assumptionAssumption remarkRemark AlgorithmAlgorithm Energy Space Newton
Differentiability for Obstacle Problems Constantin Christof and Gerd Wachsmuth

# Energy Space Newton Differentiability for Solution Maps of Unilateral and
Bilateral Obstacle Problems

Constantin Christof Technische Universität München, Faculty of Mathematics,
M17, 85748 Garching bei München, Germany,
https://www-m17.ma.tum.de/Lehrstuhl/ConstantinChristof<EMAIL_ADDRESS>Gerd Wachsmuth Brandenburgische Technische Universität Cottbus–Senftenberg,
Institute of Mathematics, 03046 Cottbus, Germany, https://www.b-tu.de/fg-
optimale-steuerung, , ORCID: 0000-0002-3098-1503<EMAIL_ADDRESS>
###### Abstract

We prove that the solution operator of the classical unilateral obstacle
problem on a nonempty open bounded set $\Omega\subset\mathbb{R}^{d}$,
$d\in\mathbb{N}$, is Newton differentiable as a function from $L^{p}(\Omega)$
to $H_{0}^{1}(\Omega)$ whenever $\max(1,2d/(d+2))<p\leq\infty$. By exploiting
this Newton differentiability property, results on angled subspaces in
$H^{-1}(\Omega)$, and a formula for orthogonal projections onto direct sums,
we further show that the solution map of the classical bilateral obstacle
problem is Newton differentiable as a function from $L^{p}(\Omega)$ to
$H_{0}^{1}(\Omega)\cap L^{q}(\Omega)$ whenever $\max(1,d/2)<p\leq\infty$ and
$1\leq q<\infty$. For both the unilateral and the bilateral case, we provide
explicit formulas for the Newton derivative. As a concrete application example
for our results, we consider the numerical solution of an optimal control
problem with $H_{0}^{1}(\Omega)$-controls and box-constraints by means of a
semismooth Newton method.

###### keywords:

obstacle problem, bilateral constraints, variational inequality,
semismoothness, Newton differentiability, optimal control, control
constraints, nonsmoothness, Newton method

35J86, 49J52, 49K20, 46G05, 49M15

##  1 Introduction and summary of results

This paper is concerned with Newton differentiability properties of solution
maps $S\colon H^{-1}(\Omega)\to H_{0}^{1}(\Omega)$, $u\mapsto y$, of obstacle-
type variational inequalities (VIs) of the form

(VI) $y\in K,\qquad\left\langle-\Delta
y-u,v-y\right\rangle_{H_{0}^{1}(\Omega)}\geq 0\qquad\forall v\in K.$

Here, $\Omega\subset\mathbb{R}^{d}$, $d\in\mathbb{N}$, is a nonempty open
bounded set; $H_{0}^{1}(\Omega)$, $H^{-1}(\Omega)$,
$\langle\cdot,\cdot\rangle_{H_{0}^{1}(\Omega)}$, and $-\Delta$ are defined as
usual (see Section 2); and $K$ is a nonempty set of the type

(1) $\displaystyle K:=\\{v\in H_{0}^{1}(\Omega)\mid\psi\leq v\leq\phi\text{
a.e.\ in }\Omega\\}$

involving measurable functions $\psi,\phi\colon\Omega\to[-\infty,\infty]$. The
main contributions of this work, that can be seen as a continuation of our
previous study [11], are as follows: First, we prove that the Newton
differentiability result in [11, Corollary 3.9], which yields that the
solution map $S\colon u\mapsto y$ of (VI) is Newton differentiable as a
function from $L^{p}(\Omega)$ to $L^{q}(\Omega)$ for certain $p$ and $q$ when
$\phi\equiv\infty$ holds, can be improved to obtain that $S$ is even Newton
differentiable as a function from $L^{p}(\Omega)$ to $H_{0}^{1}(\Omega)$
whenever $\max(1,2d/(d+2))<p\leq\infty$ and $\phi\equiv\infty$. Second, we
show that this energy space Newton differentiability result for the unilateral
case makes it possible to prove that the solution map of the bilateral
obstacle problem, i.e., the problem (VI) with $\psi\not\equiv-\infty$ and
$\phi\not\equiv\infty$, is Newton differentiable as a function from
$L^{p}(\Omega)$ to $H_{0}^{1}(\Omega)\cap L^{q}(\Omega)$ whenever
$\max(1,d/2)<p\leq\infty$ and $1\leq q<\infty$. For the precise statements of
these two main results (involving all assumptions and also the definitions of
the respective Newton derivatives), we refer the reader to Sections 4 and 6.

Before we begin with our analysis, let us briefly put our work into
perspective. For the sake of brevity, we focus on infinite-dimensional
problems. For additional information on the finite-dimensional setting, we
refer the reader to [14, 19, 22, 23, 25] and the references therein.

In optimization and optimal control, the concept of Newton differentiability
(a.k.a. semismoothness) plays a crucial role because it characterizes
nonsmooth functions that are well behaved to such a degree that classical
algorithms for the numerical solution of smooth minimization problems and
operator equations become applicable. See, e.g., [2, 6, 17, 22, 32, 36] for
various contributions on this topic. Most prominently, in the presence of
Newton differentiability, it is possible to generalize the classical Newton
algorithm to the so-called _semismooth Newton method_ which, under suitable
invertibility assumptions on the involved generalized derivatives, makes it
possible to solve nonsmooth operator equations with superlinear convergence
speed. In practice, the main difficulty in applying such a semismooth Newton
method is proving that the nonsmooth operators under consideration indeed
possess a Newton derivative. In fact, establishing Newton differentiability
results for infinite-dimensional functions is so hard that there are only a
handful of results available on this topic. The one that is most commonly used
in applications is the fact that nonsmooth superposition operators $F\colon
L^{p}(\Omega)\to L^{q}(\Omega)$, $v\mapsto f(v)$, that are induced by a
globally Lipschitz continuous and Newton differentiable function
$f\colon\mathbb{R}\to\mathbb{R}$, are Newton differentiable when $p$ and $q$
satisfy $1\leq q<p\leq\infty$ (a so-called norm gap) and the Newton derivative
of $F$ is constructed from that of $f$ by means of a pointwise-a.e. selection.
We refer to, e.g., [12, 18, 30, 31, 36], where this Newton differentiability
result is used for the numerical solution of optimal control problems with
$L^{2}$-box-constraints, and also the recent [5]. For functions that are not
of superposition type (or, more precisely, for which superposition operators
are not the sole source of nonsmoothness), Newton differentiability results
are very scarce. To the best of our knowledge, there are only two papers
containing genuine contributions on this topic. The first one, [4],
establishes the Newton differentiability of the solution mapping of a
prototypical rate-independent evolution variational inequality—the so-called
scalar stop operator—by means of a semi-explicit solution formula; see [4,
Theorem 7.15]. The second one is our previous contribution [11], which
demonstrates by means of pointwise convexity properties that solution maps of
unilateral obstacle problems are Newton differentiable as functions between
suitable Lebesgue spaces when the strong-weak Bouligand differential is used
as a generalized derivative; see [11, Theorem 2.12, Corollary 3.9]. Compare
also again with [5] in this context. Note that all of these existing
contributions have in common that they rely on pointwise properties of the
considered nonsmooth operators—either arising from a superposition structure
or a certain curvature behavior—and thus only yield Newton differentiability
results in Lebesgue spaces.

In the present paper, we demonstrate how this limitation can be overcome. More
precisely, we show how the fact that the set $K$ in (1) is polyhedric (in the
sense of [15]) can be exploited to prove that the solution operator of the
unilateral obstacle problem is Newton differentiable as a function with values
in the energy space $H_{0}^{1}(\Omega)$; see Theorem 4.4. We moreover
demonstrate that this refined Newton differentiability result for the
unilateral case and properties of projections onto angled subspaces in
$H^{-1}(\Omega)$ make it possible to establish the Newton differentiability of
the solution map of the bilateral obstacle problem as a function from
$L^{p}(\Omega)$ to $H_{0}^{1}(\Omega)\cap L^{q}(\Omega)$ for all
$\max(1,d/2)<p\leq\infty$ and $1\leq q<\infty$; see Theorem 6.32. We would
like to point out that the latter of these two main results is completely
beyond the scope of the aforementioned pointwise approaches for proving Newton
differentiability properties as the solution map of the bilateral obstacle
problem neither has a superposition structure nor possesses pointwise-a.e.
curvature properties that could be used to resort to the scalar situation. At
least to the best of our knowledge, Theorem 4.4, Theorem 6.32, and the
arguments that we employ for the proofs of these main results are new and have
no analogue in the literature. We remark that we restrict our attention to the
classical obstacle problem (VI) in our analysis mainly for simplicity and to
avoid obscuring the basic ideas of our approach with technicalities. We expect
that the techniques that we develop in the following can be extended rather
straightforwardly to, e.g., Signorini-type VIs and thin-obstacle problems.
Compare also with the general functional-analytic setting considered in [11]
in this regard.

We conclude this introduction with an overview of the content and the
structure of the remainder of the paper. Sections 2 and 3 are concerned with
preliminaries. Here, we fix the notation, introduce basic concepts, and
collect known results on the obstacle problem (VI) that are needed for our
analysis. In Section 4, we prove the first main result of this work—the
$H_{0}^{1}(\Omega)$-Newton differentiability of the solution map $S\colon
u\mapsto y$ of (VI) in the unilateral case $\phi\equiv\infty$; see Theorem
4.4. Section 5 contains several auxiliary results on orthogonal projections
and angled subspaces that are needed for the analysis of the bilateral case
$\psi\not\equiv-\infty$ and $\phi\not\equiv\infty$. The results in this
section may also be of independent interest; see, e.g., the formula for
orthogonal projections onto direct sums in Theorem 5.6. Section 6 is concerned
with the proof of our second main result—the Newton differentiability of the
solution map of the bilateral obstacle problem as a function from
$L^{p}(\Omega)$ to $H_{0}^{1}(\Omega)\cap L^{q}(\Omega)$ for all
$\max(1,d/2)<p\leq\infty$ and $1\leq q<\infty$; see Theorem 6.32. In Section
7, we conclude the paper with an explicit application example. Here, we show
that Theorem 6.32 makes it possible to prove that optimal control problems
with $H_{0}^{1}(\Omega)$-controls and box-constraints can be solved by a
semismooth Newton method in function space with superlinear convergence speed.
The results on this topic are also accompanied by numerical experiments.

##  2 Notation and basic concepts

Throughout this paper, we use the standard symbols $\|\cdot\|_{X}$,
$\langle\cdot,\cdot\rangle_{X}$, and $(\cdot,\cdot)_{X}$ to denote norms, dual
pairings, and inner products defined on a (real) vector space $X$,
respectively. The space of linear and continuous functions between two normed
spaces $(X,\|\cdot\|_{X})$ and $(Y,\|\cdot\|_{Y})$ is denoted by
$\mathscr{L}(X,Y)$. For the topological dual space, we write
$X^{\star}:=\mathscr{L}(X,\mathbb{R})$, and for the orthogonal complement of a
closed subspace $W$ in a Hilbert space $H$, we write $W^{\perp}$. The kernel
and the range of a function $G\in\mathscr{L}(X,Y)$ are denoted by
$\mathrm{ker}(G)$ and $\mathrm{range}(G)$, respectively. The identity map is
denoted by $\mathrm{Id}$. If a space $(X,\|\cdot\|_{X})$ is continuously
embedded into a space $(Y,\|\cdot\|_{Y})$, then we write $X\hookrightarrow Y$.
If the embedding is compact, then this is denoted by
$X\mathrel{\mathop{\hookrightarrow}\limits^{\vbox
to0.0pt{\kern-2.0pt\hbox{$\scriptstyle\textup{c}$}\vss}}}Y$. Weak and strong
convergences are denoted by $\rightharpoonup$ and $\to$, respectively. Given a
sequence $\\{G_{n}\\}\subset\mathscr{L}(X,Y)$, we say that $G_{n}$ converges
in the weak operator topology (WOT) to $G\in\mathscr{L}(X,Y)$, in symbols
$G_{n}\smash{\overset{\textsc{wot}}{\to}}G$, if $G_{n}h\rightharpoonup Gh$
holds in $Y$ for all $h\in X$.

A mapping $F$ from $X$ to the power set of $Y$ is denoted by $F\colon
X\rightrightarrows Y$. Recall that a function $F\colon X\to Y$ between two
normed spaces $(X,\|\cdot\|_{X})$ and $(Y,\|\cdot\|_{Y})$ is called
directionally differentiable if, for all $x,h\in X$, there exists
$F^{\prime}(x;h)\in Y$ (the directional derivative) such that

$\displaystyle\lim_{0<t\to 0}\frac{F(x+th)-F(x)}{t}=F^{\prime}(x;h).$

If $F^{\prime}(x;\cdot)$ is an element of $\mathscr{L}(X,Y)$, then $F$ is
called Gâteaux differentiable at $x$,
$F^{\prime}(x):=F^{\prime}(x;\cdot)\in\mathscr{L}(X,Y)$ is called the Gâteaux
derivative of $F$ at $x$, and $x$ is called a Gâteaux point of $F$. The set of
all Gâteaux points of a function $F\colon X\to Y$ is denoted by
$\mathcal{D}_{F}\subset X$. A directionally differentiable function $F\colon
X\to Y$ satisfying

$\\{h_{n}\\}\subset X,\\{t_{n}\\}\subset(0,\infty),h_{n}\to h,t_{n}\to
0\quad\Rightarrow\quad\frac{F(x+t_{n}h_{n})-F(x)}{t_{n}}\to F^{\prime}(x;h)$

for all $x,h\in X$ is called Hadamard directionally differentiable. Following
[8, 28], we define the strong-weak Bouligand differential
$\partial_{B}^{sw}F\colon X\rightrightarrows\mathscr{L}(X,Y)$ of a function
$F\colon X\to Y$ by

$\displaystyle\partial_{B}^{sw}F(x):=\big{\\{}G\in\mathscr{L}(X,Y)\nonscript\;\big{|}\nonscript\;\mathopen{}\exists\\{x_{n}\\}\subset\mathcal{D}_{F}\colon
x_{n}\to x\text{ in
}X,~{}F^{\prime}(x_{n})\smash{\overset{\textsc{wot}}{\to}}G\text{ in
}\mathscr{L}(X,Y)\big{\\}}.$

A function $F\colon X\to Y$ equipped with a mapping $DF\colon
X\rightrightarrows\mathscr{L}(X,Y)$ satisfying $DF(x)\neq\emptyset$ for all
$x\in X$ is called Newton differentiable if

(2) $\sup_{G\in DF(x+h)}\frac{\|F(x+h)-F(x)-Gh\|_{Y}}{\|h\|_{X}}\to
0\quad\text{ for }0<\|h\|_{X}\to 0$

for all $x\in X$; cf. [6, 17, 36]. In this case, $DF$ is called a Newton
derivative of $F$.

Given a closed convex nonempty subset $L$ of a space $(X,\|\cdot\|_{X})$ and
$x\in L$, we denote by $\mathcal{R}(x;L):=\mathbb{R}_{+}(L-x)$ the radial cone
of $L$ at $x$ and by $\mathcal{T}(x;L):=\mathrm{cl}(\mathcal{R}(x;L))$ the
tangent cone of $L$ at $x$; cf. [3, section 2.2.4]. Here, $\mathrm{cl}(\cdot)$
denotes a closure in $X$.

Given a nonempty open bounded set $\Omega\subset\mathbb{R}^{d}$,
$d\in\mathbb{N}$, we denote by $L^{p}(\Omega)$, $H^{k}(\Omega)$, and
$W^{k,p}(\Omega)$, $k\in\mathbb{N}$, $p\in[1,\infty]$, the standard real
Lebesgue and Sobolev spaces on $\Omega$, respectively. The norms
$\|\cdot\|_{L^{p}(\Omega)}$, $\|\cdot\|_{H^{k}(\Omega)}$, and
$\|\cdot\|_{W^{k,p}(\Omega)}$ are defined as usual; see [1, chapter 5].
Analogously to [1, Definition 5.1.4], we define $H_{0}^{1}(\Omega)$ to be the
closure of the set $C_{c}^{\infty}(\Omega)$ of smooth functions with compact
support on $\Omega$ in $H^{1}(\Omega)$. We equip the space $H_{0}^{1}(\Omega)$
with the energy norm of (VI), i.e.,

(3) $\lVert v\rVert_{H_{0}^{1}(\Omega)}:=\left(\int_{\Omega}\lvert\nabla
v\rvert^{2}\mathop{}\\!\mathrm{d}x\right)^{1/2}\qquad\forall v\in
H_{0}^{1}(\Omega).$

Here, $|\cdot|$ denotes the Euclidean norm and $\nabla$ is the weak gradient.
The associated inner product is denoted by
$(\cdot,\cdot)_{H_{0}^{1}(\Omega)}$. Due to Poincaré’s inequality, the norm
$\lVert\cdot\rVert_{H_{0}^{1}(\Omega)}$ is equivalent to
$\lVert\cdot\rVert_{H^{1}(\Omega)}$ on $H_{0}^{1}(\Omega)$. For the
topological dual space of $(H_{0}^{1}(\Omega),\|\cdot\|_{H_{0}^{1}(\Omega)})$,
we use the symbol $H^{-1}(\Omega)$. The Laplacian $-\Delta\colon
H_{0}^{1}(\Omega)\to H^{-1}(\Omega)$ is defined by

(4) $\langle-\Delta
v,w\rangle_{H_{0}^{1}(\Omega)}:=(v,w)_{H_{0}^{1}(\Omega)}=\int_{\Omega}\nabla
v\cdot\nabla w\mathop{}\\!\mathrm{d}x\qquad\forall v,w\in H_{0}^{1}(\Omega).$

Due to (3) and (4), $(-\Delta)^{-1}$ is precisely the Riesz map of
$H_{0}^{1}(\Omega)$. In particular, $-\Delta$ is an isometric isomorphism
between $H_{0}^{1}(\Omega)$ and $H^{-1}(\Omega)$. This also implies that the
inner product on $H^{-1}(\Omega)$ associated with the norm
$\|\cdot\|_{H^{-1}(\Omega)}$ is given by

(5)
$(z_{1},z_{2})_{H^{-1}(\Omega)}=((-\Delta)^{-1}z_{1},(-\Delta)^{-1}z_{2})_{H_{0}^{1}(\Omega)}\qquad\forall
z_{1},z_{2}\in H^{-1}(\Omega).$

For the spaces of continuous functions on $\Omega$ and
$\bar{\Omega}:=\mathrm{cl}(\Omega)$, we use the symbols $C(\Omega)$ and
$C(\bar{\Omega})$, respectively. We further define $C_{0}(\Omega):=\\{v\in
C(\bar{\Omega})\mid v=0\text{ on }\partial\Omega\\}$, where $\partial\Omega$
denotes the boundary of $\Omega$. Recall that $C(\bar{\Omega})$ and
$C_{0}(\Omega)$ are Banach when endowed with the maximum norm
$\|\cdot\|_{\infty}$. Given an element $v$ of $C(\Omega)$, we denote by
$\\{v=c\\}$, $c\in\mathbb{R}$, the level set $\\{x\in\Omega\mid v(x)=c\\}$.
Analogous abbreviations are also used for sublevel, superlevel, and
coincidence sets. For the indicator function of a set $D\subset\Omega$, we use
the notation $\mathds{1}_{D}\colon\Omega\to\\{0,1\\}$, and for the Euclidean
distance to $D$, we write $\operatorname{dist}(\cdot,D)$. Note that further
symbols etc. are introduced in the following sections wherever necessary. This
additional notation is clarified on its first appearance.

##  3 Preliminaries on the obstacle problem

In this section, we collect basic results on the obstacle problem (VI).
Throughout Section 3, we utilize the following assumptions.

[standing assumptions for Section 3]

  * •

$\Omega\subset\mathbb{R}^{d}$, $d\in\mathbb{N}$, is a nonempty, open, and
bounded set;

  * •

$\psi,\phi\colon\Omega\to[-\infty,\infty]$ are measurable functions such that
$K\neq\emptyset$.

We begin with a basic solution result for (VI).

######  Proposition 3.1 (solvability and Lipschitz continuity).

For all $u\in H^{-1}(\Omega)$, the variational inequality (VI) possesses a
unique solution $S(u):=y\in H_{0}^{1}(\Omega)$. The solution map $S\colon
H^{-1}(\Omega)\to H_{0}^{1}(\Omega)$, $u\mapsto y$, of (VI) satisfies

(6) $\left\lVert
S(u_{1})-S(u_{2})\right\rVert_{H_{0}^{1}(\Omega)}\leq\|u_{1}-u_{2}\|_{H^{-1}(\Omega)}\qquad\forall
u_{1},u_{2}\in H^{-1}(\Omega).$

######  Proof 3.2.

See [29, Theorem 4-3.1] or [20, Theorem II-2.1].

Note that $S$ is—modulo the Riesz isomorphism—precisely the metric projection
in $(H_{0}^{1}(\Omega),\|\cdot\|_{H_{0}^{1}(\Omega)})$ onto the set $K$. For
$L^{p}$-inputs, we also have the following Lipschitz continuity result for the
mapping $S$.

######  Proposition 3.3 ($L^{p}$-Lipschitz estimate).

Let $p>\max(1,2d/(d+2))$ and define

(7) $\displaystyle\mathcal{Q}_{p}:=\begin{cases}[1,\infty]&\text{ if
}p>\frac{d}{2},\\\
\left[1,\left(\frac{1}{p}-\frac{2}{d}\right)^{-1}\right)&\text{ if
}p\leq\frac{d}{2},\end{cases}$

with the convention $0^{-1}:=\infty$ in the case $p=d/2$. Then, for every
$q\in\mathcal{Q}_{p}$, there exists a constant $C>0$ such that

(8) $\displaystyle\left\lVert
S(u_{1})-S(u_{2})\right\rVert_{L^{q}(\Omega)}\leq
C\|u_{1}-u_{2}\|_{L^{p}(\Omega)}\qquad\forall u_{1},u_{2}\in L^{p}(\Omega).$

######  Proof 3.4.

This result follows from the Sobolev embeddings and classical truncation
arguments. See, e.g., [11, Lemma 3.7] where the assertion of the proposition
is proved in the unilateral case. The proof for the bilateral case is
completely analogous.

Recall that $p>\max(1,2d/(d+2))$ implies
$L^{p}(\Omega)\mathrel{\mathop{\hookrightarrow}\limits^{\vbox
to0.0pt{\kern-2.0pt\hbox{$\scriptstyle\textup{c}$}\vss}}}H^{-1}(\Omega)$ by
the Sobolev embeddings. Thus, $S(u_{i})\in H_{0}^{1}(\Omega)$ in Proposition
3.3 is well defined for $u_{i}\in L^{p}(\Omega)$. Note further that the
functions $S(u_{i})$ in (8) might not belong to $L^{q}(\Omega)$ due to the low
regularity of $\phi$ and $\psi$. Only the difference $S(u_{1})-S(u_{2})$ is
known to be in $L^{q}(\Omega)$; cf. [11, Lemma 3.7] and the comments after
[11, Corollary 4.2]. Next, we state a classical directional differentiability
result for (VI) that goes back to [15, 24].

######  Proposition 3.5 (Hadamard directional differentiability).

The solution operator $S\colon H^{-1}(\Omega)\to H_{0}^{1}(\Omega)$, $u\mapsto
y$, of (VI) is Hadamard directionally differentiable. The directional
derivative $\delta:=S^{\prime}(u;h)$ of $S$ at a point $u$ with state
$y:=S(u)$ in direction $h$ is uniquely characterized by the variational
inequality

(9)
$\delta\in\mathcal{K}(u),\qquad\left\langle-\Delta\delta-h,z-\delta\right\rangle_{H_{0}^{1}(\Omega)}\geq
0\qquad\forall z\in\mathcal{K}(u).$

Here, $\mathcal{K}(u):=\mathcal{T}(y;K)\cap\mathrm{ker}(\Delta y+u)$ is the
so-called _critical cone_ associated with $u$.

######  Proof 3.6.

See [7, Corollary 3.4.3] and also the classical works [15, 24].

######  Remark 3.7.

The proof of Proposition 3.5 relies on the so-called _polyhedricity_ of the
set $K$, i.e., the fact that, for all $u\in H^{-1}(\Omega)$ with state $y\in
H_{0}^{1}(\Omega)$, one has

(10)
$\mathrm{cl}_{H_{0}^{1}(\Omega)}\left(\mathcal{R}(y;K)\cap\mathrm{ker}(\Delta
y+u)\right)=\mathcal{T}(y;K)\cap\mathrm{ker}(\Delta y+u).$

A tangible characterization of the critical cone
$\mathcal{K}(u)=\mathcal{T}(y;K)\cap\mathrm{ker}(\Delta y+u)$ appearing on the
right-hand side of (10) can be obtained by using the notion of quasi-
everywhere and the fine-support of the multiplier $\Delta y+u\in
H^{-1}(\Omega)$. For details on this topic, we refer to [28, section 2.2],
[15, 24, 38], and the references therein.

Using (9), one can check that $u$ is an element of the set $\mathcal{D}_{S}$
of Gâteaux points of $S\colon H^{-1}(\Omega)\to H_{0}^{1}(\Omega)$ if and only
if the critical cone $\mathcal{K}(u)$ is a subspace. From the Lipschitz
estimate (6), we obtain that there are indeed many such $u\in H^{-1}(\Omega)$.

######  Theorem 3.8 (Rademacher’s theorem).

The set $\mathcal{D}_{S}$ is dense in $H^{-1}(\Omega)$.

######  Proof 3.9.

See [34, Proposition 1.3, Remark 1.3] and the references therein.

In tandem with (6), the density of $\mathcal{D}_{S}$ in $H^{-1}(\Omega)$
implies the following result for the strong-weak Bouligand differential
$\partial_{B}^{sw}S\colon
H^{-1}(\Omega)\rightrightarrows\mathscr{L}(H^{-1}(\Omega),H_{0}^{1}(\Omega))$
of the solution operator $S\colon H^{-1}(\Omega)\to H_{0}^{1}(\Omega)$ of
(VI).

######  Lemma 3.10 (boundedness and nonemptyness of $\partial_{B}^{sw}S(u)$).

The set $\partial_{B}^{sw}S(u)$ is nonempty for all $u\in H^{-1}(\Omega)$ and
it holds

(11) $\lVert G\rVert_{\mathscr{L}(H^{-1}(\Omega),H_{0}^{1}(\Omega))}\leq
1\quad\forall G\in\partial_{B}^{sw}S(u)\quad\forall u\in H^{-1}(\Omega).$

######  Proof 3.11.

From (6), it follows that $\lVert
S^{\prime}(u)\rVert_{\mathscr{L}(H^{-1}(\Omega),H_{0}^{1}(\Omega))}\leq 1$
holds for all $u\in\mathcal{D}_{S}$. In combination with the Banach–Alaoglu
theorem for the WOT, Theorem 3.8, and the definition of $\partial_{B}^{sw}S$,
this yields $\partial_{B}^{sw}S(u)\neq\emptyset$ for all $u\in
H^{-1}(\Omega)$; see [11, Theorem 2.5, Corollary 2.6]. To establish (11), it
suffices to note that, for all $u,h\in H^{-1}(\Omega)$ and all
$G\in\partial_{B}^{sw}S(u)$ with associated Gâteaux points $\\{u_{n}\\}$ as in
the definition of $\partial_{B}^{sw}S(u)$, we have

$\displaystyle\lVert
Gh\rVert_{H_{0}^{1}(\Omega)}\leq\liminf_{n\to\infty}\lVert
S^{\prime}(u_{n})h\rVert_{H_{0}^{1}(\Omega)}\leq\lVert
h\rVert_{H^{-1}(\Omega)}$

by the weak sequential lower semicontinuity of the norm. This completes the
proof.

The nonemptyness of $\partial_{B}^{sw}S(u)$ makes the strong-weak Bouligand
differential a suitable candidate for a Newton derivative of $S$; cf. Theorems
4.1, 4.4, and 6.32 below. As a final preparation for our analysis, we note the
following.

######  Lemma 3.12 (norm of derivatives).

Let $u,h\in H^{-1}(\Omega)$ be given.

  1. (i)

The derivative $\delta:=S^{\prime}(u;h)$ of $S$ at $u$ in direction $h$
satisfies

(12) $\displaystyle\lVert\delta\rVert_{H_{0}^{1}(\Omega)}^{2}=\langle
h,\delta\rangle_{H_{0}^{1}(\Omega)}.$

  2. (ii)

For all $G\in\partial_{B}^{sw}S(u)$, the function $\delta:=Gh$ satisfies

$\displaystyle\lVert\delta\rVert_{H_{0}^{1}(\Omega)}^{2}\leq\left\langle
h,\delta\right\rangle_{H_{0}^{1}(\Omega)}.$

######  Proof 3.13.

Since $\mathcal{K}(u)$ is a cone, we may choose $z=0$ and $z=2\delta$ in (9).
This yields
$\left\langle-\Delta\delta-h,\delta\right\rangle_{H_{0}^{1}(\Omega)}=0$ and,
thus, (12). Suppose now that an operator $G\in\partial_{B}^{sw}S(u)$ is given
and that $\\{u_{n}\\}\subset\mathcal{D}_{S}$ is a sequence of Gâteaux points
for $G$ as in the definition of $\partial_{B}^{sw}S(u)$. In this situation, we
have $S^{\prime}(u_{n})h\rightharpoonup Gh$ in $H_{0}^{1}(\Omega)$ and, by the
weak sequential lower semicontinuity of the norm and (i),

$\displaystyle\lVert
Gh\rVert_{H_{0}^{1}(\Omega)}^{2}\leq\liminf_{n\to\infty}\lVert
S^{\prime}(u_{n})h\rVert_{H_{0}^{1}(\Omega)}^{2}=\liminf_{n\to\infty}\left\langle
h,S^{\prime}(u_{n})h\right\rangle_{H_{0}^{1}(\Omega)}=\left\langle
h,Gh\right\rangle_{H_{0}^{1}(\Omega)}.$

This establishes the assertion of (ii) and completes the proof.

######  Remark 3.14.

Lemma 3.12 relies crucially on the polyhedricity of the set $K$. Indeed, for
nonpolyhedric sets, the variational inequality (9) typically contains an
additional curvature term that causes (12) to be invalid and whose behavior
for varying $u$ is hard to control; see [7, Theorem 1.4.1(ii)] and [9, Theorem
4.1].

##  4 Energy space Newton differentiability for the unilateral case

With the preliminaries in place, we are in the position to turn our attention
to the proof of the first main result of this paper—the Newton
differentiability of the solution operator of the unilateral obstacle problem
with values in the energy space $H_{0}^{1}(\Omega)$.

[standing assumptions for Section 4]

  * •

$\Omega\subset\mathbb{R}^{d}$, $d\in\mathbb{N}$, is a nonempty, open, and
bounded set;

  * •

$\psi\colon\Omega\to[-\infty,\infty]$ is measurable, $\phi\equiv\infty$, and
$K\neq\emptyset$.

The point of departure for our analysis is the following result that has
recently been obtained in [11].

######  Theorem 4.1 ($L^{2}$-Newton differentiability for the unilateral
obstacle problem).

The solution map $S$ of (VI) is Newton differentiable as a function $S\colon
L^{p}(\Omega)\to L^{2}(\Omega)$ for all $\max(1,2d/(d+2))<p\leq\infty$ with
Newton derivative $DS:=\partial_{B}^{sw}S$.

######  Proof 4.2.

See [11, Corollaries 3.9, 4.2].

######  Remark 4.3.

The function $S$ is even Newton differentiable with values in $L^{q}(\Omega)$
for all $q\in\mathcal{Q}_{p}\setminus\\{\infty\\}$ where $\mathcal{Q}_{p}$ is
the set in (7). Stating this generalized Newton differentiability result
requires some care though since $S$ might not map into $L^{q}(\Omega)$ due to
the (possibly) low regularity of $\psi$. The interested reader is referred to
[11].

To establish the Newton differentiability of $S$ as a function with values in
$H_{0}^{1}(\Omega)$, we combine Theorem 4.1 with Lemma 3.12.

######  Theorem 4.4 ($H_{0}^{1}$-Newton differentiability for the unilateral
obstacle problem).

The solution map $S$ of (VI) is Newton differentiable as a function $S\colon
L^{p}(\Omega)\to H_{0}^{1}(\Omega)$ for all $\max(1,2d/(d+2))<p\leq\infty$
with Newton derivative $DS:=\partial_{B}^{sw}S$.

######  Proof 4.5.

We assume without loss of generality that $p<\infty$. (If the claim is proved
for such $p$, then the assertion for $p=\infty$ follows immediately by
Hölder’s inequality.) In order to prove the theorem, it suffices to show that,
for all $u\in L^{p}(\Omega)$ and all sequences $\\{t_{n}\\}\subset(0,\infty)$,
$\\{h_{n}\\}\subset L^{p}(\Omega)$, and
$\\{G_{n}\\}\subset\mathscr{L}(H^{-1}(\Omega),H_{0}^{1}(\Omega))$ satisfying
$t_{n}\to 0$, $\lVert h_{n}\rVert_{L^{p}(\Omega)}=1$, and
$G_{n}\in\partial_{B}^{sw}S(u+t_{n}h_{n})$, we have

(13) $\frac{\lVert
S(u+t_{n}h_{n})-S(u)-G_{n}t_{n}h_{n}\rVert_{H_{0}^{1}(\Omega)}}{\lVert
t_{n}h_{n}\rVert_{L^{p}(\Omega)}}\to 0;$

cf. the definition of Newton differentiability in (2). Due to the boundedness
of $\\{h_{n}\\}$ in $L^{p}(\Omega)$ and the compact embedding
$L^{p}(\Omega)\mathrel{\mathop{\hookrightarrow}\limits^{\vbox
to0.0pt{\kern-2.0pt\hbox{$\scriptstyle\textup{c}$}\vss}}}H^{-1}(\Omega)$,
every subsequence of $\\{h_{n}\\}$ possesses a further subsequence (not
relabeled) such that $h_{n}\rightharpoonup h$ in $L^{p}(\Omega)$ and $h_{n}\to
h$ in $H^{-1}(\Omega)$ for some $h\in L^{p}(\Omega)$. From Theorem 4.1, we
further get that

$\displaystyle\frac{\lVert
S(u+t_{n}h_{n})-S(u)-G_{n}t_{n}h_{n}\rVert_{L^{2}(\Omega)}}{\|t_{n}h_{n}\|_{L^{p}(\Omega)}}\to
0,$

and from (6) and (11), it follows that there exists a constant $C>0$
satisfying

$\displaystyle\left\lVert\frac{S(u+t_{n}h_{n})-S(u)}{t_{n}}-G_{n}h_{n}\right\rVert_{H_{0}^{1}(\Omega)}\leq
C.$

Combining these two observations yields

(14) $\frac{S(u+t_{n}h_{n})-S(u)}{t_{n}}-G_{n}h_{n}\rightharpoonup 0\text{ in
}H_{0}^{1}(\Omega).$

Due to the convergence $h_{n}\to h$ in $H^{-1}(\Omega)$ and Proposition 3.5,
we know that the first term on the left-hand side of (14) converges strongly
in $H_{0}^{1}(\Omega)$ to $S^{\prime}(u;h)$. Thus,

(15) $G_{n}h_{n}\rightharpoonup S^{\prime}(u;h)\text{ in }H_{0}^{1}(\Omega).$

Define $\delta_{n}:=G_{n}h_{n}$, $\delta:=S^{\prime}(u;h)$. From Lemma 3.12,
we obtain that

$\lVert\delta_{n}\rVert_{H_{0}^{1}(\Omega)}^{2}\leq\left\langle
h_{n},\delta_{n}\right\rangle_{H_{0}^{1}(\Omega)}~{}\forall
n\in\mathbb{N}\qquad\text{and}\qquad\lVert\delta\rVert_{H_{0}^{1}(\Omega)}^{2}=\left\langle
h,\delta\right\rangle_{H_{0}^{1}(\Omega)}.$

Due to the convergence $\delta_{n}\rightharpoonup\delta$ in
$H_{0}^{1}(\Omega)$ in (15) and $h_{n}\to h$ in $H^{-1}(\Omega)$, it follows

$\displaystyle\lVert\delta-\delta_{n}\rVert_{H_{0}^{1}(\Omega)}^{2}$
$\displaystyle=\lVert\delta_{n}\rVert_{H_{0}^{1}(\Omega)}^{2}-\lVert\delta\rVert_{H_{0}^{1}(\Omega)}^{2}+2(\delta,\delta-\delta_{n})_{H_{0}^{1}(\Omega)}$
$\displaystyle\leq\left\langle
h_{n},\delta_{n}\right\rangle_{H_{0}^{1}(\Omega)}-\left\langle
h,\delta\right\rangle_{H_{0}^{1}(\Omega)}+2(\delta,\delta-\delta_{n})_{H_{0}^{1}(\Omega)}\to
0.$

Thus, $\delta_{n}\to\delta$ in $H_{0}^{1}(\Omega)$. In summary, we have now
shown that

$\frac{S(u+t_{n}h_{n})-S(u)}{t_{n}}-G_{n}h_{n}=\frac{S(u+t_{n}h_{n})-S(u)}{t_{n}}-S^{\prime}(u;h)+\delta-\delta_{n}\to
0$

holds in $H_{0}^{1}(\Omega)$ for the subsequence $\\{h_{n}\\}$ selected after
(13). A standard subsequence-subsequence argument now shows that (13) holds.
This completes the proof.

######  Remark 4.6.

Explicit formulas for certain elements of the strong-weak Bouligand
differential $\partial_{B}^{sw}S$ of the solution map $S$ of the unilateral
obstacle problem can be found in [28, Proposition 2.11(i), Theorem 4.3]. Under
mild additional assumptions, a precise characterization of the entire
differential $\partial_{B}^{sw}S$ has been obtained in [28, Theorem 5.6]. For
a tangible corollary of the results of [28] that is suitable for practical
applications, we refer the reader to Definitions 6.7 and 6.8 in Section 6.

##  5 Auxiliary results on angled subspaces and orthogonal projections

The purpose of this section is to establish several auxiliary results on
angled subspaces and orthogonal projections that are needed for the analysis
of the bilateral obstacle problem in Section 6. For the sake of reusability,
we consider a general setting.

[standing assumptions for Section 5]

  * •

$H$ is a real Hilbert space with closed unit ball $B_{H}:=\\{x\in
H\mid\|x\|_{H}\leq 1\\}$;

  * •

$W_{1},W_{2}\subset H$ are closed subspaces of $H$.

We start by defining the minimal angle between $W_{1}$ and $W_{2}$; see [13]
and the references therein for more information on this concept.

######  Definition 5.1 (minimal angle).

The minimal angle $\alpha_{0}(W_{1},W_{2})$ between $W_{1}$ and $W_{2}$ in $H$
is defined to be the angle in $[0,\pi/2]$ whose cosine is given by

$c_{0}(W_{1},W_{2}):=\sup\big{\\{}(x_{1},x_{2})_{H}\nonscript\;|\nonscript\;\mathopen{}x_{1}\in
W_{1}\cap B_{H},x_{2}\in W_{2}\cap B_{H}\big{\\}}\in[0,1].$

It is clear that $\alpha_{0}(\cdot,\cdot)$ and $c_{0}(\cdot,\cdot)$ are
symmetric. Via a simple scaling argument, it is immediate that
$c_{0}=c_{0}(W_{1},W_{2})$ is the smallest constant in $[0,1]$ for which the
following sharpened Cauchy–Bunyakovsky–Schwarz inequality holds true:

$\displaystyle\lvert(x_{1},x_{2})_{H}\rvert\leq c_{0}\lVert
x_{1}\rVert_{H}\lVert x_{2}\rVert_{H}\qquad\forall x_{1}\in W_{1},x_{2}\in
W_{2}.$

Next, we recall [13, Lemma 10 (3)]. For convenience, we also include its short
proof.

######  Lemma 5.2.

Denote by $P_{i}$ the orthogonal projection onto $W_{i}$, $i=1,2$. Then,

$\displaystyle\lVert P_{1}P_{2}\rVert_{\mathscr{L}(H,H)}=\lVert
P_{2}P_{1}\rVert_{\mathscr{L}(H,H)}=c_{0}(W_{1},W_{2}).$

######  Proof 5.3.

We have

$\displaystyle c_{0}(W_{1},W_{2})$
$\displaystyle=\sup\big{\\{}(x_{1},x_{2})_{H}\nonscript\;|\nonscript\;\mathopen{}x_{1}\in
W_{1}\cap B_{H},x_{2}\in W_{2}\cap B_{H}\big{\\}}$
$\displaystyle=\sup\big{\\{}(P_{1}x_{1},P_{2}x_{2})_{H}\nonscript\;|\nonscript\;\mathopen{}x_{1}\in
B_{H},x_{2}\in B_{H}\big{\\}}$
$\displaystyle=\sup\big{\\{}(x_{1},P_{1}P_{2}x_{2})_{H}\nonscript\;|\nonscript\;\mathopen{}x_{1}\in
B_{H},x_{2}\in B_{H}\big{\\}}=\lVert P_{1}P_{2}\rVert_{\mathscr{L}(H,H)}.$

The symmetry of $c_{0}(\cdot,\cdot)$ provides the assertion for $P_{2}P_{1}$.

The next lemma shows that the sine of the minimal angle has a geometric
meaning too. It is a quantitative version of [13, Theorem 12
(1)$\Leftrightarrow$(3)].

######  Lemma 5.4.

Denote by $s_{0}=s_{0}(W_{1},W_{2})$ the largest number in $[0,1]$ satisfying

$\displaystyle\lVert x_{1}+x_{2}\rVert_{H}\geq s_{0}\lVert
x_{1}\rVert_{H}\qquad\forall x_{1}\in W_{1},x_{2}\in W_{2}.$

Then it holds that

(16) $s_{0}(W_{1},W_{2})^{2}+c_{0}(W_{1},W_{2})^{2}=1.$

In particular, $s_{0}(W_{1},W_{2})$ is the sine of the minimal angle
$\alpha_{0}(W_{1},W_{2})$.

######  Proof 5.5.

We set $c_{0}:=c_{0}(W_{1},W_{2})$. For all $x_{1}\in W_{1}$, $x_{2}\in
W_{2}$, we have

$\displaystyle\lVert x_{1}+x_{2}\rVert_{H}^{2}$ $\displaystyle=\lVert
x_{1}\rVert_{H}^{2}+2(x_{1},x_{2})_{H}+\lVert x_{2}\rVert_{H}^{2}\geq\lVert
x_{1}\rVert_{H}^{2}-2c_{0}\lVert x_{1}\rVert_{H}\lVert x_{2}\rVert_{H}+\lVert
x_{2}\rVert_{H}^{2}$ $\displaystyle\geq\lVert x_{1}\rVert_{H}^{2}-\lVert
x_{1}\rVert_{H}^{2}-c_{0}^{2}\lVert x_{2}\rVert_{H}^{2}+\lVert
x_{2}\rVert_{H}^{2}=(1-c_{0}^{2})\lVert x_{2}\rVert_{H}^{2},$

hence $s_{0}\geq\sqrt{1-c_{0}^{2}}$. Similarly, for $x_{1}\in W_{1}\cap
B_{H}$, $x_{2}\in W_{2}\cap B_{H}$, and $t>0$, we find

$\displaystyle 2(x_{1},x_{2})_{H}$
$\displaystyle=2(t^{-1/2}x_{1},t^{1/2}x_{2})_{H}=\lVert
t^{-1/2}x_{1}\rVert_{H}^{2}+\lVert t^{1/2}x_{2}\rVert_{H}^{2}-\lVert
t^{-1/2}x_{1}-t^{1/2}x_{2}\rVert_{H}^{2}$ $\displaystyle\leq t^{-1}\lVert
x_{1}\rVert_{H}^{2}+t\lVert x_{2}\rVert_{H}^{2}-s_{0}^{2}t^{-1}\lVert
x_{1}\rVert_{H}^{2}\leq(1-s_{0}^{2})t^{-1}+t.$

Taking the supremum over all $x_{i}\in W_{i}\cap B_{H}$, $i=1,2$, and the
infimum over all $t>0$ shows that $c_{0}\leq\sqrt{1-s_{0}^{2}}$. This
establishes the claim.

Note that the symmetry of $s_{0}(\cdot,\cdot)$ is not obvious from the
definition, but it holds due to the symmetry of $c_{0}(\cdot,\cdot)$. Finally,
we show the invertibility of a certain operator that appears in Section 6.

######  Theorem 5.6.

Let $P_{i}$ denote the orthogonal projection onto $W_{i}$, $i=1,2$. Suppose
that $W_{1}$ and $W_{2}$ enclose a positive angle, i.e.,
$c_{0}(W_{1},W_{2})<1$. Then the operator

(17) $R_{1}:=\begin{pmatrix}\mathrm{Id}&P_{1}\\\
P_{2}&\mathrm{Id}\end{pmatrix}\in\mathscr{L}(H^{2},H^{2})$

is continuously invertible with $\lVert
R_{1}^{-1}\rVert_{\mathscr{L}(H^{2},H^{2})}\leq 4/(1-c_{0}(W_{1},W_{2}))$, the
subspace $W_{1}+W_{2}$ is closed, and the orthogonal projection $P$ onto
$W_{1}+W_{2}$ is given by

(18)
$P=\begin{pmatrix}\mathrm{Id}&\mathrm{Id}\end{pmatrix}R_{1}^{-1}\begin{pmatrix}P_{1}\\\
P_{2}\end{pmatrix}.$

######  Proof 5.7.

By a standard Schur complement approach, we find the factorization

$\displaystyle R_{1}=\begin{pmatrix}\mathrm{Id}&P_{1}\\\
P_{2}&\mathrm{Id}\end{pmatrix}=\begin{pmatrix}\mathrm{Id}&0\\\
P_{2}&\mathrm{Id}\end{pmatrix}\begin{pmatrix}\mathrm{Id}&0\\\
0&\mathrm{Id}-P_{2}P_{1}\end{pmatrix}\begin{pmatrix}\mathrm{Id}&P_{1}\\\
0&\mathrm{Id}\end{pmatrix}.$

Since $\lVert P_{2}P_{1}\rVert_{\mathscr{L}(H,H)}=c_{0}(W_{1},W_{2})<1$ holds
by Lemma 5.2, it follows from the Neumann series that the operator
$\mathrm{Id}-P_{2}P_{1}$ is invertible with
$\lVert(\mathrm{Id}-P_{2}P_{1})^{-1}\rVert_{\mathscr{L}(H,H)}\leq
1/(1-c_{0}(W_{1},W_{2}))$. This implies that $R_{1}$ is continuously
invertible too with

(19) $R_{1}^{-1}=\begin{pmatrix}\mathrm{Id}&-P_{1}\\\
0&\mathrm{Id}\end{pmatrix}\begin{pmatrix}\mathrm{Id}&0\\\
0&(\mathrm{Id}-P_{2}P_{1})^{-1}\end{pmatrix}\begin{pmatrix}\mathrm{Id}&0\\\
-P_{2}&\mathrm{Id}\end{pmatrix}$

and

$\displaystyle\lVert R_{1}^{-1}\rVert_{\mathscr{L}(H^{2},H^{2})}\leq
2\cdot\lVert(\mathrm{Id}-P_{2}P_{1})^{-1}\rVert_{\mathscr{L}(H,H)}\cdot
2\leq\frac{4}{1-c_{0}(W_{1},W_{2})}.$

It remains to check that $W_{1}+W_{2}$ is closed and that the map
$P\in\mathscr{L}(H,H)$ in (18) is the orthogonal projection onto
$W_{1}+W_{2}$. By combining (18) with (19), we get that

(20) $\displaystyle P$
$\displaystyle=\begin{pmatrix}\mathrm{Id}&\mathrm{Id}-P_{1}\end{pmatrix}\begin{pmatrix}\mathrm{Id}&0\\\
0&(\mathrm{Id}-P_{2}P_{1})^{-1}\end{pmatrix}\begin{pmatrix}P_{1}\\\
P_{2}-P_{2}P_{1}\end{pmatrix}$
$\displaystyle=P_{1}+(\mathrm{Id}-P_{1})(\mathrm{Id}-P_{2}P_{1})^{-1}P_{2}(\mathrm{Id}-P_{1}).$

Since $P_{i}=P_{i}^{2}$ and
$P_{i}(\mathrm{Id}-P_{i})=(\mathrm{Id}-P_{i})P_{i}=0$ for $i=1,2$, (20) yields
that

$\displaystyle P^{2}$
$\displaystyle=P_{1}+(\mathrm{Id}-P_{1})(\mathrm{Id}-P_{2}P_{1})^{-1}P_{2}(\mathrm{Id}-P_{1})(\mathrm{Id}-P_{2}P_{1})^{-1}P_{2}(\mathrm{Id}-P_{1})$
$\displaystyle=P_{1}+(\mathrm{Id}-P_{1})(\mathrm{Id}-P_{2}P_{1})^{-1}P_{2}(\mathrm{Id}-P_{2}P_{1})(\mathrm{Id}-P_{2}P_{1})^{-1}P_{2}(\mathrm{Id}-P_{1})$
$\displaystyle=P.$

Hence, $P$ is idempotent. Next, we note that

(21)
$(\mathrm{Id}-P_{2}P_{1})^{-1}=\sum_{k=0}^{\infty}(P_{2}P_{1})^{k}=\mathrm{Id}+P_{2}(\mathrm{Id}-P_{1}P_{2})^{-1}P_{1}.$

Plugging this series expansion into (20) gives

(22)
$P=P_{1}+(\mathrm{Id}-P_{1})\sum_{k=0}^{\infty}(P_{2}P_{1})^{k}P_{2}(\mathrm{Id}-P_{1}).$

As the orthogonal projections $P_{1}$ and $P_{2}$ are self-adjoint, (22)
implies that $P$ is self-adjoint too. Thus, $P$ is the orthogonal projection
onto $\mathrm{range}(P)$ by [13, Lemma 4]. In particular, $\mathrm{range}(P)$
is closed. It remains to show that $\mathrm{range}(P)=W_{1}+W_{2}$. From (20),
we obtain that $PP_{1}=P_{1}$ holds and that

$\displaystyle PP_{2}$
$\displaystyle=P_{1}P_{2}+(\mathrm{Id}-P_{1})(\mathrm{Id}-P_{2}P_{1})^{-1}P_{2}(\mathrm{Id}-P_{1})P_{2}$
$\displaystyle=P_{1}P_{2}+(\mathrm{Id}-P_{1})(\mathrm{Id}-P_{2}P_{1})^{-1}(\mathrm{Id}-P_{2}P_{1})P_{2}=P_{2}.$

Hence, $W_{1}+W_{2}\subset\mathrm{range}(P)$. From (20) and (21), we further
obtain that

$\displaystyle P$
$\displaystyle=P_{1}-P_{1}(\mathrm{Id}-P_{2}P_{1})^{-1}P_{2}(\mathrm{Id}-P_{1})+(\mathrm{Id}-P_{2}P_{1})^{-1}P_{2}(\mathrm{Id}-P_{1})$
$\displaystyle=P_{1}-P_{1}(\mathrm{Id}-P_{2}P_{1})^{-1}P_{2}(\mathrm{Id}-P_{1})+\left(\mathrm{Id}+P_{2}(\mathrm{Id}-P_{1}P_{2})^{-1}P_{1}\right)P_{2}(\mathrm{Id}-P_{1})$
$\displaystyle=P_{1}-P_{1}(\mathrm{Id}-P_{2}P_{1})^{-1}P_{2}(\mathrm{Id}-P_{1})+P_{2}\left(\mathrm{Id}+(\mathrm{Id}-P_{1}P_{2})^{-1}P_{1}P_{2}\right)(\mathrm{Id}-P_{1}).$

This shows that $\mathrm{range}(P)\subset W_{1}+W_{2}$. Thus,
$\mathrm{range}(P)=W_{1}+W_{2}$, the operator $P$ is the orthogonal projection
onto $W_{1}+W_{2}$, and $W_{1}+W_{2}$ is a closed subspace.

######  Remark 5.8.

  * •

The closedness of the set $W_{1}+W_{2}$ in Theorem 5.6 can also be derived
directly from the condition $c_{0}(W_{1},W_{2})<1$; cf. [13, Theorem 12]. It
is interesting to see that it is also obtained as a byproduct when analyzing
the formula (18).

  * •

From (22), we obtain the series representation

$\displaystyle P$
$\displaystyle=\sum_{k=0}^{\infty}\Big{(}P_{1}(P_{2}P_{1})^{k}+(P_{2}P_{1})^{k}P_{2}-(P_{1}P_{2})^{k+1}-(P_{2}P_{1})^{k+1}\Big{)}$
$\displaystyle=(P_{1}+P_{2})-(P_{1}P_{2}+P_{2}P_{1})+(P_{1}P_{2}P_{1}+P_{2}P_{1}P_{2})-\ldots$

for the orthogonal projection $P$ onto the sum $W_{1}+W_{2}$ of two closed
angled subspaces $W_{i}$, $i=1,2$, in a Hilbert space $H$ in terms of the
orthogonal projections $P_{i}$, $i=1,2$, onto the spaces $W_{i}$. Note that,
in contrast to the formulas in (20) and (22), this representation is symmetric
in $P_{1}$ and $P_{2}$.

##  6 Energy space Newton differentiability for the bilateral case

We are now in the position to prove the Newton differentiability of the
solution operator of (VI) in the case $\psi\not\equiv-\infty$ and
$\phi\not\equiv\infty$. For the analysis of this bilateral case, we require
slightly more restrictive assumptions than in Section 4.

[standing assumptions for Section 6]

  * •

$\Omega\subset\mathbb{R}^{d}$, $d\in\mathbb{N}$, is a nonempty, open, and
bounded set;

  * •

$p>\max(1,d/2)$ is a given, fixed exponent;

  * •

$\Omega$ is such that it holds

(23) $v\in H_{0}^{1}(\Omega),-\Delta v\in L^{p}(\Omega)\qquad\Rightarrow\qquad
v\in C_{0}(\Omega);$

  * •

$\psi$ and $\phi$ satisfy $\psi,\phi\in C(\bar{\Omega})\cap H^{1}(\Omega)$ and
$\Delta\psi,\Delta\phi\in L^{p}(\Omega)$ (where $\Delta$ is understood
distributionally), and there exists a number $c>0$ such that

(24) $\psi\leq\phi-c\text{ in }\Omega\qquad\text{and}\qquad\psi\leq-c\leq
c\leq\phi\text{ on }\partial\Omega.$

######  Remark 6.1.

If $\Omega$ is a bounded Lipschitz domain, then (23) holds automatically; see
[33, Theorem 1.1]. For arbitrary nonempty, open, bounded sets, (23) may fail
(as one may easily check by means of the example
$\Omega=\\{x\in\mathbb{R}^{d}\nonscript\;|\nonscript\;\mathopen{}0<|x|<1\\}$,
$d>1$).

For later reference, we provide a small lemma.

######  Lemma 6.2.

Let $v\in H_{0}^{1}(\Omega)$ be given. Then, $\min(v,\phi),\max(v,\psi)\in
H_{0}^{1}(\Omega)$.

######  Proof 6.3.

We only prove the claim for $\min(v,\phi)$. From Stampacchia’s lemma, see,
e.g., [1, Theorem 5.8.2], we get that $\min(v,\phi)\in H^{1}(\Omega)$. By
definition, there exists a sequence $\\{v_{n}\\}\subset
C_{c}^{\infty}(\Omega)$ with $v_{n}\to v$ in $H^{1}(\Omega)$. By using
Stampacchia’s lemma again, we get $\min(v_{n},\phi)\in H^{1}(\Omega)\cap
C_{c}(\Omega)$. A standard mollification argument provides us with
$\min(v_{n},\phi)\in H_{0}^{1}(\Omega)$. It is easy to check that
$\min(v_{n},\phi)\to\min(v,\phi)$ in $L^{2}(\Omega)$. Further, this sequence
is bounded in $H^{1}(\Omega)$ which yields
$\min(v_{n},\phi)\rightharpoonup\min(v,\phi)$ in $H^{1}(\Omega)$. Since
$H_{0}^{1}(\Omega)$ is a closed subspace, we infer $\min(v,\phi)\in
H_{0}^{1}(\Omega)$.

Note that Lemma 6.2 implies in particular that $\min(\phi,\max(0,\psi))\in
H_{0}^{1}(\Omega)$ holds. Since $\psi\leq\min(\phi,\max(0,\psi))\leq\phi$,
this shows that the admissible set $K$ is nonempty in the situation of Section
6 and that the results of Sections 3 and 4 are applicable.

The basic idea of the following analysis is to establish the Newton
differentiability of the solution operator of the bilateral obstacle problem
by means of the results for the unilateral case in Section 4 and an
argumentation similar to that in the proof of the classical implicit function
theorem. To pursue this approach, we require some additional notation.
Henceforth, we denote by $S_{\psi}$ the solution operator of the obstacle
problem (VI) with lower obstacle $\psi$ and upper obstacle $\infty$, by
$S^{\phi}$ the solution operator of the obstacle problem (VI) with lower
obstacle $-\infty$ and upper obstacle $\phi$, and by $S_{\psi}^{\phi}$ the
solution operator of the bilateral obstacle problem with lower obstacle $\psi$
and upper obstacle $\phi$. Note that all of these operators are well defined
and globally Lipschitz continuous as functions from $H^{-1}(\Omega)$ to
$H_{0}^{1}(\Omega)$; see Proposition 3.1. We further introduce the symbols
$M_{\psi}$, $M^{\phi}$, and $M_{\psi}^{\phi}$ to denote the multiplier maps
associated with the functions $S_{\psi}$, $S^{\phi}$, and $S_{\psi}^{\phi}$,
respectively, i.e., $M_{\psi},M^{\phi},M_{\psi}^{\phi}\colon H^{-1}(\Omega)\to
H^{-1}(\Omega)$ are defined by

$\displaystyle
M_{\psi}(u):=u-(-\Delta)S_{\psi}(u),~{}~{}\,M^{\phi}(u):=u-(-\Delta)S^{\phi}(u),~{}~{}\,M_{\psi}^{\phi}(u):=u-(-\Delta)S_{\psi}^{\phi}(u),$

for all $u\in H^{-1}(\Omega)$. Clearly, the maps
$M_{\psi},M^{\phi},M_{\psi}^{\phi}\colon H^{-1}(\Omega)\to H^{-1}(\Omega)$ are
globally Lipschitz continuous too. From (23) and classical results, we obtain
the following.

######  Lemma 6.4 (higher regularity of multipliers and states).

For all $u\in L^{p}(\Omega)$, we have
$M_{\psi}(u),M^{\phi}(u),M_{\psi}^{\phi}(u)\in L^{p}(\Omega)$ and
$S_{\psi}(u),S^{\phi}(u),S_{\psi}^{\phi}(u)\in C_{0}(\Omega)$. Moreover, the
maps $S_{\psi}$, $S^{\phi}$, and $S_{\psi}^{\phi}$ are globally Lipschitz as
functions from $L^{p}(\Omega)$ to $C_{0}(\Omega)$, and the maps $M_{\psi}$,
$M^{\phi}$, and $M_{\psi}^{\phi}$ are globally Lipschitz as functions from
$L^{p}(\Omega)$ to $L^{1}(\Omega)$.

######  Proof 6.5.

The assertions of the lemma follow from the so-called Lewy–Stampacchia
inequalities; see [35, Theorem 4.35]. We include a proof for the convenience
of the reader. We begin with the bilateral case. Given a function $u\in
L^{p}(\Omega)$ with state $y:=S_{\psi}^{\phi}(u)$, we consider the solution
$\tilde{y}\in H_{0}^{1}(\Omega)$ of the variational inequality

(25) $\tilde{y}\geq
y,\qquad\left\langle-\Delta\tilde{y}-\min(-\Delta\phi,u),\tilde{v}-\tilde{y}\right\rangle_{H_{0}^{1}(\Omega)}\geq
0\quad\forall\tilde{v}\in H_{0}^{1}(\Omega),\tilde{v}\geq y.$

Here, the inequalities $\tilde{y}\geq y$ and $\tilde{v}\geq y$ are to be
understood in the a.e.-sense. In order to show $\tilde{y}\leq\phi$, we note
that $y\leq\min(\tilde{y},\phi)\in H_{0}^{1}(\Omega)$ holds; see Lemma 6.2.
Choosing $\min(\tilde{y},\phi)$ as the test function in (25) yields

$\displaystyle 0$
$\displaystyle\leq\int_{\Omega}\nabla\tilde{y}\cdot\nabla(\min(\tilde{y},\phi)-\tilde{y})-\min(-\Delta\phi,u)(\min(\tilde{y},\phi)-\tilde{y})\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\int_{\Omega}\nabla\tilde{y}\cdot\nabla(\min(\tilde{y},\phi)-\tilde{y})-(-\Delta\phi)(\min(\tilde{y},\phi)-\tilde{y})\mathop{}\\!\mathrm{d}x$
$\displaystyle=\int_{\Omega}\nabla(\tilde{y}-\phi)\cdot\nabla(\min(\tilde{y},\phi)-\tilde{y})\mathop{}\\!\mathrm{d}x=-\lVert\min(0,\phi-\tilde{y})\rVert_{H_{0}^{1}(\Omega)}^{2}.$

Here, we have again used [1, Theorem 5.8.2]. The above shows that
$\min(0,\phi-\tilde{y})=0$. Thus, $\psi\leq y\leq\tilde{y}\leq\phi$ a.e. in
$\Omega$. By choosing $\tilde{y}$ as the test function in the VI for $y$, by
setting $\tilde{v}=y$ in (25), and by adding the resulting inequalities, we
obtain that

$\displaystyle\lVert
y-\tilde{y}\rVert_{H_{0}^{1}(\Omega)}^{2}\leq\int_{\Omega}(u-\min(-\Delta\phi,u))(y-\tilde{y})\mathop{}\\!\mathrm{d}x\leq
0.$

This proves $y=\tilde{y}$. From (25), we now get

$\displaystyle-\Delta
y-\min(-\Delta\phi,u)=-\Delta\tilde{y}-\min(-\Delta\phi,u)\geq 0,$

where the inequality is to be understood in the sense of $H^{-1}(\Omega)$; cf.
[28, Lemma 2.5]. Hence, $\min(-\Delta\phi,u)\leq-\Delta y$. Analogously, one
also shows that $-\Delta y\leq\max(-\Delta\psi,u)$. Thus, $-\Delta y\in
L^{p}(\Omega)$ and, as a consequence, $M_{\psi}^{\phi}(u)=u-(-\Delta)y\in
L^{p}(\Omega)$. Due to (23), this also yields $S_{\psi}^{\phi}(u)\in
C_{0}(\Omega)$. The asserted Lipschitz continuity of $S_{\psi}^{\phi}$ as a
function from $L^{p}(\Omega)$ to $C_{0}(\Omega)$ now follows immediately from
Proposition 3.3. For the unilateral cases, one easily obtains from Proposition
3.3 that $S_{\psi}(u),S^{\phi}(u)\in L^{\infty}(\Omega)$ for $u\in
L^{p}(\Omega)$. This allows us to add artificial, inactive, constant obstacles
to the VIs satisfied by $S_{\psi}(u)$ and $S^{\phi}(u)$ and to deduce the
regularities $M_{\psi}(u),M^{\phi}(u)\in L^{p}(\Omega)$ and
$S_{\psi}(u),S^{\phi}(u)\in C_{0}(\Omega)$ from the already established result
for the bilateral case. The Lipschitz continuity of $S_{\psi}$ and $S^{\phi}$
then again follows from Proposition 3.3. To finally establish the Lipschitz
continuity of $M_{\psi}$, $M^{\phi}$, and $M_{\psi}^{\phi}$ from
$L^{p}(\Omega)$ to $L^{1}(\Omega)$, one can argue along the exact same lines
as in [10, proof of Theorem 2.3].

To be able to define suitable Newton derivatives for $S_{\psi}$ and
$S^{\phi}$, we introduce some additional notation.

######  Definition 6.6 (solution maps of Poisson problems).

Given an open nonempty set $O\subset\Omega$, we denote by
$\mathcal{S}(O)\colon H^{-1}(\Omega)\to H_{0}^{1}(\Omega)$ the solution map
$h\mapsto\delta$ of the Poisson problem

$\displaystyle\delta\in
H_{0}^{1}(O),\qquad\langle-\Delta\delta-h,v\rangle_{H_{0}^{1}(\Omega)}=0\qquad\forall
v\in H_{0}^{1}(O).$

Here, $H_{0}^{1}(O)\subset H_{0}^{1}(\Omega)$ is the completion of
$C_{c}^{\infty}(O)$ in $H^{1}(\Omega)$; cf. Section 2.

Using Definition 6.6, we can define the set-valued functions that serve as the
Newton derivatives for $S_{\psi}$, $S^{\phi}$, $M_{\psi}$, and $M^{\phi}$ in
this section.

######  Definition 6.7 (Newton derivatives for $S_{\psi}$, $S^{\phi}$,
$M_{\psi}$, and $M^{\phi}$).

Given $u\in L^{p}(\Omega)$, we define:

$\displaystyle DS_{\psi}(u)$
$\displaystyle:=\big{\\{}\mathcal{S}(O)\,\left|~{}O\text{ open and
}\\{S_{\psi}(u)>\psi\\}\subset
O\subset\Omega\setminus\mathrm{supp}\left(M_{\psi}(u)\right)\big{\\}},\right.$
$\displaystyle DS^{\phi}(u)$
$\displaystyle:=\big{\\{}\mathcal{S}(O)\,\left|~{}O\text{ open and
}\\{S^{\phi}(u)<\phi\\}\subset
O\subset\Omega\setminus\mathrm{supp}\left(M^{\phi}(u)\right)\big{\\}},\right.$
$\displaystyle DM_{\psi}(u)$
$\displaystyle:=\mathrm{Id}-(-\Delta)DS_{\psi}(u),$ $\displaystyle
DM^{\phi}(u)$ $\displaystyle:=\mathrm{Id}-(-\Delta)DS^{\phi}(u).$

Here, the sets $\\{S_{\psi}(u)>\psi\\}$ and $\\{S^{\phi}(u)<\phi\\}$ are
defined with respect to (w.r.t.) the $C(\Omega)$-representatives of the
functions $S_{\psi}(u)$, $S^{\phi}(u)$, $\psi$, and $\phi$, respectively, and
the symbol $\mathrm{supp}(\cdot)$ denotes the support in the sense of Borel
measures on $\Omega$.

We note that the sets in Definition 6.7 are nonempty since we can always
choose $O=\\{S_{\psi}(u)>\psi\\}$ and $O=\\{S^{\phi}(u)<\phi\\}$,
respectively. From the results of [28], we obtain the following.

######  Lemma 6.8.

Let $u\in L^{p}(\Omega)$. Then
$DS_{\psi}(u)\subset\partial_{B}^{sw}S_{\psi}(u)$ and
$DS^{\phi}(u)\subset\partial_{B}^{sw}S^{\phi}(u)$.

######  Proof 6.9.

This follows from [28, Theorem 4.3], the fact that open sets are quasi-open,
the fact that the $H_{0}^{1}(\Omega)$-fine-support of a measure is contained
in its ordinary support (see [37, Lemma A.4]), and the inclusions in [28,
Proposition 2.11]. Note that the result for $S^{\phi}$ can be trivially
obtained from that for $S_{\psi}$ by a simple switch of signs so that the
results of [28] indeed also apply to $S^{\phi}$.

In combination with Theorem 4.4, Lemma 6.8 shows that the maps
$DS_{\psi},DS^{\phi}\colon$
$L^{p}(\Omega)\rightrightarrows\mathscr{L}(H^{-1}(\Omega),H_{0}^{1}(\Omega))$
and $DM_{\psi},DM^{\phi}\colon
L^{p}(\Omega)\rightrightarrows\mathscr{L}(H^{-1}(\Omega),H^{-1}(\Omega))$ can
be used as Newton derivatives for $S_{\psi}$, $S^{\phi}$, $M_{\psi}$, and
$M^{\phi}$, respectively.

######  Corollary 6.10.

The operators $S_{\psi}$ and $S^{\phi}$ are Newton differentiable as functions
from $L^{p}(\Omega)$ to $H_{0}^{1}(\Omega)$ when equipped with the derivatives
$DS_{\psi}$ and $DS^{\phi}$, respectively, and the operators $M_{\psi}$ and
$M^{\phi}$ are Newton differentiable as functions from $L^{p}(\Omega)$ to
$H^{-1}(\Omega)$ when equipped with the derivatives $DM_{\psi}$ and
$DM^{\phi}$, respectively.

######  Proof 6.11.

For $S_{\psi}$, the assertion follows from Theorem 4.4, Lemma 6.8, the
estimate $p>\max(1,d/2)\geq\max(1,2d/(d+2))$, the fact that $DS_{\psi}(u)$ is
nonempty for all $u\in L^{p}(\Omega)$ (cf. Definition 6.7), and the definition
of Newton differentiability. The result for $S^{\phi}$ is obtained from that
for $S_{\psi}$ by an elementary sign transformation. As the maps $M_{\psi}$
and $M^{\phi}$ arise from $S_{\psi}$ and $S^{\phi}$, respectively, by applying
the Laplacian and by adding the identity, the assertions for $M_{\psi}$ and
$M^{\phi}$ follow from the results for $S_{\psi}$ and $S^{\phi}$ and the
definition of Newton differentiability; cf. Definition 6.7.

######  Remark 6.12.

  * •

As the reader may have noticed, we do not endow the maps $S_{\psi}$ and
$S^{\phi}$ with their full strong-weak Bouligand differentials
$\partial_{B}^{sw}S_{\psi}$ and $\partial_{B}^{sw}S^{\phi}$ as Newton
derivatives in this section although we know from Theorem 4.4 that we could do
so and still obtain Newton differentiability. The reason for this is that we
require additional properties of the Newton derivatives of $S_{\psi}$ and
$S^{\phi}$ in the following to be able to invoke the results of Section 5; cf.
Lemma 6.24.

  * •

The subsequent analysis still works when the concepts of openness, inclusion,
and support in Definition 6.7 are replaced by their capacitary counterparts;
cf. [28, Theorem 4.3]. We omit this generalization here for the sake of
simplicity.

We can now start proving the Newton differentiability of $S_{\psi}^{\phi}$ and
$M_{\psi}^{\phi}$. We begin with the following observation.

######  Proposition 6.13.

Suppose that $u\in L^{p}(\Omega)$ and $y\in H_{0}^{1}(\Omega)$ are given. Then
it holds $y=S_{\psi}^{\phi}(u)$ if and only if there exist
$\lambda_{\psi},\lambda^{\phi}\in L^{p}(\Omega)$ satisfying

(26) $\begin{gathered}-\Delta
y=u-\lambda_{\psi}-\lambda^{\phi},\qquad\lambda_{\psi}=M_{\psi}(u-\lambda^{\phi}),\qquad\lambda^{\phi}=M^{\phi}(u-\lambda_{\psi}).\end{gathered}$

If $y=S_{\psi}^{\phi}(u)$ holds, then the functions $\lambda_{\psi}$ and
$\lambda^{\phi}$ are unique and one has

$\lambda_{\psi}=\min(0,M_{\psi}^{\phi}(u)),\quad\lambda^{\phi}=\max(0,M_{\psi}^{\phi}(u)),\quad\text{and}\quad
y=S_{\psi}(u-\lambda^{\phi})=S^{\phi}(u-\lambda_{\psi}).$

######  Proof 6.14.

Suppose that $y=S_{\psi}^{\phi}(u)$ holds. Then $-\Delta
y=u-M_{\psi}^{\phi}(u)$ is in $L^{p}(\Omega)$ by Lemma 6.4 and the VI for $y$
reads

(27) $\int_{\Omega}(-\Delta y-u)(v-y)\mathop{}\\!\mathrm{d}x\geq
0\qquad\forall v\in H_{0}^{1}(\Omega)\text{ with }\psi\leq v\leq\phi\text{
a.e.\ in }\Omega.$

By exploiting our assumptions on $\psi$ and $\phi$ and the continuity of $y$,
and by choosing functions of the form $v=y+z$, $z\in C_{c}^{\infty}(\Omega)$,
in (27), one easily checks that

(28) $-M_{\psi}^{\phi}(u)=-\Delta y-u\begin{cases}=0&\text{ a.e.\ in
}\\{\psi<y<\phi\\},\\\ \geq 0&\text{ a.e.\ in }\\{y=\psi\\},\\\ \leq 0&\text{
a.e.\ in }\\{y=\phi\\},\end{cases}$

where the sets on the right are defined w.r.t. the $C(\Omega)$-representatives
of $y$, $\psi$, and $\phi$. If we set
$\lambda_{\psi}:=\mathds{1}_{\\{y=\psi\\}}(u-(-\Delta)y)\in L^{p}(\Omega)$ and
$\lambda^{\phi}:=\mathds{1}_{\\{y=\phi\\}}(u-(-\Delta)y)\in L^{p}(\Omega)$,
then (28) yields that

(29) $\begin{gathered}-\Delta y=u-\lambda_{\psi}-\lambda^{\phi},\\\
\lambda_{\psi}\leq 0\text{ a.e.\ in }\Omega,\qquad y-\psi\geq 0\text{ a.e.\ in
}\Omega,\qquad\lambda_{\psi}(y-\psi)=0\text{ a.e.\ in }\Omega,\\\
\lambda^{\phi}\geq 0\text{ a.e.\ in }\Omega,\qquad y-\phi\leq 0\text{ a.e.\ in
}\Omega,\qquad\lambda^{\phi}(y-\phi)=0\text{ a.e.\ in }\Omega.\end{gathered}$

This implies $y=S_{\psi}(u-\lambda^{\phi})$, $y=S^{\phi}(u-\lambda_{\psi})$,
$\lambda_{\psi}=M_{\psi}(u-\lambda^{\phi})$, and
$\lambda^{\phi}=M^{\phi}(u-\lambda_{\psi})$ and proves the implication
“$\Rightarrow$” as well as the last sentence of the assertion.

To see that “$\Leftarrow$” also holds, let us assume that
$u,\lambda_{\psi},\lambda^{\phi}\in L^{p}(\Omega)$ and $y\in
H_{0}^{1}(\Omega)$ satisfying (26) are given. Then the identities
$\lambda_{\psi}=M_{\psi}(u-\lambda^{\phi})$,
$\lambda^{\phi}=M^{\phi}(u-\lambda_{\psi})$, and $-\Delta
y=u-\lambda_{\psi}-\lambda^{\phi}$ imply that
$y=S_{\psi}(u-\lambda^{\phi})=S^{\phi}(u-\lambda_{\psi})$ holds. By the
definitions of $S_{\psi}$, $M_{\psi}$, $S^{\phi}$, and $M^{\phi}$, this
entails that $y$, $\lambda_{\psi}$, $u$, and $\lambda^{\phi}$ satisfy (29).
This complementarity system implies (28) which yields $y=S_{\psi}^{\phi}(u)$
as desired.

Consider now the Hilbert space

$\displaystyle\mathcal{H}:=H^{-1}(\Omega)\times H^{-1}(\Omega)$

(endowed with its canonically induced norm) and the functions defined by

$\begin{gathered}F\colon\mathcal{H}\times
L^{p}(\Omega)\to\mathcal{H},\qquad\begin{pmatrix}\lambda_{\psi}\\\
\lambda^{\phi}\\\
u\end{pmatrix}\mapsto\begin{pmatrix}\lambda_{\psi}-M_{\psi}(u-\lambda^{\phi})\\\
\lambda^{\phi}-M^{\phi}(u-\lambda_{\psi})\end{pmatrix},\end{gathered}$

and

(30) $\Phi\colon L^{p}(\Omega)\to L^{p}(\Omega)^{2},\qquad
u\mapsto\begin{pmatrix}\min(0,M_{\psi}^{\phi}(u))\\\
\max(0,M_{\psi}^{\phi}(u))\end{pmatrix}=:\begin{pmatrix}\Phi_{\psi}(u)\\\
\Phi^{\phi}(u)\end{pmatrix}.$

Then we have the following.

######  Lemma 6.15.

It holds $F(\Phi(u),u)=0\in\mathcal{H}$ for all $u\in L^{p}(\Omega)$.

######  Proof 6.16.

See (30) and Proposition 6.13.

In what follows, our plan is to argue as in the proof of the implicit function
theorem to establish the Newton differentiability of $\Phi$ as a function from
$L^{p}(\Omega)$ to $\mathcal{H}$. To pursue this approach, we require a
generalized derivative for $F$.

######  Definition 6.17 (Newton derivative of $F$).

We denote by $DF$ the set-valued map from $L^{p}(\Omega)^{3}$ to
$\mathscr{L}(\mathcal{H},\mathcal{H})\times\mathscr{L}(L^{p}(\Omega),\mathcal{H})$
given by

$\displaystyle
DF(\lambda_{\psi},\lambda^{\phi},u):=\left\\{\mathcal{G}=\left(\begin{pmatrix}\mathrm{Id}&G_{\psi}\\\
G^{\phi}&\mathrm{Id}\end{pmatrix},\begin{pmatrix}-G_{\psi}\\\
-G^{\phi}\end{pmatrix}\right)\nonscript\;\middle|\nonscript\;\mathopen{}\begin{aligned}
G_{\psi}&\in DM_{\psi}(u-\lambda^{\phi}),\\\ G^{\phi}&\in
DM^{\phi}(u-\lambda_{\psi})\end{aligned}\right\\}.$

The two components of $\mathcal{G}$ will be denoted by
$\mathcal{G}_{1}\in\mathscr{L}(\mathcal{H},\mathcal{H})$ and
$\mathcal{G}_{2}\in\mathscr{L}(L^{p}(\Omega),\mathcal{H})$.

Now, one is tempted to use a standard estimate to establish that the implicit
function $\Phi$ is Newton differentiable from $L^{p}(\Omega)$ to $\mathcal{H}$
with Newton derivative
$D\Phi(u):=\\{-\mathcal{G}_{1}^{-1}\mathcal{G}_{2}\mid\mathcal{G}\in
DF(\Phi(u),u)\\}$. Given $u\in L^{p}(\Omega)$, $h\in
L^{p}(\Omega)\setminus\\{0\\}$, and
$G:=-\mathcal{G}_{1}^{-1}\mathcal{G}_{2}\in D\Phi(u+h)$ with $\mathcal{G}\in
DF(\Phi(u+h),u+h)$, one would try

(31)
$\displaystyle\frac{\lVert\Phi(u+h)-\Phi(u)-Gh\rVert_{\mathcal{H}}}{\lVert
h\rVert_{L^{p}(\Omega)}}=\frac{\lVert\Phi(u+h)-\Phi(u)+\mathcal{G}_{1}^{-1}\mathcal{G}_{2}h\rVert_{\mathcal{H}}}{\lVert
h\rVert_{L^{p}(\Omega)}}$ $\displaystyle\leq
C\frac{\lVert\mathcal{G}_{1}\left(\Phi(u+h)-\Phi(u)\right)+\mathcal{G}_{2}h\rVert_{\mathcal{H}}}{\lVert
h\rVert_{L^{p}(\Omega)}}=C\frac{\left\|(\mathcal{G}_{1}~{}~{}\mathcal{G}_{2})\begin{pmatrix}\Phi(u+h)-\Phi(u)\\\
h\end{pmatrix}\right\|_{\mathcal{H}}}{\lVert h\rVert_{L^{p}(\Omega)}}$
$\displaystyle=C\frac{\left\|F(\Phi(u+h),u+h)-F(\Phi(u),u)-\mathcal{G}\begin{pmatrix}\Phi(u+h)-\Phi(u)\\\
h\end{pmatrix}\right\|_{\mathcal{H}}}{\lVert h\rVert_{L^{p}(\Omega)}}$
$\displaystyle\leq
CL_{\mathcal{X}}\frac{\left\|F(\Phi(u+h),u+h)-F(\Phi(u),u)-\mathcal{G}\begin{pmatrix}\Phi(u+h)-\Phi(u)\\\
h\end{pmatrix}\right\|_{\mathcal{H}}}{\left\|\begin{pmatrix}\Phi(u+h)-\Phi(u)\\\
h\end{pmatrix}\right\|_{\mathcal{X}}}.$

Here, $C>0$ denotes a bound on the $\mathscr{L}(\mathcal{H},\mathcal{H})$-norm
of the inverse $\mathcal{G}_{1}^{-1}$ that is (at least for all small enough
$h$) independent of $h$ and $\mathcal{G}_{1}$ (and whose existence, of course,
has to be proved); $\mathcal{X}$ is a suitable normed space; and
$L_{\mathcal{X}}$ is a (local) Lipschitz constant of the function
$(\Phi,\mathrm{Id})^{\top}\colon L^{p}(\Omega)\to\mathcal{X}$; cf. [21] and
the references therein. Ideally, the last term in (31) would converge to zero
for $0<\|h\|_{L^{p}(\Omega)}\to 0$.

Unfortunately, these standard arguments fail for all $d>1$ as they require a
space $\mathcal{X}$ such that $F$ is Newton differentiable as a map from
$\mathcal{X}$ to $\mathcal{H}$ and such that $(\Phi,\mathrm{Id})^{\top}$ is
locally Lipschitz as a function from $L^{p}(\Omega)$ to $\mathcal{X}$. To the
best of our knowledge, such a space $\mathcal{X}$ is not available for $d>1$
as the maps $M_{\psi}$ and $M^{\phi}$ are Newton differentiable only from
$L^{q}(\Omega)$ to $H^{-1}(\Omega)$ and since the map $M_{\psi}^{\phi}$ is
Lipschitz only from $L^{q}(\Omega)$ to $H^{-1}(\Omega)\cap L^{1}(\Omega)$ for
all $q>\max(1,2d/(d+2))$ and $d>1$. (For $d=1$, the arguments in (31) can be
shown to work with $\mathcal{X}=L^{1}(\Omega)^{3}$.) We will therefore argue
differently and exploit the structure of $F$ and $\Phi$ to show directly that

(32) $\sup_{\mathcal{G}\in
DF(\Phi(u+h),u+h)}\frac{\lVert\mathcal{G}_{1}(\Phi(u+h)-\Phi(u))+\mathcal{G}_{2}h\rVert_{\mathcal{H}}}{\lVert
h\rVert_{L^{p}(\Omega)}}\to 0~{}~{}\text{for }0<\|h\|_{L^{p}(\Omega)}\to 0.$

The first main step of our analysis is proving that, for every $u\in
L^{p}(\Omega)$, there indeed exist constants $r>0$ and $C>0$ satisfying

(33) $\sup_{\mathcal{G}\in
DF(\Phi(u+h),u+h)}\|\mathcal{G}_{1}^{-1}\|_{\mathscr{L}(\mathcal{H},\mathcal{H})}\leq
C\qquad\forall h\in L^{p}(\Omega),\|h\|_{L^{p}(\Omega)}\leq r.$

To establish this locally uniform invertibility, we verify that the operators
$G_{\psi}$ and $G^{\phi}$ appearing in the definition of $DF(\Phi(u),u)$ are
orthogonal projections in $H^{-1}(\Omega)$.

######  Lemma 6.18.

Let $u\in L^{p}(\Omega)$. Let
$G_{\psi}=\mathrm{Id}-(-\Delta)\mathcal{S}(O_{\psi})\in
DM_{\psi}(u-\Phi^{\phi}(u))$ and
$G^{\phi}=\mathrm{Id}-(-\Delta)\mathcal{S}(O^{\phi})\in
DM^{\phi}(u-\Phi_{\psi}(u))$ be given, where $\Phi_{\psi}(u),\Phi^{\phi}(u)\in
L^{p}(\Omega)$ denote the components of $\Phi(u)$ and $O_{\psi}$ and
$O^{\phi}$ the open sets that generate $G_{\psi}$ and $G^{\phi}$,
respectively, according to Definition 6.7. Define

(34) $\displaystyle W(G_{\psi})$ $\displaystyle:=\left[-\Delta
H_{0}^{1}(O_{\psi})\right]^{\perp}\qquad\text{and}\qquad$ $\displaystyle
W(G^{\phi})$ $\displaystyle:=\left[-\Delta
H_{0}^{1}(O^{\phi})\right]^{\perp},$

where the orthogonal complement is taken in $H^{-1}(\Omega)$, and denote the
$(\cdot,\cdot)_{H^{-1}(\Omega)}$-orthogonal projection onto a closed subspace
$W\subset H^{-1}(\Omega)$ by $P_{W}$. Then it holds

$\displaystyle G_{\psi}=P_{W(G_{\psi})}\qquad\text{and}\qquad
G^{\phi}=P_{W(G^{\phi})}.$

######  Proof 6.19.

From Definition 6.6, it follows that $\mathcal{S}(O_{\psi})(-\Delta)$ is the
$H_{0}^{1}(\Omega)$-orthogonal projection onto $H_{0}^{1}(O_{\psi})$. In view
of (5), this implies that $(-\Delta)\mathcal{S}(O_{\psi})$ is the
$H^{-1}(\Omega)$-orthogonal projection onto $(-\Delta)H_{0}^{1}(O_{\psi})$.
Thus, $G_{\psi}=\mathrm{Id}-(-\Delta)\mathcal{S}(O_{\psi})$ is the
$H^{-1}(\Omega)$-orthogonal projection onto $W(G_{\psi})=[-\Delta
H_{0}^{1}(O_{\psi})]^{\perp}$ as claimed. The assertion for
$G^{\phi}=\mathrm{Id}-(-\Delta)\mathcal{S}(O^{\phi})$ is proved analogously.

Next, we are going to show that the angle between the spaces $W(G_{\psi})$ and
$W(G^{\phi})$ in (34) is positive. To this end, we need a lemma on the
stability of contact sets.

######  Lemma 6.20.

Let $u\in L^{p}(\Omega)$ be given. Then there exist a constant $r>0$ and
nonempty disjoint compact sets $Z_{\psi},Z^{\phi}\subset\Omega$ such that it
holds

$\displaystyle\\{S_{\psi}^{\phi}(u+h)=\psi\\}\subset
Z_{\psi}\qquad\text{and}\qquad\\{S_{\psi}^{\phi}(u+h)=\phi\\}\subset Z^{\phi}$

for all $h\in L^{p}(\Omega)$ with $\|h\|_{L^{p}(\Omega)}\leq r$. Here, the
contact sets are again defined w.r.t. the $C(\Omega)$-representatives of the
functions $S_{\psi}^{\phi}(u+h)$, $\phi$, and $\psi$.

######  Proof 6.21.

If $\\{S_{\psi}^{\phi}(u)=\psi\\}=\\{S_{\psi}^{\phi}(u)=\phi\\}=\emptyset$
holds, then the continuity of $S_{\psi}^{\phi}(u)$, $\phi$, and $\psi$ implies
that there exists $m>0$ with $\psi+m\leq S_{\psi}^{\phi}(u)\leq\phi-m$ in
$\bar{\Omega}$ and it follows from the Lipschitz continuity of
$S_{\psi}^{\phi}$ as a map from $L^{p}(\Omega)$ to $C_{0}(\Omega)$ in Lemma
6.4 that the contact sets of $S_{\psi}^{\phi}(u+h)$ are empty for all small
enough $\|h\|_{L^{p}(\Omega)}$. In this case, the assertion holds with any
disjoint nonempty compact $Z_{\psi},Z^{\phi}\subset\Omega$.

Let us now suppose that $\\{S_{\psi}^{\phi}(u)=\psi\\}\neq\emptyset$ and let
$c>0$ be the constant from (24). Then our assumptions on $\psi$, the
continuity of $S_{\psi}^{\phi}(u)$, and the homogeneous Dirichlet boundary
conditions imply that $Z_{\psi}:=\\{x\in\Omega\mid
S_{\psi}^{\phi}(u)(x)\leq\psi(x)+c/4\\}$ is a nonempty and compact set. Again
due to the Lipschitz continuity of $S_{\psi}^{\phi}$ as a function from
$L^{p}(\Omega)$ to $C_{0}(\Omega)$, we obtain that we can find $r>0$ such
that, for all $h\in L^{p}(\Omega)$ with $\|h\|_{L^{p}(\Omega)}\leq r$, we have
$S_{\psi}^{\phi}(u+h)(x)>\psi(x)+c/8$ for all $x\in\Omega\setminus Z_{\psi}$.
This implies that $\\{S_{\psi}^{\phi}(u+h)=\psi\\}\subset Z_{\psi}$ holds for
all such $h$ as desired. The set $Z^{\phi}$ is constructed analogously (or, in
the case $\\{S^{\phi}_{\psi}(u)=\phi\\}=\emptyset$, chosen as an arbitrary
nonempty compact subset of $\Omega\setminus Z_{\psi}\neq\emptyset$ after
potentially making $r$ smaller).

Via Lemma 5.4, we can now show that the spaces $W(G_{\psi})$ and $W(G^{\phi})$
in (34) indeed enclose a positive angle that is stable w.r.t. small
perturbations of $u$.

######  Lemma 6.22.

For every $u\in L^{p}(\Omega)$, there exist constants $r>0$,
$\bar{s}_{0}\in(0,1]$ with

(35) $\inf_{\begin{subarray}{c}h\in L^{p}(\Omega),\\\
\|h\|_{L^{p}(\Omega)}\leq
r\end{subarray}}\;\inf_{\begin{subarray}{c}G_{\psi}\in
DM_{\psi}(u+h-\Phi^{\phi}(u+h)),\\\ G^{\phi}\in
DM^{\phi}(u+h-\Phi_{\psi}(u+h))\end{subarray}}s_{0}(W(G_{\psi}),W(G^{\phi}))\geq\bar{s}_{0}.$

Here, $W(G_{\psi})$ and $W(G^{\phi})$ are defined in (34) and $s_{0}$ is
defined in Lemma 5.4.

######  Proof 6.23.

Let $u\in L^{p}(\Omega)$ be fixed and let $r>0$, $Z_{\psi}$, and $Z^{\phi}$ be
chosen for $u$ as in Lemma 6.20. Let $\zeta\in C_{c}^{\infty}(\Omega)$ be a
function that satisfies $\zeta\equiv 1$ on $Z_{\psi}$ and $\zeta\equiv 0$ on
$Z^{\phi}$. We define $J\colon H^{-1}(\Omega)\to H^{-1}(\Omega)$ to be the
operator that maps a functional $v\in H^{-1}(\Omega)$ to the element $J(v)$ of
$H^{-1}(\Omega)$ defined by

(36) $\left\langle J(v),z\right\rangle_{H_{0}^{1}(\Omega)}:=\left\langle
v,\zeta z\right\rangle_{H_{0}^{1}(\Omega)}\qquad\forall z\in
H_{0}^{1}(\Omega).$

Note that $J\in\mathscr{L}(H^{-1}(\Omega),H^{-1}(\Omega))\setminus\\{0\\}$.
Consider now some $h\in L^{p}(\Omega)$ satisfying $\|h\|_{L^{p}(\Omega)}\leq
r$ and let $G_{\psi}\in DM_{\psi}(u+h-\Phi^{\phi}(u+h))$ and $G^{\phi}\in
DM^{\phi}(u+h-\Phi_{\psi}(u+h))$ be given. Denote the open sets that generate
$G_{\psi}$ and $G^{\phi}$ according to Definition 6.7 by $O_{\psi}$ and
$O^{\phi}$, respectively. Then it follows from the inclusions
$\\{S_{\psi}^{\phi}(u+h)=\psi\\}\subset Z_{\psi}$,
$\\{S_{\psi}^{\phi}(u+h)=\phi\\}\subset Z^{\phi}$,
$\\{S_{\psi}^{\phi}(u+h)>\psi\\}\subset O_{\psi}$, and
$\\{S_{\psi}^{\phi}(u+h)<\phi\\}\subset O^{\phi}$ (see Definition 6.7,
Proposition 6.13, (30), and Lemma 6.20); the properties of $\zeta$; and the
definitions in (34) that, for all $v\in W(G_{\psi})$, $w\in W(G^{\phi})$, and
$z\in H_{0}^{1}(\Omega)$, we have

$\displaystyle\left\langle J(v+w),z\right\rangle_{H_{0}^{1}(\Omega)}$
$\displaystyle=\left\langle v+w,\zeta z\right\rangle_{H_{0}^{1}(\Omega)}$ by
(36) $\displaystyle=\left(v+w,-\Delta(\zeta z)\right)_{H^{-1}(\Omega)}$ by (5)
$\displaystyle=\left(v,-\Delta(\zeta z)\right)_{H^{-1}(\Omega)}$ by (34) and
$\zeta z\in H_{0}^{1}(O^{\phi})$ $\displaystyle=\left(v,-\Delta
z\right)_{H^{-1}(\Omega)}$ by (34) and $(1-\zeta)z\in H_{0}^{1}(O_{\psi})$
$\displaystyle=\left\langle v,z\right\rangle_{H_{0}^{1}(\Omega)}$ by (5).

Thus, $J(v+w)=v$ and, as a consequence,

$\displaystyle\|J\|_{\mathscr{L}(H^{-1}(\Omega),H^{-1}(\Omega))}\|v+w\|_{H^{-1}(\Omega)}\geq\|v\|_{H^{-1}(\Omega)}\quad\forall
v\in W(G_{\psi}),w\in W(G^{\phi}).$

Due to the definition of $s_{0}(W(G_{\psi}),W(G^{\phi}))$ in Lemma 5.4, this
yields

$\displaystyle
s_{0}(W(G_{\psi}),W(G^{\phi}))\geq\min\big{(}1,\|J\|_{\mathscr{L}(H^{-1}(\Omega),H^{-1}(\Omega))}^{-1}\big{)}>0.$

As $h$, $G_{\psi}$, and $G^{\phi}$ were arbitrary, the estimate (35) now
follows immediately (with
$\bar{s}_{0}:=\min(1,\|J\|_{\mathscr{L}(H^{-1}(\Omega),H^{-1}(\Omega))}^{-1})>0$).
This completes the proof.

As a consequence of Theorems 5.6, 6.22, and 6.18, we now obtain the locally
uniform invertibility in (33).

######  Lemma 6.24.

For every $u\in L^{p}(\Omega)$, there exist constants $r,C>0$ such that, for
all $h\in L^{p}(\Omega)$ with $\|h\|_{L^{p}(\Omega)}\leq r$ and for all
$\mathcal{G}\in DF(\Phi(u+h),u+h)$, the operator
$\mathcal{G}_{1}\in\mathscr{L}(\mathcal{H},\mathcal{H})$ is continuously
invertible and

(37) $\sup_{\mathcal{G}\in
DF(\Phi(u+h),u+h)}\|\mathcal{G}_{1}^{-1}\|_{\mathscr{L}(\mathcal{H},\mathcal{H})}\leq
C\qquad\forall h\in L^{p}(\Omega),\|h\|_{L^{p}(\Omega)}\leq r.$

######  Proof 6.25.

From Definitions 6.17 and 6.18, we obtain that the operators $\mathcal{G}_{1}$
appearing in $DF(\Phi(u+h),u+h)$ have precisely the form (17). From Lemma
6.22, it follows further that we can find a constant $r>0$ such that the
subspaces of $H^{-1}(\Omega)$ appearing in the formulas for the operators
$\mathcal{G}_{1}$ enclose an angle whose sine is bounded from below by a
number $\bar{s}_{0}\in(0,1]$ for all $h\in L^{p}(\Omega)$,
$\|h\|_{L^{p}(\Omega)}\leq r$. By invoking Theorem 5.6 and (16), the assertion
of the lemma now follows immediately with
$C:=4(1-(1-\bar{s}_{0}^{2})^{1/2})^{-1}$. This completes the proof.

Having established (37), we can turn our attention to the proof of (32). We
begin with an auxiliary result on the regularity properties of the solution of
the Poisson problem away from the support of the right-hand side.

######  Lemma 6.26.

Let $Z\subset\Omega$ be nonempty and compact and let $\zeta\in
C_{c}^{\infty}(\Omega)$ be given such that $\zeta\equiv 0$ holds on a
neighborhood of $Z$. Identify $L^{p}(Z)$ with a subset of
$L^{p}(\Omega)\subset H^{-1}(\Omega)$ via the canonical embedding
$L^{p}(Z)\hookrightarrow L^{p}(\Omega)$. Then there exists a constant $C>0$
such that

$\displaystyle\lVert\zeta(-\Delta)^{-1}f\rVert_{W^{2,p}(\Omega)}\leq C\lVert
f\rVert_{L^{1}(Z)}\qquad\forall f\in L^{p}(Z).$

######  Proof 6.27.

We assume that $\zeta\not\equiv 0$. (For $\zeta\equiv 0$, the assertion is
trivial.) Let $f\in L^{p}(Z)$ be given and define $y:=(-\Delta)^{-1}f\in
H_{0}^{1}(\Omega)$. By a duality argument, we get

(38) $\displaystyle\lVert y\rVert_{L^{1}(\Omega)}$ $\displaystyle=\sup_{\lVert
z\rVert_{L^{\infty}(\Omega)}\leq
1}\int_{\Omega}yz\mathop{}\\!\mathrm{d}x=\sup_{\lVert
z\rVert_{L^{\infty}(\Omega)}\leq
1}\int_{\Omega}f(-\Delta)^{-1}z\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\sup_{\lVert z\rVert_{L^{\infty}(\Omega)}\leq 1}\lVert
f\rVert_{L^{1}(\Omega)}\lVert(-\Delta)^{-1}z\rVert_{L^{\infty}(\Omega)}\leq
C\lVert f\rVert_{L^{1}(\Omega)},$

where, in the last step, we used the estimate
$\lVert(-\Delta)^{-1}z\rVert_{L^{\infty}(\Omega)}\leq C\lVert
z\rVert_{L^{\infty}(\Omega)}$ that is a special case of Proposition 3.3. As
the support $\mathrm{supp}(\zeta)$ of $\zeta$ and the set $Z$ are nonempty,
disjoint, and compact, we can further find $\varepsilon>0$ such that the set

$\displaystyle
U:=\\{x\in\Omega\nonscript\;|\nonscript\;\mathopen{}\operatorname{dist}(x,\mathrm{supp}(\zeta))<\varepsilon\\}$

does not intersect $\partial\Omega\cup Z$. Now, the solution
$y=(-\Delta)^{-1}f\in H_{0}^{1}(\Omega)$ satisfies

$\displaystyle(y,-\Delta
v)_{L^{2}(\Omega)}=(y,v)_{H_{0}^{1}(\Omega)}=(f,v)_{L^{2}(\Omega)}=0\qquad\forall
v\in C_{c}^{\infty}(U).$

By Weyl’s lemma, see [26, Proposition 2.14], this implies that $y$ is harmonic
on $U$. From the mean value property of harmonic maps, it now follows that the
convolution $\eta\mathbin{\star}y$ of $y$ with a rotationally symmetric
mollifier $\eta$ of radius smaller $\varepsilon$ satisfies
$y=\eta\mathbin{\star}y$ on $\mathrm{supp}(\zeta)$ and that the partial
derivatives of $y$ satisfy
$\partial^{\alpha}y=(\partial^{\alpha}\eta)\mathbin{\star}y$ on
$\mathrm{supp}(\zeta)$ for all $\alpha\in\mathbb{N}_{0}^{d}$. Owing to Young’s
inequality for convolutions, this gives

$\displaystyle\lVert\partial^{\alpha}y\rVert_{L^{\infty}(\mathrm{supp}(\zeta))}\leq\lVert
y\rVert_{L^{1}(U)}\lVert\partial^{\alpha}\eta\rVert_{L^{\infty}(\mathbb{R}^{d})}=C_{\alpha}\lVert
y\rVert_{L^{1}(U)}.$

Together with the product rule and (38), this finishes the proof.

Next, we prove a result that allows us to bridge the regularity gap in (31).

######  Lemma 6.28.

Let $u\in L^{p}(\Omega)$ be given and let $r$, $Z_{\psi}$, and $Z^{\phi}$ be
as in Lemma 6.20. Then there exist $\zeta_{\psi},\zeta^{\phi}\in
C_{c}^{\infty}(\Omega)$ satisfying $0\leq\zeta_{\psi},\zeta^{\phi}\leq 1$ in
$\Omega$, $\zeta_{\psi}\equiv 1-\zeta^{\phi}\equiv 0$ on a neighborhood of
$Z_{\psi}$, and $\zeta^{\phi}\equiv 1-\zeta_{\psi}\equiv 0$ on a neighborhood
of $Z^{\phi}$ such that

(39) $\displaystyle M_{\psi}(u+h-\Phi^{\phi}(u+h))$
$\displaystyle=M_{\psi}(u+h-(-\Delta)\zeta^{\phi}(-\Delta)^{-1}\Phi^{\phi}(u+h)),$
$\displaystyle M^{\phi}(u+h-\Phi_{\psi}(u+h))$
$\displaystyle=M^{\phi}(u+h-(-\Delta)\zeta_{\psi}(-\Delta)^{-1}\Phi_{\psi}(u+h)),$
$\displaystyle DM_{\psi}(u+h-\Phi^{\phi}(u+h))$
$\displaystyle=DM_{\psi}(u+h-(-\Delta)\zeta^{\phi}(-\Delta)^{-1}\Phi^{\phi}(u+h)),$
$\displaystyle DM^{\phi}(u+h-\Phi_{\psi}(u+h))$
$\displaystyle=DM^{\phi}(u+h-(-\Delta)\zeta_{\psi}(-\Delta)^{-1}\Phi_{\psi}(u+h)),$

for all $h\in L^{p}(\Omega)$ with $\lVert h\rVert_{L^{p}(\Omega)}\leq r$.

######  Proof 6.29.

As $Z_{\psi}$ and $Z^{\phi}$ are nonempty, compact, and disjoint subsets of
$\Omega$, we can find $\zeta_{\psi}\in C_{c}^{\infty}(\Omega)$ such that
$0\leq\zeta_{\psi}\leq 1$ in $\Omega$, $\zeta_{\psi}\equiv 1$ on a
neighborhood of $Z^{\phi}$, and $\zeta_{\psi}\equiv 0$ on a neighborhood of
$Z_{\psi}$. Suppose that $h\in L^{p}(\Omega)$ satisfying $\lVert
h\rVert_{L^{p}(\Omega)}\leq r$ is given. We define
$y(h):=S_{\psi}^{\phi}(u+h)$, $\lambda_{\psi}(h):=\Phi_{\psi}(u+h)$, and
$\lambda^{\phi}(h):=\Phi^{\phi}(u+h)$. Then it follows from (30) that $-\Delta
y(h)=u+h-\lambda_{\psi}(h)-\lambda^{\phi}(h).$ Let $\tilde{y}(h)\in
H_{0}^{1}(\Omega)$ be the solution of

$\displaystyle-\Delta\tilde{y}(h)=u+h-(-\Delta)\zeta_{\psi}(-\Delta)^{-1}\lambda_{\psi}(h)-\lambda^{\phi}(h).$

Then it holds

$\displaystyle
y(h)-\tilde{y}(h)=-(1-\zeta_{\psi})(-\Delta)^{-1}\lambda_{\psi}(h).$

In particular, $y(h)=\tilde{y}(h)$ a.e. on $Z^{\phi}$ since
$\zeta_{\psi}\equiv 1$ on $Z^{\phi}$. Moreover, the comparison principle and
the inequality $\lambda_{\psi}(h)\leq 0$ a.e. in $\Omega$ imply that
$y(h)-\tilde{y}(h)\geq 0$ a.e. in $\Omega$. Thus, $\tilde{y}(h)\leq
y(h)\leq\phi$ a.e. in $\Omega$. From (29), we further get
$(y(h)-\phi)\lambda^{\phi}(h)=0$ a.e. in $\Omega$. Since
$\\{y(h)=\phi\\}\subset Z^{\phi}$, $y(h)=\tilde{y}(h)$ a.e. on $Z^{\phi}$, and
$\mathrm{supp}(\lambda^{\phi}(h))\subset\\{y(h)=\phi\\}$, this implies
$(\tilde{y}(h)-\phi)\lambda^{\phi}(h)=0$ a.e. on $\Omega$. Thus,

$\displaystyle\lambda^{\phi}(h)\geq 0\text{ a.e.\ in
}\Omega,\qquad\tilde{y}(h)-\phi\leq 0\text{ a.e.\ in
}\Omega,\qquad(\tilde{y}(h)-\phi)\lambda^{\phi}(h)=0\text{ a.e.\ in }\Omega.$

Together with the definition of $\tilde{y}(h)$, it follows that

(40) $\displaystyle\tilde{y}(h)$
$\displaystyle=S^{\phi}(u+h-(-\Delta)\zeta_{\psi}(-\Delta)^{-1}\lambda_{\psi}(h)),$
$\displaystyle\lambda^{\phi}(h)$
$\displaystyle=M^{\phi}(u+h-(-\Delta)\zeta_{\psi}(-\Delta)^{-1}\lambda_{\psi}(h)).$

Due to the definitions $\lambda_{\psi}(h):=\Phi_{\psi}(u+h)$ and
$\lambda^{\phi}(h):=\Phi^{\phi}(u+h)$ and the identity

$\displaystyle\Phi^{\phi}(u+h)=\max(0,M_{\psi}^{\phi}(u+h))=M^{\phi}\left(u+h-\Phi_{\psi}(u+h)\right)$

obtained from (30) and Proposition 6.13, this proves the second line of (39).
Since $\\{y(h)=\phi\\}\subset Z^{\phi}$, $y(h)=\tilde{y}(h)$ on $Z^{\phi}$,
and $\tilde{y}(h)\leq y(h)\leq\phi$ in $\Omega$, we further have
$\\{y(h)=\phi\\}=\\{\tilde{y}(h)=\phi\\}$ and, as a consequence,
$\\{y(h)<\phi\\}=\\{\tilde{y}(h)<\phi\\}$. In view of Definition 6.7, (40),
and again Proposition 6.13, this yields

$\displaystyle DS^{\phi}(u+h-\lambda_{\psi}(h))$
$\displaystyle=\big{\\{}\mathcal{S}(O)\,\left|~{}O\text{ open and
}\\{y(h)<\phi\\}\subset
O\subset\Omega\setminus\mathrm{supp}\left(\lambda^{\phi}(h)\right)\big{\\}}\right.$
$\displaystyle=\big{\\{}\mathcal{S}(O)\,\left|~{}O\text{ open and
}\\{\tilde{y}(h)<\phi\\}\subset
O\subset\Omega\setminus\mathrm{supp}\left(\lambda^{\phi}(h)\right)\big{\\}}\right.$
$\displaystyle=DS^{\phi}(u+h-(-\Delta)\zeta_{\psi}(-\Delta)^{-1}\lambda_{\psi}(h)).$

Due to Definition 6.7, this establishes the fourth line of (39). The
assertions for $M_{\psi}$ in the first and the third line of (39) are proved
completely analogously.

By combining Lemmas 6.26 and 6.28, we now obtain (32) and, ultimately, the
Newton differentiability of $\Phi$ as a function from $L^{p}(\Omega)$ to
$\mathcal{H}$.

######  Proposition 6.30.

The function $\Phi$ in (30) is Newton differentiable as a map from
$L^{p}(\Omega)$ to $\mathcal{H}$ with Newton derivative

(41) $\displaystyle D\Phi\colon
L^{p}(\Omega)\rightrightarrows\mathscr{L}(L^{p}(\Omega),\mathcal{H}),\quad
u\mapsto\\{-\mathcal{G}_{1}^{-1}\mathcal{G}_{2}\mid\mathcal{G}\in
DF(\Phi(u),u)\\}.$

######  Proof 6.31.

Let $u\in L^{p}(\Omega)$ be given and let $r>0$ be chosen such that the
assertions of Lemmas 6.20, 6.24, and 6.28 hold for some
$Z_{\psi},Z^{\phi}\subset\Omega$, $C>0$, and $\zeta_{\psi},\zeta^{\phi}\in
C_{c}^{\infty}(\Omega)$. Suppose further that $h\in L^{p}(\Omega)$ with
$0<\|h\|_{L^{p}(\Omega)}\leq r$ is given. Then we obtain from the arguments in
the first half of (31) and Definition 6.17 that

(42) $\displaystyle\sup_{G\in
D\Phi(u+h)}\frac{\lVert\Phi(u+h)-\Phi(u)-Gh\rVert_{\mathcal{H}}}{\lVert
h\rVert_{L^{p}(\Omega)}}$ $\displaystyle\leq C\sup_{\mathcal{G}\in
DF(\Phi(u+h),u+h)}\frac{\lVert\mathcal{G}_{1}(\Phi(u+h)-\Phi(u))+\mathcal{G}_{2}h\rVert_{\mathcal{H}}}{\lVert
h\rVert_{L^{p}(\Omega)}}$ $\displaystyle=C\sup_{\begin{subarray}{c}G_{\psi}\in
DM_{\psi}(u+h-\Phi^{\phi}(u+h)),\\\ G^{\phi}\in
DM^{\phi}(u+h-\Phi_{\psi}(u+h))\end{subarray}}\frac{\left\|\begin{pmatrix}\Phi_{\psi}(u+h)-\Phi_{\psi}(u)+G_{\psi}(\Phi^{\phi}(u+h)-\Phi^{\phi}(u)-h)\\\
\Phi^{\phi}(u+h)-\Phi^{\phi}(u)+G^{\phi}(\Phi_{\psi}(u+h)-\Phi_{\psi}(u)-h)\end{pmatrix}\right\|_{\mathcal{H}}}{\lVert
h\rVert_{L^{p}(\Omega)}}$

holds. Let us again define $\lambda^{\phi}(h):=\Phi^{\phi}(u+h)$,
$\lambda^{\phi}(0):=\Phi^{\phi}(u)$, and $y(h):=S_{\psi}^{\phi}(u+h)$. Due to
Lemmas 6.26 and 6.28, (30), and Proposition 6.13, we have

$\Phi_{\psi}(u+h)-\Phi_{\psi}(u)=M_{\psi}(u+h-\lambda^{\phi}(h))-M_{\psi}(u-\lambda^{\phi}(0))=M_{\psi}(\tilde{u}+\tilde{g}(h))-M_{\psi}(\tilde{u})$

and

$\displaystyle
DM_{\psi}(u+h-\lambda^{\phi}(h))=DM_{\psi}(\tilde{u}+\tilde{g}(h))$

with

$\displaystyle\tilde{u}$
$\displaystyle:=u-(-\Delta)\zeta^{\phi}(-\Delta)^{-1}\lambda^{\phi}(0)\in
L^{p}(\Omega),$ $\displaystyle\tilde{g}(h)$
$\displaystyle:=h+(-\Delta)\zeta^{\phi}(-\Delta)^{-1}(\lambda^{\phi}(0)-\lambda^{\phi}(h))\in
L^{p}(\Omega).$

For every $G_{\psi}\in DM_{\psi}(u+h-\lambda^{\phi}(h))$ with generating set
$O_{\psi}$ as in Definition 6.7, we further obtain from $1-\zeta^{\phi}\equiv
0$ on $Z_{\psi}$, $\\{y(h)>\psi\\}\subset O_{\psi}$, and
$\\{y(h)=\psi\\}\subset Z_{\psi}$ that
$(1-\zeta^{\phi})(-\Delta)^{-1}(\lambda^{\phi}(h)-\lambda^{\phi}(0))\in
H_{0}^{1}(O_{\psi})$ and, thus,

$\displaystyle\mathcal{S}(O_{\psi})(-\Delta)(1-\zeta^{\phi})(-\Delta)^{-1}(\lambda^{\phi}(h)-\lambda^{\phi}(0))=(1-\zeta^{\phi})(-\Delta)^{-1}(\lambda^{\phi}(h)-\lambda^{\phi}(0)).$

Due to $G_{\psi}=\mathrm{Id}-(-\Delta)\mathcal{S}(O_{\psi})$, this yields

$\displaystyle G_{\psi}(\lambda^{\phi}(h)-\lambda^{\phi}(0)-h)$
$\displaystyle=G_{\psi}(-\Delta)\zeta^{\phi}(-\Delta)^{-1}(\lambda^{\phi}(h)-\lambda^{\phi}(0))-G_{\psi}h$
$\displaystyle=-G_{\psi}\tilde{g}(h).$

If we put all of the above together, then it follows that

(43) $\displaystyle\sup_{G_{\psi}\in
DM_{\psi}(u+h-\Phi^{\phi}(u+h))}\left\|\Phi_{\psi}(u+h)-\Phi_{\psi}(u)+G_{\psi}\left(\Phi^{\phi}(u+h)-\Phi^{\phi}(u)-h\right)\right\|_{H^{-1}(\Omega)}$
$\displaystyle=\sup_{G_{\psi}\in
DM_{\psi}(\tilde{u}+\tilde{g}(h))}\left\|M_{\psi}(\tilde{u}+\tilde{g}(h))-M_{\psi}(\tilde{u})-G_{\psi}\tilde{g}(h)\right\|_{H^{-1}(\Omega)}.$

Note that, due to the Lipschitz continuity of $\Phi^{\phi}$ from
$L^{p}(\Omega)$ to $L^{1}(\Omega)$ (see Lemma 6.4 and (30)), the fact that
$\lambda^{\phi}(h)-\lambda^{\phi}(0)=\Phi^{\phi}(u+h)-\Phi^{\phi}(u)$ is
supported only on $Z^{\phi}$, and Lemma 6.26, we know that there exist
constants $C_{1},C_{2}>0$ such that

$\displaystyle\|\tilde{g}(h)\|_{L^{p}(\Omega)}$
$\displaystyle\leq\|h\|_{L^{p}(\Omega)}+\|(-\Delta)\zeta^{\phi}(-\Delta)^{-1}(\lambda^{\phi}(0)-\lambda^{\phi}(h))\|_{L^{p}(\Omega)}$
$\displaystyle\leq\|h\|_{L^{p}(\Omega)}+C_{1}\|\lambda^{\phi}(0)-\lambda^{\phi}(h)\|_{L^{1}(\Omega)}$
$\displaystyle\leq(1+C_{1}C_{2})\|h\|_{L^{p}(\Omega)}.$

In combination with the Newton differentiability of $M_{\psi}$ as a function
from $L^{p}(\Omega)$ to $H^{-1}(\Omega)$, (43), and the fact that $h$ was
arbitrary, this allows us to conclude that

$\displaystyle\sup_{G_{\psi}\in
DM_{\psi}(u+h-\Phi^{\phi}(u+h))}\frac{\left\|\Phi_{\psi}(u+h)-\Phi_{\psi}(u)+G_{\psi}\left(\Phi^{\phi}(u+h)-\Phi^{\phi}(u)-h\right)\right\|_{H^{-1}(\Omega)}}{\|h\|_{L^{p}(\Omega)}}$
$\displaystyle\leq(1+C_{1}C_{2})\sup_{G_{\psi}\in
DM_{\psi}(\tilde{u}+\tilde{g}(h))}\frac{\left\|M_{\psi}(\tilde{u}+\tilde{g}(h))-M_{\psi}(\tilde{u})-G_{\psi}\tilde{g}(h)\right\|_{H^{-1}(\Omega)}}{\|\tilde{g}(h)\|_{L^{p}(\Omega)}}\to
0$

holds for $(0,r]\ni\|h\|_{L^{p}(\Omega)}\to 0$, where, in the case
$\tilde{g}(h)=0$, the second fraction is to be understood as zero. By
proceeding analogously with the second component in (42) and by plugging
everything in, we now obtain that

$\sup_{G\in
D\Phi(u+h)}\frac{\lVert\Phi(u+h)-\Phi(u)-Gh\rVert_{\mathcal{H}}}{\lVert
h\rVert_{L^{p}(\Omega)}}\to 0\quad\text{ for }0<\|h\|_{L^{p}(\Omega)}\to 0$

holds as desired. This completes the proof.

As $\Phi$ is the function that maps $u\in L^{p}(\Omega)$ to the positive and
the negative part of the multiplier $M_{\psi}^{\phi}(u)\in
L^{p}(\Omega)\subset H^{-1}(\Omega)$ and since the description of $D\Phi$ in
(41) is cumbersome, Proposition 6.30 is not well suited for applications. To
refine it, we can use (18). Doing so yields the following second main result
of this paper which shows that the solution mapping $S_{\psi}^{\phi}$ of the
bilateral obstacle problem is Newton differentiable with a nicely describable
Newton derivative in the situation of Section 6.

######  Theorem 6.32 ($H_{0}^{1}$-Newton differentiability for the bilateral
obstacle problem).

The operator $S_{\psi}^{\phi}$ is Newton differentiable as a function
$S_{\psi}^{\phi}\colon L^{p}(\Omega)\to H_{0}^{1}(\Omega)\cap L^{q}(\Omega)$
for all $1\leq q<\infty$ with the derivative $\smash{DS_{\psi}^{\phi}\colon
L^{p}(\Omega)\rightrightarrows}\mathscr{L}(L^{p}(\Omega),H_{0}^{1}(\Omega))$
given by

(44) $\displaystyle DS_{\psi}^{\phi}(u):=\Big{\\{}\mathcal{S}(O)\,\Big{|}~{}O$
$\displaystyle\text{ open and }\\{\psi<S_{\psi}^{\phi}(u)<\phi\\}\subset
O\subset\Omega\setminus\mathrm{supp}\big{(}M_{\psi}^{\phi}(u)\big{)}\Big{\\}}.$

Here, the inactive set is defined w.r.t. the $C(\Omega)$-representatives of
$S_{\psi}^{\phi}(u)$, $\psi$, and $\phi$, and the symbol
$\mathrm{supp}(\cdot)$ denotes the support in the sense of Borel measures on
$\Omega$.

######  Proof 6.33.

From Proposition 6.30 and (30), we obtain that $M_{\psi}^{\phi}\colon
L^{p}(\Omega)\to H^{-1}(\Omega)$ is Newton differentiable with derivative

(45) $\displaystyle
DM_{\psi}^{\phi}(u):=\left\\{-\begin{pmatrix}\mathrm{Id}&\mathrm{Id}\end{pmatrix}\mathcal{G}_{1}^{-1}\mathcal{G}_{2}\mid\mathcal{G}\in
DF(\Phi(u),u)\right\\}.$

Suppose now that an open set $O$ satisfying the conditions in (44) is given
and define

(46) $\displaystyle
O_{\psi}:=O\cup\\{S_{\psi}^{\phi}(u)>\psi\\}\qquad\text{and}\qquad
O^{\phi}:=O\cup\\{S_{\psi}^{\phi}(u)<\phi\\}.$

Then $O_{\psi}$ and $O^{\phi}$ satisfy the conditions in Definition 6.7 and
give rise to operators $G_{\psi}\in DM_{\psi}(u-\Phi^{\phi}(u))$ and
$G^{\phi}\in DM^{\phi}(u-\Phi_{\psi}(u))$, respectively; cf. Proposition 6.13.

Due to Lemma 6.18, $G_{\psi}$ and $G^{\phi}$ are precisely the orthogonal
projections $P_{W(G_{\psi})}$ and $P_{W(G^{\phi})}$ onto the spaces
$W(G_{\psi})$ and $W(G^{\phi})$ in (34). In combination with Definition 6.17,
(45), Lemma 6.22, and Theorem 5.6, this yields

$\displaystyle DM_{\psi}^{\phi}(u)$
$\displaystyle\ni\begin{pmatrix}\mathrm{Id}&\mathrm{Id}\end{pmatrix}\begin{pmatrix}\mathrm{Id}&P_{W(G_{\psi})}\\\
P_{W(G^{\phi})}&\mathrm{Id}\end{pmatrix}^{-1}\begin{pmatrix}P_{W(G_{\psi})}\\\
P_{W(G^{\phi})}\end{pmatrix}=P_{W(G_{\psi})+W(G^{\phi})},$

where $P_{W(G_{\psi})+W(G^{\phi})}$ denotes the orthogonal projection onto
$W(G_{\psi})+W(G^{\phi})$. From (34), [16, Theorem 4.5], and (46), it follows
further that

$\displaystyle W(G_{\psi})+W(G^{\phi})$ $\displaystyle=\left[-\Delta
H_{0}^{1}(O_{\psi})\right]^{\perp}+\left[-\Delta
H_{0}^{1}(O^{\phi})\right]^{\perp}$
$\displaystyle=\left[-\Delta\left(H_{0}^{1}(O_{\psi})\cap
H_{0}^{1}(O^{\phi})\right)\right]^{\perp}$ $\displaystyle=\left[-\Delta
H_{0}^{1}(O_{\psi}\cap O^{\phi})\right]^{\perp}$ $\displaystyle=\left[-\Delta
H_{0}^{1}(O)\right]^{\perp}.$

This shows that $\mathrm{Id}-P_{W(G_{\psi})+W(G^{\phi})}$ is the orthogonal
projection onto $-\Delta H_{0}^{1}(O)$ and, in view of (5) and Definition 6.6,
that $(-\Delta)^{-1}(\mathrm{Id}-P_{W(G_{\psi})+W(G^{\phi})})=\mathcal{S}(O)$.
If we combine all of the above, then we obtain that
$\mathcal{S}(O)\in(-\Delta)^{-1}(\mathrm{Id}-DM_{\psi}^{\phi}(u))$. Since
$S_{\psi}^{\phi}=(-\Delta)^{-1}(\mathrm{Id}-M_{\psi}^{\phi})$ holds by the
definition of $M_{\psi}^{\phi}$, the Newton differentiability of the operator
$S_{\psi}^{\phi}$ as a function from $L^{p}(\Omega)$ to $H_{0}^{1}(\Omega)$
with the derivative in (44) now follows immediately from the Newton
differentiability properties of $M_{\psi}^{\phi}$. It remains to prove that
the operator $S_{\psi}^{\phi}$ is also Newton differentiable as a function
from $L^{p}(\Omega)$ to $L^{q}(\Omega)$ for all $1\leq q<\infty$. This,
however, follows straightforwardly from the definition of Newton
differentiability, the fact that $S_{\psi}^{\phi}$ is globally Lipschitz from
$L^{p}(\Omega)$ to $L^{\infty}(\Omega)$ by Lemma 6.4, the fact that
$\\{\mathcal{S}(O)\mid O\subset\Omega\text{ nonempty, open}\\}$ is a bounded
subset of $\mathscr{L}(L^{p}(\Omega),L^{\infty}(\Omega))$ (which can easily be
checked by means of Definition 6.6 and the classical truncation arguments in
[20, Theorem B.2]), the Newton differentiability of $S_{\psi}^{\phi}$ as a map
from $L^{p}(\Omega)$ to $H_{0}^{1}(\Omega)$, and the dominated convergence
theorem; cf. the arguments in the proof of Theorem 4.4.

######  Remark 6.34.

  * •

By reversing the calculations in the proof of Theorem 6.32, one, of course,
also obtains that the map $M_{\psi}^{\phi}$ is Newton differentiable as a
function from $L^{p}(\Omega)$ to $H^{-1}(\Omega)$ with Newton derivative
$DM_{\psi}^{\phi}(u):=\mathrm{Id}-(-\Delta)DS_{\psi}^{\phi}(u)$.

  * •

The Newton derivative $DS_{\psi}^{\phi}(u)$ in (44) contains the elements of
$\partial_{B}^{sw}S_{\psi}^{\phi}(u)$ identified in [27, Theorem 6.2] as
special cases. Using the implicit function approach developed in this section,
it might be possible to show that all elements of the set
$DS_{\psi}^{\phi}(u)$ in (44) are, in fact, generalized derivatives in the
strong-weak Bouligand sense. We leave this topic for future research.

##  7 An application example from optimal control

We conclude this paper with an application example from optimal control which
illustrates how Theorem 6.32 can be used in practice. As the following
analysis parallels that in [11, sections 5, 6], we do not aim for the highest
level of generality and keep the discussion concise. The model problem that we
consider reads as follows:

(M) $\left\\{\,\,\begin{aligned} \text{Minimize}\quad&\frac{1}{2}\left\lVert
y-y_{D}\right\rVert_{L^{2}(\Omega)}^{2}+\frac{\nu}{2}\|u\|_{H_{0}^{1}(\Omega)}^{2}\\\
\text{w.r.t.}\quad&u,y\in H_{0}^{1}(\Omega),\\\
\text{s.t.}\quad&\mathopen{}-\Delta y=u\text{ and }\psi\leq u\leq\phi\text{ in
}\Omega.\end{aligned}\right.$

[standing assumptions for Section 7]

  * •

$\Omega\subset\mathbb{R}^{d}$, $d\in\\{1,2,3\\}$, is a bounded Lipschitz
domain;

  * •

$\nu>0$ and $y_{D}\in L^{2}(\Omega)$ are given;

  * •

$\psi$ and $\phi$ satisfy $\psi,\phi\in C(\bar{\Omega})\cap H^{1}(\Omega)$ and
$\Delta\psi,\Delta\phi\in L^{2}(\Omega)$, and there exists a number $c>0$ such
that (24) holds.

Note that the standing assumptions of Section 6 hold in the above situation
with $p=2$; cf. Remark 6.1. From standard arguments, we obtain the following
theorem.

######  Theorem 7.1.

Problem (M) possesses a unique optimal control $\bar{u}$ with associated state
$\bar{y}:=(-\Delta)^{-1}\bar{u}$. The optimal state $\bar{y}$ is uniquely
characterized by the equation

(47)
$\bar{y}-(-\Delta)^{-1}S_{\psi}^{\phi}\left(\nu^{-1}(-\Delta)^{-1}(y_{D}-\bar{y})\right)=0,$

where $S_{\psi}^{\phi}\colon H^{-1}(\Omega)\to H_{0}^{1}(\Omega)$ denotes the
solution map of (VI).

######  Proof 7.2.

The unique solvability follows from the direct method of the calculus of
variations and the strict convexity of (M). Equation (47) is obtained from the
exact same arguments as in [11, section 5]; see [11, Proposition 5.2, Equation
(5.4)].

The next lemma shows that (47) is amenable to a semismooth Newton method.

######  Lemma 7.3.

Let $DS_{\psi}^{\phi}$ be defined as in (44). The function

$\displaystyle Q\colon L^{2}(\Omega)\to L^{2}(\Omega),\qquad y\mapsto
y-(-\Delta)^{-1}S_{\psi}^{\phi}\left(\nu^{-1}(-\Delta)^{-1}(y_{D}-y)\right),$

is Newton differentiable with derivative

$\displaystyle
DQ(y):=\mathrm{Id}+\nu^{-1}(-\Delta)^{-1}DS_{\psi}^{\phi}\left[\nu^{-1}(-\Delta)^{-1}(y_{D}-y)\right](-\Delta)^{-1}\subset\mathscr{L}(L^{2}(\Omega),L^{2}(\Omega)).$

Further, for all $y\in L^{2}(\Omega)$ and $G\in DQ(y)$, $G^{-1}$ exists and
$\|G^{-1}\|_{\mathscr{L}(L^{2}(\Omega),L^{2}(\Omega))}\leq 1$.

######  Proof 7.4.

The Newton differentiability of $Q$ as a function from $L^{2}(\Omega)$ to
$L^{2}(\Omega)$ with derivative $DQ$ follows from the estimate
$2>\max(1,d/2)$, the linearity and continuity of $(-\Delta)^{-1}\colon
L^{2}(\Omega)\to H_{0}^{1}(\Omega)\hookrightarrow L^{2}(\Omega)$, and Theorem
6.32. To see that the elements of $DQ(y)$ are invertible for all $y\in
L^{2}(\Omega)$, we note that (44) and Definition 6.6 imply that, for every
$G\in DQ(y)$, there exists an open nonempty set $O\subset\Omega$ such that

$\displaystyle\left(z,Gz\right)_{L^{2}(\Omega)}$
$\displaystyle=\|z\|_{L^{2}(\Omega)}^{2}+\nu^{-1}\left(z,(-\Delta)^{-1}\mathcal{S}(O)(-\Delta)^{-1}z\right)_{L^{2}(\Omega)}$
$\displaystyle=\|z\|_{L^{2}(\Omega)}^{2}+\nu^{-1}\left((-\Delta)^{-1}z,\mathcal{S}(O)(-\Delta)^{-1}z\right)_{L^{2}(\Omega)}$
$\displaystyle=\|z\|_{L^{2}(\Omega)}^{2}+\nu^{-1}\left(\mathcal{S}(O)(-\Delta)^{-1}z,\mathcal{S}(O)(-\Delta)^{-1}z\right)_{H_{0}^{1}(\Omega)}$
$\displaystyle\geq\|z\|_{L^{2}(\Omega)}^{2}\qquad\forall z\in L^{2}(\Omega).$

In combination with the lemma of Lax–Milgram, this establishes the claim.

The fact that the lemma of Lax–Milgram can be used in the situation of (47) to
establish the invertibility of the Newton derivatives is the main reason why
we have formulated the first-order optimality condition of (M) in $y$ and not
in $u$. Lemma 7.3 shows that it makes sense to consider a semismooth Newton
method for the solution of (47). The resulting algorithm can be seen in 7.4.

[semismooth Newton method for the solution of (M)]

1: Choose an initial guess $y_{0}\in L^{2}(\Omega)$ and a tolerance
$\texttt{tol}\geq 0$.

2: for $i=0,1,2,3,\ldots$ do

3: Calculate $z_{i}:=(-\Delta)^{-1}(y_{D}-y_{i})/\nu$,
$u_{i}:=S_{\psi}^{\phi}(z_{i})$, and $\tilde{y}_{i}:=(-\Delta)^{-1}u_{i}$.

4: if $\|y_{i}-\tilde{y}_{i}\|_{L^{2}(\Omega)}\leq\texttt{tol}$ then

5: STOP the iteration (convergence reached).

6: else

7: Choose an open set $O$ satisfying
$\\{\psi<S_{\psi}^{\phi}(z_{i})<\phi\\}\subset
O\subset\Omega\setminus\mathrm{supp}\big{(}M_{\psi}^{\phi}(z_{i})\big{)}$ with
$S_{\psi}^{\phi}$ and $M_{\psi}^{\phi}$ as in Theorem 6.32 and determine
$y_{i+1}\in L^{2}(\Omega)$ by solving  $\displaystyle
y_{i+1}+\nu^{-1}(-\Delta)^{-1}\mathcal{S}(O)(-\Delta)^{-1}y_{i+1}=\tilde{y}_{i}+\nu^{-1}(-\Delta)^{-1}\mathcal{S}(O)(-\Delta)^{-1}y_{i}.$

8: end if

9: end for

From the standard convergence analysis for the semismooth Newton method, we
now obtain the following result.

######  Theorem 7.5 (local $q$-superlinear convergence).

Let $\bar{u}$ be the optimal control of (M) and
$\bar{y}:=(-\Delta)^{-1}\bar{u}$ the associated optimal state. There exists
$r>0$ such that, for every $y_{0}\in L^{2}(\Omega)$ with
$\|y_{0}-\bar{y}\|_{L^{2}(\Omega)}<r$, 7.4 with $\texttt{tol}=0$ either
terminates after finitely many steps with $\bar{y}$ or produces a sequence
$\\{y_{i}\\}\subset L^{2}(\Omega)$ that converges $q$-superlinearly to
$\bar{y}$ in $L^{2}(\Omega)$.

######  Proof 7.6.

The update step in 7.4 is equivalent to $G_{i}(y_{i+1}-y_{i})=-Q(y_{i})$ with
$G_{i}\in DQ(y_{i})$ and with $Q$ and $DQ$ as in Lemma 7.3. From Lemma 7.3, we
further know that $Q$ is Newton differentiable from $L^{2}(\Omega)$ to
$L^{2}(\Omega)$ with derivative $DQ$ and that all $G$ in the image of $DQ$ are
invertible with $\|G^{-1}\|_{\mathscr{L}(L^{2}(\Omega),L^{2}(\Omega))}\leq 1$.
With this information at hand, the local $q$-superlinear convergence of the
iterates $\\{y_{i}\\}$ in $L^{2}(\Omega)$ to $\bar{y}$ follows completely
analogously to, e.g., the proof of [6, Theorem 3.4].

The results that are obtained when piecewise linear finite elements are used
to discretize (M) and a semismooth Newton method along the lines of 7.4 is
applied to the resulting finite-dimensional problem (analogously to [11,
section 6, Algorithm 6.1]) can be seen in Table 1 and Fig. 1. Here, we
considered the situation $\Omega=(0,1)^{2}$, $\nu=10^{-5}$,
$y_{D}(x_{1},x_{2})=10(-x_{1}-x_{2}+1)$, $\psi(x_{1},x_{2})=-5$,
$\phi(x_{1},x_{2})=5$, $y_{0}\equiv 0$, and $\texttt{tol}=10^{-12}$; the
underlying meshes were Friedrichs–Keller triangulations of width $h$; and the
inactive set was used as the set $O$ in step 7 of the algorithm. As Table 1
shows, the number of iterations that our Newton method requires to drive the
residue below $\texttt{tol}=10^{-12}$ does not change in our experiment as $h$
tends to zero. This so-called mesh-independence is characteristic for
algorithms whose convergence can be proved not only in the discrete but also
in the infinite-dimensional setting.

Table 1: Number of Newton iterations needed to drive the
$L^{2}(\Omega)$-residue below $\texttt{tol}=10^{-12}$ for various $h$.

mesh width $h$ | $\frac{1}{32}$ | $\frac{1}{64}$ | $\frac{1}{128}$ | $\frac{1}{256}$ | $\frac{1}{512}$ | $\frac{1}{1024}$
---|---|---|---|---|---|---
number of iterations | $2$ | $2$ | $2$ | $2$ | $2$ | $2$

Figure 1: Optimal control (left) and associated Lagrange multiplier of the bi-
obstacle problem (right) obtained from a finite element discretization of 7.4
with mesh width $h=1/64$.

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