^{\prime}(\mu,f,u)=P(\mu,f,u)$.

Step 1.  Let $\\{(\phi_{i},\psi_{i})\\}\subset\mathcal{D}^{\prime}$ be a
maximizing sequence for $D^{\prime}(\mu,f,u)$, in particular, $\phi_{i}\leq
0$. Because $\tilde{D}(\mu,f,u)=D^{\prime}(\mu,f,u)$ from the proof of Theorem
4.1, we may assume that $\phi_{i}\in C(\bar{U})$ and
$\psi_{i}=(\phi_{i}-u)^{sp}$.

Now we increase $\phi_{i}\leq 0$ point-wise as much as possible without
changing $\psi_{i}$, which is favored for the maximization of the functional
$\phi\to\int\phi df-\int\psi d\mu$. Then since $\psi_{i}=(\phi_{i}-u)^{sp}$ is
the smallest superharmonic function $\geq\phi_{i}-u$, the largest $\phi_{i}$
(holding the same $\psi_{i}$) satisfies

  * (a)

$\phi_{i}(x)=0$ (where $0$ is the upper bound of $\phi_{i}$), if
$(\phi_{i}-u)^{sp}(x)>-u(x)$;

  * (b)

$(\phi_{i}-u)(x)=(\phi_{i}-u)^{sp}(x)$, otherwise.

Hence we may assume without loss of generality that

$\phi_{i}-u=\min[(\phi_{i}-u)^{sp},-u].$

From this we see that

$\displaystyle 0$ $\displaystyle\leq(\phi_{i}-u)^{sp}-(\phi_{i}-u)$
$\displaystyle=[(\phi_{i}-u)^{sp}-(-u)]\chi_{\\{(\phi_{i}-u)^{sp}>-u\\}}$
$\displaystyle\leq[(\phi_{i}-u)^{sp}-(-u)]\chi_{\\{\phi_{i}=0\\}}$
$\displaystyle\leq u\chi_{\\{\phi_{i}=0\\}}\quad$ $\displaystyle\leq u$

where the third inequality follows since $(\phi_{i}-u)^{sp}\leq 0$ (here
recall that $(\phi_{i}-u)^{sp}$ is the smallest superharmonic function above
$\phi_{i}-u$, while $\phi_{i}-u\leq 0$).

For simplicity of notation, we define

$\hat{\phi}_{i}:=(\phi_{i}-u)^{sp}-(\phi_{i}-u).$

Then from the above estimate, we have

$\displaystyle 0\leq\hat{\phi}_{i}\leq
u\qquad\hbox{and}\quad\int|\hat{\phi}_{i}|\leq C.$

Step 2.

We will show that

(5.1) $\displaystyle\int|\phi_{i}|dx\leq C^{\prime}\hbox{ for all large }i$

where $C^{\prime}$ is a constant independent of $i$ (but on $\delta$ among
others). To see this first note

$\int\phi_{i}df-\int(\phi_{i}-u)^{sp}d\mu\rightarrow
D^{\prime}(\mu,f,u)=D(\mu,f,u).$

Recall that by our assumption there is a measure $\nu$ with $\nu\leq\delta
f\chi_{U}$, $0<\delta<1$, and a stopping time $\tau$ with
$W_{0}\sim\mu,W_{\tau}\sim\nu$. Then consider

$\displaystyle\int\phi_{i}df-\int(\phi_{i}-u)^{sp}d\mu$ $\displaystyle=\int
udf+\int(\phi_{i}-u)df-\int(\phi_{i}-u)^{sp}d\mu$ $\displaystyle=\int
udf+\int(\phi_{i}-u)(df-d\nu)+\int(\phi_{i}-u)d\nu-\int(\phi_{i}-u)^{sp}d\mu$
$\displaystyle\leq\int
udf+\int(\phi_{i}-u)(df-d\nu)+\left[\int(\phi_{i}-u)^{sp}d\nu-\int(\phi_{i}-u)^{sp}d\mu\right]$
(since $(\phi_{i}-u)^{sp}\geq(\phi_{i}-u)$) $\displaystyle\leq\int
udf+\int(\phi_{i}-u)(df-d\nu)\hbox{ (from \eqref{eqn:maximum-good})}$
$\displaystyle=\int ud\nu+\int\phi_{i}(df-d\nu)$ $\displaystyle\leq\int
udf+\int\phi_{i}(df-d\nu).$

Since $\int udf$ and $D(\mu,f,u)$ are finite and $\int\phi_{i}(df-d\nu)\leq
0$, this verifies $\int|\phi_{i}|(df-d\nu)\leq C$ for a constant $C$
independent of $i$, and since $df-d\nu\geq(1-\delta)fdx$, this shows (5.1) as
desired.

Step 3. From the above $L^{1}$ bounds for $\phi_{i}$ and $\hat{\phi}_{i}$
obtained in steps 1 and 2, we can apply Komlós theorem to find a.e. convergent
sequences $\phi^{*}_{i}$ and $\hat{\phi^{*}}_{i}$, where $\phi^{*}_{i}$ and
$\hat{\phi^{*}}_{i}$ are obtained as the average of some subsequences
$\phi_{i_{j}}$ and $\hat{\phi}_{i_{j}}$ in such a way that
$\phi^{*}_{k}=\frac{1}{k}\sum_{j=1}^{k}\phi_{i_{j}}$ and
$\hat{\phi^{*}}_{k}=\frac{1}{k}\sum_{j=1}^{k}\hat{\phi}_{i_{j}}$. Let us
denote their limits as $\phi^{*}$ and $\hat{\phi^{*}}$, namely

$\phi^{*}_{i}\to\phi^{*},\quad\hat{\phi^{*}}_{i}\to\hat{\phi^{*}}\hbox{ a.e.}$

Observe that we can keep the same estimates

$\displaystyle 0\leq\hat{\phi^{*}}_{i}\leq u\hbox{ for all
$i$}\quad\&\quad\hbox{ $\int|\phi^{*}_{i}-u|\leq C^{\prime}$ for all large
$i$}.$

Step 4. We now show that $\hat{\phi^{*}}+(\phi^{*}-u)$ is superharmonic. and
$\hat{\phi^{*}}+(\phi^{*}-u)\geq(\phi^{*}-u)^{sp}$. Indeed notice that (a) and
(b) in step 1 imply

$\displaystyle\hat{\phi^{*}}_{k}+(\phi^{*}_{k}-u)=\frac{1}{k}\sum_{j=1}^{k}(\phi_{i_{j}}-u)^{sp},$

where the latter is superharmonic, and has uniformly bounded $L^{1}$ integral
on $U$, which is because $(\phi_{i}-u)^{sp}=\phi_{i}-u+\hat{\phi}_{i}$ and
both $\hat{\phi_{i}}$ and $\phi_{i}$ have uniform $L^{1}$ bounds. By Lemma
2.6, $(\phi_{i_{j}}-u)^{sp}$ is LSC for each $i$. Moreover,

$\displaystyle\hat{\phi^{*}}_{k}+(\phi^{*}_{k}-u)\rightarrow\hat{\phi^{*}}+(\phi^{*}-u).$

Therefore from Proposition 5.2, we see that
$\psi^{*}:=\hat{\phi^{*}}+(\phi^{*}-u)$ is superharmonic in the sense of
Definition 2.2. Since $\hat{\phi^{*}}\geq 0$, we see

$\psi^{*}\geq(\phi^{*}-u).$

Also, $\phi^{*}\leq 0$ as the limit of $\phi^{*}_{k}\leq 0$, therefore, the
pair
$(\phi^{*},\psi^{*})=(\phi^{*},\hat{\phi^{*}}+(\phi^{*}-u))\in\mathcal{D}$.

Step 5. Now observe

$\displaystyle D(\mu,f,u)$
$\displaystyle=\lim_{k}\left[\int({\phi^{*}}_{k}-u)(df-d\mu)-\int{\hat{\phi^{*}}_{k}}d\mu+\int
udf.\right]$
$\displaystyle=\lim_{k}\left[\int\phi^{*}_{k}df-\int(\hat{\phi^{*}}_{k}+\phi^{*}_{k}-u)d\mu\right]$
$\displaystyle\leq\int\phi^{*}df-\int(\hat{\phi^{*}}+\phi^{*}-u)d\mu$

where the inequality follows by Fatou’s lemma for the first integral, and the
dominated convergence theorem for the second integral (note that
$|{\hat{\phi}}_{i}+\phi_{i}-u|=|(\phi_{i}-u)^{sp}|\leq|(-u)^{sp}|$ and hence
$|\hat{\phi^{*}}_{k}+\phi^{*}_{k}-u|\leq|(-u)^{sp}|$).

On the other hand, from Step 4, the pair
$(\phi^{*},\hat{\phi^{*}}+(\phi^{*}-u))\in\mathcal{D}$, so admissible for
$D(\mu,f,u)$, therefore, the above line is less or equal than $D(\mu,f,u).$ ∎

######  Remark 5.4.

The proof of dual attainment is the only place where we need $\delta$ in
$(C_{0})$ to be strictly less than $1$.

##  6\. Saturation property for the primal optimal solution

We prove in this section that the optimal target measure $\nu^{*}$ of
$P(\mu,f,u)$ saturates the density constraint wherever it is positive. To
motivate the statement of the main result of this section (Theorem 6.5) and
also to demonstrate why the optimality of $\nu^{*}$ is related to its
saturation, we first give the following (easy) proposition before proving the
main theorem; this proposition will not be used anywhere else in the paper.

######  Proposition 6.1.

Let $\nu^{*}$ be an optimal solution to $P(\mu,f,u)$. Then, for each
$\epsilon>0$ there is no open set $V\subset\mathbb{R}^{d}$ such that

$\displaystyle 0<\nu^{*}(x)<f(x)-\epsilon\quad\hbox{ for a.e. $x\in V$.}$

###### Proof.

Let $\tau^{*}$ denote the optimal stopping time (via [1, 18]) with
$W_{\tau^{*}}\sim\nu^{*}$ and $W_{0}\sim\mu$. Suppose by contradiction that
there is $\epsilon>0$ and an open set $V$ with $0<\nu^{*}(x)<f(x)-\epsilon$
for a.e. $x\in V$. Pick $x\in V$, and consider small $r>0$ such that
$B_{2r}(x)\subset V$. Then, there exists a stopping time $\tau$ with initial
distribution $\chi_{B_{r}(x)}$ and final distribution $\delta\chi_{B_{2r}(x)}$
for some dimensional constant $\delta>0$. Then because of the assumption on
$V$, there exists $\delta_{1}>0$, depending on $\epsilon$ and $r$, such that
the restriction of $\tau$, say, $\tau_{1}$, to the initial distribution
$\delta_{1}\nu^{*}|_{B_{r}(x)}$ has final distribution, say, $\nu_{1}$ on
$B_{2r}(x)$, with $\nu_{1}\leq(f-\nu^{*})|_{B_{2r}(x)}$. We extend the
stopping time $\tau_{1}$ to be $0$ for the initial distribution
$\nu^{*}-\delta_{1}\nu^{*}|_{B_{r}(x)}$. In other words,

$\displaystyle\tau_{1}:=\begin{cases}\tau_{1}&\text{for Brownian motion
starting from the distribution $\delta_{1}\nu^{*}|_{B_{r}(x)}$ },\\\
0&\text{otherwise, i.e. for Brownian motion starting from the distribution
$\nu^{*}-\delta_{1}\nu^{*}|_{B_{r}(x)}$}.\end{cases}$

Let $\nu_{1}^{*}$ denote the corresponding final distribution
$W_{\tau_{1}}\sim\nu_{1}^{*}$ with $W_{0}\sim\nu^{*}$ . Because of the
construction we have $\nu^{*}_{1}\leq f$. Notice that for $\nu^{*}_{1}$, there
is a corresponding stopping time from the initial distribution $\mu$, obtained
by concatenating the stopping time $\tau^{*}$ for $\nu^{*}$ and $\tau_{1}$. So
$\nu^{*}_{1}$ is admissible for the problem $P(\mu,f,u)$ (see (3.1)). Also,
notice that $Prob[\tau^{*}_{1}>0]>0$. Since $u$ is strictly superharmonic
($\Delta u<0$), this then implies

$\displaystyle\int ud\nu^{*}_{1}<\int ud\nu^{*},$

which contradicts with the optimality of $\nu^{*}$ for $P(\mu,f,u)$. This
completes the proof. ∎

######  Remark 6.2.

Proposition 6.1 gives a version of the saturation result on open sets. It
therefore would give the main result below (Theorem 6.5), assuming $f$ is
continuous, if the positive set of $\nu^{*}$ were open a.e., which is not
known to us. We expect it to be false given that, formally, our target measure
$\nu^{*}$ describes the shape of ice crystals generated by supercooling
process.

Before we present the main theorem of this section, using the following
remarkable result proven in [13]:

######  Lemma 6.3 (Theorem 16 of [13]).

Let $U\subset\mathbb{R}^{d}$ be open and $\psi\in L^{1}_{loc}(U)$ be strictly
subharmonic in the sense that $\int_{E}\Delta\psi>0$ for any measurable
$E\subset U$ with $|E|>0$. Then $|\\{\psi=r\\}|=0$ for any $r\in\mathbb{R}$.
(Here $\int_{E}\Delta\psi$ should be understood as $\mu_{\psi}(E)$ for the
Borel measure $\mu_{\psi}$ which is generated by the nonnegative distribution
$\Delta\psi$.)

From this we immediately deduce the following proposition that we need:

######  Proposition 6.4.

Let $u_{1},u_{2}:U\to\mathbb{R}$ be two functions such that $u_{1}$ is
superharmonic as in (2.1) with $u_{1}\in L^{1}_{loc}(U)$ and that $u_{2}\in
C^{2}(U)$ and $\Delta u_{2}>0$ in $U$. Then,

$\displaystyle|\\{x\in U\ |\ u_{1}(x)=u_{2}(x)\\}|=0.$

###### Proof.

Note that $u_{2}-u_{1}$ satisfies the condition in Lemma 6.3. Therefore the
result follows. ∎

We now prove the main theorem of this section, whose proof crucially uses the
dual attainment (Theorem 5.3) as well as Proposition 6.4.

######  Theorem 6.5 (Saturation).

Let $\mu$ be as given in Theorem 5.3. Let $\nu^{*}$ be an optimal measure for
$P(\mu,f,u)$ and $(\phi^{*},\psi^{*})$ an optimal solution to $D(\mu,f,u)$.
Then

$\nu^{*}=f\chi_{\\{\phi^{*}<0\\}}.$

###### Proof.

Consider an optimal primal/dual pair $(\nu^{*},(\phi^{*},\psi^{*}))$ for
$P(\mu,f,u)$ and $D(\mu,f,u)$ from Theorem 5.3, respectively; Recall from
Theorem 5.3 that $\psi^{*}$ is chosen to satisfy Definition 2.2 so satisfies
(2.1) with $\psi^{*}\geq\phi^{*}-u$. Let $\tau^{*}$ be the optimal stopping
time with $W_{0}\sim\mu$ and $W_{\tau^{*}}\sim\nu^{*}$. Then, proceeding
exactly as in the proof of Theorem 4.1 using Lemma 2.3 and Lemma 2.4, we see
that

(6.1) $\int\psi^{*}d\mu\geq\int\psi^{*}d\nu^{*}\geq\int(\phi^{*}-u)d\nu^{*}.$

From Theorem 4.2 we then see that (as in the proof of Theorem 4.1)

$\displaystyle\int ud\nu^{*}=P(\mu,f,u)$
$\displaystyle=D(\mu,f,u)=\int\phi^{*}df-\int\psi^{*}d\mu$
$\displaystyle\leq\int\phi^{*}df-\int(\phi^{*}-u)d\nu^{*}$
$\displaystyle=\int\phi^{*}(df-d\nu^{*})+\int ud\nu^{*}$
$\displaystyle\leq\int ud\nu^{*}\hbox{ (since $\phi^{*}\leq 0$ and
$\nu^{*}\leq f$)}.$

It follows that we have

  * $(i)$

$\int\psi^{*}d\mu=\int(\phi^{*}-u)d\nu^{*}$ and

  * $(ii)$

$\int\phi^{*}(df-d\nu^{*})=0$.

From $(ii)$ and the fact that $\phi^{*}\leq 0$ and $\nu^{*}\leq f$, the
following is immediate:

$\nu^{*}=f\hbox{ a.e. wherever }\phi^{*}<0.$

Next we claim that $\nu^{*}=0$ wherever $\phi^{*}=0$, namely

(6.2) $\nu^{*}[\\{\phi^{*}=0\\}]=0,$

which will yield the desired saturation property,
$\nu^{*}=f\chi_{\\{\phi^{*}<0\\}}$.

The rest of the proof presents the proof of (6.2). First from $(i)$ and (6.1)
and the fact that $\psi^{*}\geq\phi^{*}-u$, it follows

(6.3) $\displaystyle\nu^{*}[\\{x\ |\ \psi^{*}(x)>(\phi^{*}-u)(x)\\}]=0.$

To show (6.2) using (6.3), it is sufficient to prove:

$\displaystyle|\\{\phi^{*}=0\\}\setminus\\{\psi^{*}>(\phi^{*}-u)\\}|=0.$

Since $\psi^{*}\geq\phi^{*}-u$, the above set is the same as
$\\{\phi=0\\}\cap\\{\psi^{*}=\phi^{*}-u\\}$. Notice that

$\displaystyle\\{\phi^{*}=0\ \&\
\psi^{*}=(\phi^{*}-u)\\}=\\{\psi^{*}=(\phi^{*}-u)=-u\\}.$

Now, $\psi^{*}$ is superharmonic as Definition 2.2 and $-u$ is smooth and
strictly subharmonic, therefore by Proposition 6.4, the set
$\\{\psi^{*}=-u\\}$ has zero Lebesgue measure. This completes the proof of
(6.2) and we conclude. ∎

The above saturation result gives a characterization of the optimal target
$\nu^{*}$ as well as its uniqueness.

######  Corollary 6.6 (Uniqueness with fixed $u$).

For $\mu$ as given in Theorem 5.3, the primal problem $P(\mu,f,u)$ has a
unique optimal solution.

###### Proof.

This follows from Theorem 6.5 because $\nu^{*}=f\chi_{\\{\phi^{*}<0\\}}$ is
completely determined by any optimal dual solution $(\phi^{*},\psi^{*})$. This
also shows that the set $\\{\phi^{*}<0\\}$, equivalently $\\{\phi^{*}=0\\}$,
is determined $f$-a.e. by the primal solution $\nu^{*}$. ∎

It is not clear whether the optimal $\nu^{*}$ can even be determined
independent of the choice of $u$. Also, at this moment it is not clear whether
the dual problem $D(\mu,f,u)$ has unique optimal solution, though all its
optimal solutions have the same zero set (of $\phi^{*}$, up to $f$-measure
zero set).

##  7\. Existence for the supercooled Stefan problem

In this section we focus on the case $f\equiv 1$ to prove our main theorem,
Theorem A, for the supercooled Stefan problem. The results in the previous
sections together with [22] will give us the existence of a weak solution of
$(St)$, with the initial data $\eta_{0}$ and initial domain
$\\{\eta_{0}>0\\}$.

Let us recall the definition of a weak solution to the supercooled Stefan
problems with initial density $\eta_{0}\in L^{1}(\mathbb{R}^{d})$ and initial
domain $E\subset\mathbb{R}^{d}$:

$None$
$(\eta-\chi_{\\{\eta>0\\}})_{t}-\frac{1}{2}\Delta\eta=0,\qquad\eta(\cdot,0)=\eta_{0}\in
L^{1}(\mathbb{R}^{d}),\quad E:=\limsup_{t\to 0^{+}}\\{\eta(\cdot,t)>0\\}.$

######  Definition 7.1 (See e.g. [22]).

A nonnegative function $\eta\in L^{1}(\mathbb{R}^{d}\times[0,\infty))$ is a
weak solution of $(St)$ with initial data $(\eta_{0},E)$ with $\eta_{0}\in
L^{1}(\mathbb{R}^{d})$ and $\\{\eta_{0}>0\\}\subset E$ if

  * (a)

the set $\\{\eta(\cdot,t)>0\\}$ decreases in $t$;

  * (b)

$E=\limsup_{t\to 0^{+}}\\{\eta(\cdot,t)>0\\}$ ;

  * (c)

for any test function $\varphi\in
C^{\infty}_{c}(\mathbb{R}^{d}\times[0,\infty))$,

(7.1)
$\int_{0}^{\infty}\int_{\mathbb{R}^{d}}[(\eta-\chi_{\\{\eta>0\\}})\varphi_{t}+\frac{1}{2}\eta\Delta\varphi]dxdt=\int_{\mathbb{R}^{d}}[(-\eta_{0}+\chi_{E})\varphi](x,0)dx.$

First we point out that the condition $(C)$ given in Section 1 is necessary in
the existence of $(St)$ (as in Definition 7.1) with initial domain
$E=\\{\eta_{0}>0\\}$.

######  Theorem 7.2.

Suppose $\eta$ is a weak solution of $(St)$, Then $\eta_{0}$ satisfies the
condition (C), namely, it is strictly subharmonically ordered to
$\nu:=\chi_{A}$, where $A$ is a subset of $E$.

###### Proof.

From [22, Theorem 5.6] it follows that $\eta_{0}$ is subharmonically ordered
to $\nu\chi_{\\{s(x)<\infty\\}}$ for a function $s(x)$. Moreover the weak
solution $\eta(\cdot,t)$ represents the active distribution of heat particles
at time $t$. One can thus deduce that there is no instantly stopped particles,
namely $\tau>0$ a.e., since one can easily check from the weak solution
definition that $\lim_{t\to 0}\eta(\cdot,t)=\eta_{0}$ a.e. ∎

We now prove Theorem A, which is restated here in the context of previous
sections.

######  Theorem 7.3 (Global existence).

For $\mu$ and $U$ as given in section 2, let us denote $\eta_{0}$ as the
density of $\mu$. Suppose that $\\{\eta_{0}>0\\}=U$. Further suppose that
$\eta_{0}$ satisfies $(C_{0})$ in Definition 1.1 with $|\\{\eta_{0}=1\\}|=0$.

Then there exists a weak solution $\eta$ to $(St)$ with the initial data
$\eta_{0}$ and the initial domain $E=U$, corresponding to the Eulerian
variable associated to the optimal solution $\nu^{*}$ to $P(\mu,f=1,u)$.
Moreover, $\eta$ is maximal in the sense of Definition 1.2.

###### Proof.

In the following proof we use the results of [18, 22]. Let $u$ be a
superharmonic function and let $\nu^{*}$ be the optimal solution (target
distribution) to $P(\mu,f=1,u)$ as given in Theorem 3.1. Then we have
$\nu^{*}=\chi_{\\{\phi^{*}<0\\}}$ from Theorem 6.5 for a dual optimal solution
$(\phi^{*},\psi^{*})\in\mathcal{D}$ given as in Theorem 5.3; note that $\mu$
satisfies (3.2) as it satisfies the stronger condition $(C_{0})$. Let
$\tau^{*}$ be the optimal stopping time for $\nu^{*}$ obtained via the optimal
Skorokhod problem for a type (I) cost as in [18]. Note that the value of
$P(\mu,f=1,u)$ depends only on the optimal target distribution $\nu^{*}$: thus
it follows that the pair $(\tau^{*},\nu^{*})$ is an optimal solution of
$P(\mu,f=1,u)$.

Since $|\\{\eta_{0}=1\\}|=0$ by our assumption, and since the density of
$\nu^{*}$ is a characteristic function, $\tau^{*}>0$ almost surely. This can
be seen from [22, Section 8]. Moreover from [22, Section 5] we have the
corresponding Eulerian flow $(\eta,\rho)$ as given in [18] (or see [22,
Section 2]) yielding a solution to the weighted supercooled problem, namely,

(7.2)
$(\eta-\nu^{*}\chi_{\\{\eta>0\\}})_{t}-\frac{1}{2}\Delta\eta=0,\qquad\eta(\cdot,0)=\eta_{0}\in
L^{1}(\mathbb{R}^{d}),\quad E:=\limsup_{t\to 0^{+}}\\{\eta(\cdot,t)>0\\}.$

We mention that $E=U$ a.e. follows from the fact $\tau^{*}>0$ and the
assumption that $\eta_{0}>0$ on $U$. It is important to note that the equation
(7.2) depends only on the value of $\nu^{*}$ on the set $\\{x\ |\
(x,t)\in\partial\\{\eta>0\\}\\}$, where the free boundary goes through for
some time, on the support of the final distribution $\nu^{*}$ of $\tau^{*}$;
see [22, Remark 5.2]. Because $\nu^{*}=\chi_{\\{\phi^{*}<0\\}}$, this implies
that the equation (7.2) is reduced to $(St)$, proving that the Eulerian
variable for $\tau^{*}$ gives the solution to $(St)$.

Now let us consider the maximality (see Definition 1.2) of our solution
$\eta$. It can be proven again by using the results of [18, 22]: Suppose there
is a weak solution $\tilde{\eta}$ with the same initial data
$(\tilde{\eta}_{0},E)=(\mu,U)$ such that
$\\{\eta>0\\}\subseteq\\{\tilde{\eta}>0\\}.$ Our goal is to show that
$\tilde{\eta}=\eta$. Following [22, Theorem 5.6], let us define $\tilde{s}$
and $\tilde{\rho}$ by

$\displaystyle\tilde{s}(x):=\sup\\{t:\tilde{\eta}(x,t)>0\\}\hbox{ and }$
$\displaystyle\int_{0}^{\infty}\int_{\mathbb{R}^{d}}\tilde{\rho}(x)\varphi(x,t)\,dxdt=\int\tilde{\nu}(x)\varphi(x,\tilde{s}(x))dx$

for any test function $\varphi\in
C^{\infty}_{c}(\mathbb{R}^{d}\times[0,\infty))$; here $\tilde{\nu}$ is the
measure corresponding to $\tilde{\eta}$ as in Theorem 7.2.

Then [22, Theorem 5.6] implies $\mu\leq_{SH}\tilde{\nu}$ for
$\tilde{\nu}:=\chi_{\\{\tilde{s}(x)<\infty\\}\cap U}$, and
$(\tilde{\eta},\tilde{\rho})$ is the Eulerian variables between
$\tilde{\eta}_{0}=\mu$ and $\tilde{\nu}$, generated by the optimal stopping
time, say $\tilde{\tau}$, of the optimal Skorokhod problem in the sense of
[18] for costs of type (I).

In particular, from [18], we have that $\tilde{\tau}$ is the hitting time to
the barrier set $\\{(x,t)\ |\ t\geq\tilde{s}(x)\\}$ which is equal to the
compliment of $\\{\tilde{\eta}>0\\}$. Notice that the same relation applies to
the stopping time $\tau^{*}$ (corresponding to $\eta$) and the set
$\\{\eta>0\\}$. Therefore, the condition
$\\{\eta>0\\}\subseteq\\{\tilde{\eta}>0\\}$ implies that

$\displaystyle\tau^{*}\leq\tilde{\tau}.$

And from the strict superharmonicity $\Delta u<0$ in $U$, we have that

if $Prob[\tau^{*}<\tilde{\tau}]>0$ then $\int ud\nu^{*}>\int ud\tilde{\nu}$.

On the other hand, $\tilde{\nu}\leq 1$ and also $\tilde{\tau}\leq\tau^{U}$. So
$\tilde{\nu}$ is admissible for $P(\mu,f=1,u)$. Now, due to the optimality of
$\nu^{*}$ we have $\int ud\nu^{*}\leq\int ud\tilde{\nu}$. Therefore, we have
$Prob[\tau^{*}<\tilde{\tau}]=0$, that is, $\tau^{*}=\tilde{\tau}$ almost
surely. This implies that their corresponding Eulerian variables coincide, in
particular, $\eta=\tilde{\eta}$, and we conclude. ∎

######  Remark 7.4.

Note that in the proof of Theorem 7.3, The assumption $|\\{\eta_{0}=1\\}|=0$
is used to ensure that the associated optimal stopping time is strictly
positive. This is necessary to ensure that the active particle density $\eta$
that is obtained from $P(\mu,f=1,u)$, which is our solution of $(St)$, assumes
its initial data as $\eta_{0}$.

######  Remark 7.5.

Note that our definition requires $\eta$ to solve the heat equation in the
interior of its positive set in $\mathbb{R}^{d}\times[0,\infty)$. Thus,
formally if the set $\\{\eta>0\\}$ discontinuously shrink at time $t=t_{0}$,
then $\eta(\cdot,t_{0}^{-})\leq\eta(\cdot,0)<1$. On the other hand
$\\{\eta(\cdot,t_{0}^{-})>0\\}\setminus\\{\eta(\cdot,t_{0}^{+})>0\\}$ is a
subset of $\\{s(x)=t_{0}\\}$, where
$s(x):=\inf\\{t:(x,t)\in\\{\eta>0\\}^{c}\\}.$ Thus there
$\eta(\cdot,t_{0}^{-})$ equals $\nu=1$. Hence we conclude that the set
$\\{\eta(\cdot,t)>0\\}$ cannot decrease discontinuously. Of course this
heuristic argument does not carefully consider the potential irregularity of
$\partial\\{\eta(\cdot,t)>0\\}$. The precise statement of the “no-jump”
property remains to be established.

###  7.1. Irregular evolution is generic for $(St)$

The free boundary for solutions of $(St)$ can be highly irregular in its
dynamic evolution, if the initial free boundary $\partial\\{\eta_{0}>0\\}$ is
less than $C^{\infty}$. More precisely, any presence of a non-smooth point on
the initial free boundary may cause either global-time waiting time phenomena
or instant nucleation of additional boundary points in its evolution. We
explain it below:

In [22, Theorem 4.15] it is shown that the optimal stopping time $\tau$
between a given initial and final measures $\mu$ and $\nu$, is determined by
its potential flow $U_{t}$ (with $\Delta U_{t}=\mu_{t}$ and $\Delta
U_{\infty}=\nu$, for $W_{t\wedge\tau}\sim\mu_{t}$). This implies that if
$\eta$ is the Eulerian variable ([22, Definition 4.20]) for the initial
distribution $\mu$ and target measure $\nu$, then the active region
$\\{x:\eta(x,t)>0\hbox{ for some }t>0\\}$ is given by $\\{w>0\\}$, where $w$
solves

$\Delta w=\nu-\mu.$

If the associated optimal stopping time $\tau$ is positive, then both $\nu$
and $\mu$ are supported in $\\{w>0\\}$, and thus one can write above equation
as

$\Delta w=(\nu-\mu)\chi_{\\{w>0\\}}.$

In our problem, $U=\\{\eta_{0}>0\\}$ equals the active region of $\eta$, which
is $\\{w>0\\}$. Moreover we show that the optimal target measure $\nu=\nu^{*}$
is a characteristic function $\chi_{\Sigma}$,where $\Sigma$ is the trace of
the free boundary in space, namely $\Sigma=\\{x:\eta(x,t)=0\hbox{ for some
}0<t<\infty\\}\cap U$. Thus if we suppose

(7.3) $\partial\Sigma\cap B_{r}(x_{0})=\partial U\hbox{ for some }r>0\hbox{
and }x_{0}\in\partial U,$

then we conclude that $w$ solves

$\Delta w=(1-\eta_{0})\chi_{\\{w>0\\}}\hbox{ in }B_{r}(x_{0}),$

where now $\eta_{0}$ denotes its density. Suppose that $\eta_{0}$ is in
$C^{k}(\bar{U})$ for some $k\in\mathbb{N}$ and strictly less than $1$. Then
this problem is now the classical obstacle problem with smooth data. Hence the
local regularity theory available for this problem conditions the geometry of
$\partial\\{w>0\\}=\partial U$ near $x_{0}$. Since $U$ is a domain with
Lipschitz boundary, it follows from classical literature on the obstacle
problem (for instance [23]) that $\partial\\{w>0\\}$ is $C^{k+1}$ in
$B_{r}(x_{0})$. So we conclude that, for instance, (7.3) cannot be true for
any boundary points of $U$ that does not satisfy this regularity. For the
simple case where $\eta_{0}$ is a constant in $U$, then $U$ must have
$C^{\infty}$ boundary to satisfy (7.3), otherwise we expect instant cracks
appear on the free boundary, when the initial flux $|D\eta_{0}|$ at $x_{0}$ is
nonzero. To give more explicit examples, let us assume that $U$ has a corner
at $x_{0}$ and $\eta_{0}$ is harmonic in $U$ with zero boundary conditions.
When the corner at $x_{0}$ points outward (sharp corner), then the initial
flux of $\eta$ is zero at $x_{0}$ and we expect $\\{\eta(\cdot,t)>0\\}$ to
still contain $x_{0}$ for all times. When the corner at $x_{0}$ points toward
inward (obtuse corner), then the initial flux of $\eta$ is infinite at $x_{0}$
and we expect an instant slit to form on the boundary of
$\\{\eta(\cdot,t)>0\\}$ propagating from $x_{0}$. In two space dimensions,
this slit blocks Brownian particles, and they stop before hitting it.
Therefore, the initial domain may not be the actual region for the Brownian
particle dynamics modulo a measure zero set. Similar phenomena may occur near
non-smooth boundary point, in space dimension $d\geq 3$, instantly generating
a $d-1$ dimensional slit inside the original domain, that blocks Brownian
motion to pass.

##  8\. Examples of initial data for existence and non-existence results.

This section discusses examples of initial data $\eta_{0}$ to which our theory
applies. More precisely we provide concrete examples of $\eta_{0}$ which
either fails $(C)$ or satisfies $(C_{0})$ but is larger than $1$ in some
parts. Due to our results in the previous section, these examples yield in
turn the existence and non-existence results for $(St)$. The examples we
provide are given in terms of elliptic obstacle problems associated to
$\eta_{0}$ (see also Remark 7.1).

First we present a concrete condition on initial data $\eta_{0}$, in terms of
an associated elliptic obstacle problem, so that it fails condition $(C)$ in
the introduction. Theorem 7.2 then yields a nonexistence result.

######  Corollary 8.1 (Nonexistence).

Given $E\subset\mathbb{R}^{d}$ and $\eta_{0}\in C(\mathbb{R}^{d})$ supported
in $E$, consider the solution $w$ of the obstacle problem for a given measure
$\mu$:

$\displaystyle\Delta w=(1-\eta_{0})\chi_{\\{w>0\\}},\quad w\geq 0.$

Suppose that

(8.1) $\emptyset\neq\\{w>0\\}\not\subseteq E.$

Then $\eta_{0}$ fails $(C)$, and there is no weak solution to $(St)$ with the
initial data $(\eta_{0},E)$.

###### Proof.

We show that the inclusion $\\{w>0\\}\subseteq E$ is necessary for the
existence of weak solutions. Suppose $(St)$ has a weak solution $\eta$ in this
setting. Theorem 7.2 implies that, condition $(C)$ holds for $\eta_{0}$, and
thus there is a target measure $\nu=\chi_{A}$ with a stopping time $\tau$ with
$W_{0}\sim\eta_{0}$ and $W_{\tau}\sim\nu$. Now use [22, Corollary 8.5] to see
that

$\bar{\nu}:=\chi_{\\{w>0\\}}+\eta_{0}\chi_{\\{w=0\\}}$

is the unique solution for the optimization problem considered in [22, Theorem
7.4] with the choice $\mu=\eta_{0}$ and the upper bound constraint $f=1$
there. In particular $\bar{\nu}\leq 1$. Let $\bar{\tau}$ be the optimal
stopping time (in the sense of [22, Theorems 7.4]) between $\eta_{0}$ and
$\bar{\nu}$; from [22, Theorems 8.3] we see that the active region for the
associated active particle density $\bar{\eta}$, namely the set
$\\{\bar{\eta}>0\\}$, satisfies $\lim_{\\{t\to
0^{+}\\}}\\{\bar{\eta}(\cdot,t)>0\\}=\\{w>0\\}$. Since
$\eta_{0}\chi_{\\{w>0\\}}\leq\eta_{0}$, the monotonicity result [22, Theorem
7.1(1)] gives that

$\bar{\tau}\leq\tau\hbox{ almost surely.}$

Thus it follows that $\\{\bar{\eta}>0\\}\subset\\{\eta>0\\}$. By sending $t\to
0^{+}$, it follows that

$\\{w>0\\}\subseteq E.$

This completes the proof. ∎

######  Remark 8.2.

Note that if $\eta_{0}\leq 1$, then $w\equiv 0$ and thus (8.1) fails. On the
other hand it is easy to see from the elliptic equation that $w$ should be
differentiable and in particular has its slope zero when it hits zero value.
Hence if $\eta_{0}$ is large on most of $E$, $w$ is very concave in most of
$E$. For instance in the radial case, one can imagine that $w$ has large
downward slope, growing linearly away from the center point of $E$. From this
and the fact that $w$ is $C^{2}$, it is easy to check that $\\{w>0\\}$ needs
to be larger than $E$.

We next provide conditions under which $\eta_{0}$ satisfies condition
$(C_{0})$ with some $\eta$, based on the results from [22]. Existence results
follow due to Theorem A.

######  Corollary 8.3 (Existence).

Given $\eta_{0}\in L^{\infty}(\mathbb{R}^{d})$, assume that the set
$\\{\eta_{0}>0\\}$ is a bounded open set with locally Lipschitz boundary with
the property $|\\{\eta_{0}=1\\}|=0$. Suppose further that there exists
$0<\delta<1$ such that for the solution $w$ of the obstacle problem

$\displaystyle\Delta w=(\delta-\eta_{0})\,\chi_{\\{w>0\\}},\quad w\geq 0,$

satisfies

$\displaystyle\\{w>0\\}\subseteq\\{\eta_{0}>0\\}.$

Then, there is a weak solution to $(St)$ with the initial data $\eta_{0}$ and
initial domain $\\{\eta_{0}>0\\}$ .

###### Proof.

The proof is in the same spirit of the one for Corollary 8.1. Use [22,
Corollary 8.5] to see that

$\eta_{0}\leq_{SH}\bar{\nu}:=\delta\chi_{\\{w>0\\}}+\eta_{0}\chi_{\\{w=0\\}}$

where $\bar{\nu}$ is the unique optimal solution in the sense of [22, Theorem
7.4] with the choice $\mu=\eta_{0}$ and $f=\delta$ there; so
$\bar{\nu}\leq\delta$. Moreover, from the condition
$\\{w>0\\}\subseteq\\{\eta_{0}>0\\}$ it holds that
$\\{\bar{\nu}>0\\}=\\{\eta_{0}>0\\}$. We thus can apply Theorem A as
$\eta_{0}$ and $\eta_{0}=\bar{\nu}$ satisfy the condition $(\mathcal{C}_{0})$.
This completes the proof. ∎

######  Remark 8.4.

Notice that the condition of Corollary 8.3 can easily be satisfied by many
$\eta_{0}$ that takes values larger than $1$, for instance when $\eta_{0}$ is
large in the center of its support, far away from its boundary.

## References

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