Uniqueness Results of Semilinear Parabolic Equations in Infinite-Dimensional Hilbert Spaces

Abstract

This paper is devoted to the uniqueness of solutions for a class of nonhomogeneous stationary partial differential equations related to Hamilton–Jacobi-type equations in infinite-dimensional Hilbert spaces. Specifically, the uniqueness of the viscosity solution is established by employing the inf/sup-convolution approach in a separable infinite-dimensional Hilbert space. The proof is based on the Faedo–Galerkin approximate method by assuming the existence of a Hilbert–Schmidt operator and by employing modulus continuity and Lipschitz arguments. The results are of interest regarding the stochastic optimal control problem.

1. Introduction

The mathematical analysis of problems based on partial differential equations requires considerable attention when the dependent variable belongs to infinite-dimensional spaces, particularly Hilbert spaces [1,2]. In this context, it is important to establish the meanings of the various differential operators, such as Laplacian [3,4]. Propagation techniques need to be developed as well; see [5,6,7] and the reference cited therein.

This paper completes the program laid out in [8], which contains the existing result of the Hamilton–Jacobi-type equation. Specifically, the paper is devoted to the uniqueness solution result of the following Hamilton–Jacobi-type equation:

$$u-\Delta u+F\left(Du\right)=f\left(x\right),\mathrm{for}x\in H,$$

(1)

where H is the separable Hilbert space, x denotes a generic point, F and f are functions from H to $\mathbb{R}$, $Du\left(x\right)$ stands for the first-order Fréchet derivative of u, $\Delta$ denotes the N-dimensional Laplacian, and $\Delta u=\mathrm{Tr}\left(D^{2}u\right)=$ trace of the Hessian of u. Equation (1) is a special case of the following equation:

$$-\Delta u+\mathcal{F}(y,u(y),Du(y\left)\right)=0,$$

(2)

with $y=x$ and $\mathcal{F}(y,p,r)=r+F\left(p\right)-f\left(y\right)$.

The motivation for the study of the above equation derives from its connection with the partial stochastic differential equation. A typical example (meaningful for the control of stochastic differential equations) is provided by (see, for instance, [9])

$$F(x,\nabla \varphi )={\displaystyle \frac{1}{2}}{|\nabla \varphi |}^2-g\left(x\right),$$

(3)

where F is a smooth mapping from $H\times H$ into $\mathbb{R}$, $\varphi$ a mapping from H into $\mathbb{R}$, g are convex (with polynomial growth to infinity), $\nabla \varphi$ denotes the gradient of variables x with respect to $\varphi$, and the notation $|·|$ means the standard Euclidean norm on ${\mathbb{R}}^n$, i.e., $\left|p\right|={\left({\sum }_{i=1}^n{p}_i\right)}^{1/2}$ for all $p=(p_1,\dots ,p_n)\in {\mathbb{R}}^n$.

An example of (1) is the following Kolmogorov equation:

$$u_t-\mathrm{Tr}\left(A_0{D}^{2}u\right)+F\left(Du\right)=0\mathrm{for}x\in H,$$

(4)

that in infinite dimension has been extensively studied by many scholars, starting with the pioneering papers by Y Dalecky (see the monograph [10]). In (4), $\mathrm{Tr}$ denotes the trace, and $A_0$ may be a strictly positive linear operator (for instance, $A_0=I$). The operator $\mathrm{Tr}\left(A_0{D}^{2}u\right)$ is the natural generalization of the Laplacian to an infinite-dimensional space. The main difficulty in the study of this kind of equation is the fact that, in a Hilbert space H, the compactness is lost.

It is worth stressing that the most important and original point of this work is the presence of the Laplacian in (1) in the infinite-dimensional case and a priori. It is not possible to know how this term will behave.

The uniqueness of the viscosity solution of (1) is established by means of a comparison principle between the subsolution and supersolution. It is well known that first-order equations generally do not admit classical solutions due to the possibility of crossing characteristics. On the other hand, there are infinitely many Lipschitz continuous functions that satisfy the equation almost everywhere. Since the equation is nonlinear, we cannot define weak solutions via integration by parts. Thus, because of this possible degenerate (and the highly nonlinear form of the equations considered here), a classical solution (smooth) cannot be expected to exist. In this setting, the correct notion of weak solutions is the so-called viscosity solutions discovered by Crandall, Evans, and Lions [11,12]. Therefore, we work with the so-called viscosity solutions, which, roughly speaking, are functions that satisfy all the inequalities expected from the classical maximum principle when compared with smooth test functions. Precise definitions are provided below.

There are several motivations to develop infinite-dimensional theory. First of all, the theory is a natural part of functional analysis. It is worth stressing that G Da Prato and J. Zabczyk, et al. [5,13,14,15,16,17,18] have studied this kind of problem in relation to stochastic differential equations (under different assumptions) in a Hilbert space. In [19,20,21,22,23,24], the authors consider the Hamilton–Jacobi equation in connection with the stochastic partial differential equation

$$\left\{\begin{array}{c}dX\left(t\right)=\left[AX\right(t)+F(t,X\left(t\right),u\left(t\right)\left)\right]dt+dW\left(t\right),t⩾0\hfill \\ X\left(0\right)=x,x\in H\hfill \end{array}\right.$$

(5)

where H is a separable Hilbert space. Under suitable assumptions, Equation (5) has a unique mild solution $x\left(t\right)$. If $R_t$, $t⩾0$ denotes the corresponding transition semigroup $R_t\phi \left(x\right)=\mathbb{E}\left[\phi \left(X(t,x)\right)\right]$, then $\nu (t,x)=R_t\phi \left(x\right)$ is a solution to the following semilinear parabolic equation in infinitely many variables

$$\left\{\begin{array}{c}{\nu }_t+{\displaystyle \frac{1}{2}}\mathrm{Tr}\left[Q{D}_x^2\nu \right]+⟨Ax,D\nu ⟩-⟨F(\left(x\right),u_x(t,x))⟩=0\hfill \\ \nu (0,x)=\phi \left(x\right),x\in H,\hfill \end{array}\right.$$

(6)

Q is a positive self-adjoint operator in H, A is the infinitesimal generator of a strongly continuous semigroup in H, and $\phi :H\to \mathbb{R}$ is a continuous function.

In all these articles, a probabilistic approach or a semigroup approach using only functional analysis and PDE arguments has been used. As an example, the authors consider Q to be a positive nuclear operator in H (i.e., Hilbert–Schmidt operators of order 2) so that there exists a complete orthonormal system in H. By the closed graph theorem, it follows that, for each $t>0$, A is Hilbert–Schmidt, and, consequently, by the semigroup property, $e^{tA}$ is of trace class.

Notice that, from the point of view of Da Prato–Zabczyk theory, the stochastic differential Equation (4) is heavily degenerate since the noise is concentrated in a finite-dimensional subspace. We require diffusion matrices that probe Hilbert–Schmidt type and such that $\sum {\lambda }_i$ has a zero trace. These authors did not examine the impure Brownian but focused on the Brownian multiplied by the matrix, which seems to provide meaning to the equation regarding applying the variational type. Another approach is framed in semigroup theory [23,25]. For other literature on infinite-dimensional Hamilton–Jacobi equations derived from the connection with the stochastic partial differential equation, we refer the reader to [19,20,21,22,23,25,26,27,28,29].

All this being said, we note that the methods used in the above references to prove the existence of viscosity solutions are purely variational.

The novelty of our work is based on the analysis of the regularized problem of type (6) in which one has both the existence and uniqueness of solutions, solved by the so-called viscosity solutions; there are two scales: the regularity in B in this metric and the regularity x, but the regularity in B is not satisfactory.

In the present work, and differently from [5,13,14,15,16,17,18], the viscosity solutions are “B-continuous” ([30,31] and ([32], Chapter 3)) instead of weakly sequentially continuous. We extend to infinite-dimensional equations some techniques that are well known in finite dimensions. We also require some uniform continuity conditions on F. In order to simplify the proof, we assume that u and v are Lipschitz. It is worth noting that F may depend on the variable x. However, for the purposes of this work, we wish to limit the technical details. In order to prove comparison theorems for such viscosity solutions, we combine a perturbed optimization technique of finite-dimensional methods of maximum principle with the finite-dimensional reduction technique to obtain a kind of infinite-dimensional maximum principle, which is the main tool in the proof of the comparison theorem. Galerkin-type approximations are employed. We project the equation into the finite dimension in order to properly measure the chains of derivative $H_N$; see, among others, papers [33,34]. Galerkin’s method consists of approximating the functional space by a finite-dimensional Hilbert space for the same scalar product. Note that, for the purpose of studying (1), we need only to solve the transformed Equation (27) in $H_N$.

It is worth noting that, for studying the viscosity solutions for infinite-dimensional problems, there exist two methods: weak continuity or the localization technique. The latter is used in the proof of the main theorem.

The reduction to finite-dimensional spaces to produce appropriate test functions follows (this reduction technique was first introduced in [35]). The proof strategy is fairly clear: it consists of stating that we are able to project our equation on $H_N$ to essentially construct a $u_N$ from u and a $v_N$, which are essentially subsolutions and supersolutions of the projected equation, and we apply comparison results in the infinite dimension and deduce a comparison result.

It is worth stressing that the class of the problem (1) studied in the present paper has interesting probabilistic interpretations as Dynamic Programming for the control problem of the Bellman equation types (see, for example, [2,36,37,38]) and geographical storage problems [39]. Readers interested in mathematical problems based on Equation (1) are referred to papers [40,41,42] and the references therein.

The present paper is laid out as follows:

-

Section 2 contains preliminary material regarding the infinite-dimensional Hamilton–Jacobi equation, including the passage from finite-dimensional to infinite-dimensional problems. The section also contains some technical assumptions regarding the Hamiltonian F and the function f; some a priori estimations on $Du$ are also recalled.

-

Section 3 deals with the statements of our results, including the main theorem.

-

Section 4 presents the proofs of the main results of the paper. Note that the appropriate mathematical framework for the study of infinite-dimensional processes is the theory of Hilbert–Schmidt operators. Thereby, our proof is obtained by assuming the existence of a Hilbert–Schmidt operator and by employing modulus continuity and Lipschitz arguments [43].

2. Preliminaries

2.1. Notations and Setting of the Problem

In this section, we introduce our main notation and recall some basic facts about infinite-dimensional problems. Throughout this paper, $H=(H,⟨·,·⟩,\parallel ·\parallel )$ is a real separable infinite-dimensional Hilbert space. H is identified with its dual $H^{*}$ (Riesz Theorem), this for the convenience of notation. A point x of H is identified with ${\displaystyle x=\sum _{i=1}^{\infty }{x}_i{e}_i}$, where $\left(e_i\right)$ is a canonical orthonormal basis.

Let $H_1\subset H_2\subset \dots$ be finite-dimensional subspaces of H such that $\overline{{\bigcup }_{N=1}^{\infty }{H}_N}=H$. Given $N\in \mathbb{N}$, $N>1$, denote by $P_N$ the orthogonal projection onto $H_N$, let $Q_N=I-P_N$, and $H_N^{\perp }=Q_{N}H$. We then have an orthogonal decomposition $H=H_N\times H_N^{\perp }$, and we denote by $x_N$ an element of $H_N$ and by $x_N^{\prime }$ an element of $H_N^{\perp }$. Actually, $H_N$, coincides with the vector space generated by $e_1,\dots ,e_N$; that is, $H_N=\mathrm{Span}(e_1,\dots ,e_N)$ is the vector space generated by the N-first coordinates.

In addition, the spaces $H_N$ and ${\mathbb{R}}^N$ are isomorphic spaces. Note that completeness implies that, for every $x\in H$, one has $P_{N}x\to x$ as $N\to \infty$. With this identification we can interpret the Fréchet derivatives $Du\left(x\right)$ as an element of H equipped with norm ${\parallel x\parallel }_B={⟨Bx,x⟩}^{{\displaystyle \frac{1}{2}}}$. Finally, we write indifferently ${|·|}_B$ or $\parallel ·\parallel$. If Y, Z are real separable Hilbert spaces, a bounded linear operator $\mathcal{O}:Z\to Y$ is Hilbert–Schmidt, ${\displaystyle \sum _{i=0}^{\infty }{\left|\mathcal{O}f\right|}_Y^2<\infty }$, for any orthonormal basis $\{f_1,f_2,\dots \}$ of Z.

Reduction to Finite Dimensions

The solutions $u_N$ of Equation (1) are not quite defined in $H_N$ because we have $H_N$, which is the vector space generated by $\mathrm{Span}(e_1,\dots ,e_N)$. It is worth noting that a point $x=(x_1,\dots ,x_N)$ in the Hilbert space is a point in which all the coordinates after $x_N$ are equal to 0:

$$x=(x_1,\dots ,x_N),x=(x_1,\dots ,x_N,0,\dots ).$$

More specifically, if we take a point y in $H_N$, there is a canonical injection that consists of completing it with 0

$$i:y\in H_N⟶(y,0,\dots ,0).$$

Each time we are in $H_N$, we consider that we are in ${\mathbb{R}}^N$ with $x=(x_1,\dots ,x_N)$.

We conclude this section by briefly listing some definitions and notations that are used throughout this paper. In view of the fact that we are dealing with semicontinuous functions, it necessary to have a notation that records function values.

Definition 1. 

Let B be a bounded linear operator in Ω.

A continuous function $u:\Omega \to \mathbb{R}$ is said to be B-upper-semicontinuous on $\Omega \subset [0,T]\times H$ if $u(t,x)⩾lim\underset{n\to \infty }{sup}u(t_n,x_n)$ whenever $t_n\to t$, $x_n\rightharpoonup x$, $B{x}_n\to Bx$, $(t,x)\in \Omega$.

A continuous function $u:\Omega \to \mathbb{R}$ is said to be B-lower-semicontinuous on $\Omega \subset [0,T]\times H$ if $u(t,x)⩾lim\underset{n\to \infty }{inf}u(t_n,x_n)$ whenever $t_n\to t$, $x_n\rightharpoonup x$, $B{x}_n\to Bx$, $(t,x)\in \Omega$.

A function u is said to be B-continuous on Ω if it is B-upper-semicontinuous and B-lower-semicontinuous on Ω.

We use ⇀ to denote the weak convergence in H. Let $\Omega$ be a subset of a normed vector space. We employ the following functions spaces:

• $C(\Omega )=\{u:\Omega ⟼\mathbb{R}|u \mathrm{is} \mathrm{continuous}\}$
• Let $UC(\Omega )=\{u:\Omega ⟼\mathbb{R}|u\mathrm{be} \mathrm{uniformly} \mathrm{continuous}\}$
• Let $BUC(\Omega )=\{u\in UC(\Omega \left)\right|u \mathrm{be} \mathrm{bounded}\}$
• $C_b(\Omega )=\{u:\Omega ⟼\mathbb{R}|\mathrm{continuous} \mathrm{and} \mathrm{bounded} \mathrm{on} \mathrm{bounded} \mathrm{subsets} \mathrm{of} \Omega \}$
• $C^{1,1}(\Omega )=\{u:\Omega ⟼\mathbb{R}|\nabla u\mathrm{is} \mathrm{Lipschitz} \mathrm{continuous}\}$
• $C_b^1(\Omega )=\{u\in C^1(\Omega |v\nabla v\mathrm{bounded} \mathrm{on} \Omega \}$
• $C_b^{1,1}(\Omega )=\{u\in C_b^1(\Omega )|\nabla u\mathrm{is} \mathrm{Lipschitz} \mathrm{in} \Omega \}$.
• In particular, $C_b^{1,1}\left(H\right)$ is dense in $C_b\left(H\right)$; see [44,45].

A nondecreasing, continuous, and subadditive function $\omega :[0,\infty )\to [0,\infty )$ satisfying $\omega \left(0\right)=0$ is called a modulus of continuity.

2.2. Statement of the Main Results

By means of explicit inf-sup-convolution formulas, we prove that bounded uniformly continuous scalar functions defined in a Hilbert space H can be uniformly approximated by functions belonging to class $C^{1,1}\left(H\right)$.

We focus on the uniqueness of viscosity solutions of (1). The uniqueness of solutions in the viscosity theory is typically a consequence of a comparison theorem, which ensures that, under certain conditions, a subsolution is always less than or equal to a supersolution. Our definition is based on the notion of the so-called B-continuous viscosity solution, which was introduced for first-order equations in [30,31].

As far as Hamiltonian F and function f are concerned, we assume the following structure conditions:

(H1) 

The Hamiltonian F and f are regular in the Hilbert space H, at least $C^{1,1}(H,\mathbb{R})$; and in addition

(H2) 

$\exists \mu \in C^2\left(H\right)$, radial, nondecreasing, non-negative, $\mu \to \infty$ as $\parallel x\parallel \to \infty$, and $D\mu$ is bounded and

$$F(Du+\gamma D\mu (x\left)\right)⩾F\left(Du\right)-\omega \left(\gamma \right),$$

(7)

for all x, $\forall \gamma ⩾0$, where $\omega$ is a local modulus.

(H3) 

There exists a strictly positive symmetric operator B of Hilbert–Schmidt type (i.e., $\mathrm{Tr}\left(B^2\right)<\infty$) such that

$${\displaystyle \underset{\epsilon \to 0^{+}}{lim}sup\{f\left(x\right)-f\left(y\right)|\left|\right|B(x-y)|⩽\epsilon \}=0,}$$

that is

$$\left|f\right(x)-f(y\left)\right|⩽\omega \left(\right|B(x-y)\left|\right),\forall x,y\in H,$$

(8)

where $\omega$ is continuously increasing on $[0,+\infty [$ with $\omega \left(0\right)=0$.

If the modulus of continuity $\omega$ can be chosen as linear, then $B^{-1}\nabla u$ is bounded on H and $\mathrm{Tr}\left(\right|D^{2}u\left|\right)$ is bounded on H.

For control of the Laplacian, we make the following assumption:

(H4) 

For a regular test function $\phi \in H$ and Hilbert–Schmidt operator B, such that $\mathrm{Tr}\left(B^2\right)<\infty$, there exists a constant $C<\infty$ such that

$$|B^{-1}{D}^2\phi B^{-1}|⩽C.$$

(9)

In other words,

$$|D^2\phi |⩽C{B}^2.$$

(10)

• Essentially, this property holds that the Laplacian is a bounded function; $D\phi \in BUC\left(H\right)$ since a finite sum of derivatives remain in $BUC\left(H\right)$. This condition is crucial to pass to the limit in the Laplacian.

(H5) 

We assume that $\omega$ is a linear continuity modulus. More specifically,

$$\omega \left(z\right)=Lz,L>0,$$

(11)

where L is the Lipschitz modulus.

We always make this tacit Lipschitz condition in the sequel.

Attenuation Assumption. A major role in what follows is played by the attenuation operator C, which we need in our investigation. Key to our proof is the fact that

(H6) 

There exists a diagonalizable Hilbert–Schmidt type operator C with respect to the orthonormal basis with the corresponding eigenvalues (${\nu }_1,\dots ,{\nu }_N,\dots )$ such that, if $({\mu }_1,\dots ,{\mu }_N,\dots )$ are eigenvalues of the operator B, then

$${\displaystyle {\nu }_i>0,\frac{{\nu }_i}{{\mu }_i}\to +\infty \mathrm{as}i\to \infty ,}$$

(12)

and ${\nu }_i$ values are square-summable but less summable than the ${\mu }_i$.

In order to indicate why (H6) is an interesting hypothesis, let us quickly recall some facts. The assumption (H6) deserves some remarks:

(A₁)

An immediate consequence of the attenuation assumption (H6) is to ensure that, if C is bounded, then B is compact and the maxima/minima occur in the definition of a viscosity solution since u and B are continuous.

(A₂)

The main idea behind (H6) is the following. Let us fix $\delta$ and start at $(N+1)$ in $H_N$. Since ${\displaystyle Cy=\sum {\nu }_i^2{y}_i^2}$, this implies

$$\sum _{N+1}^{\infty }{\nu }_i^2{y}_i^2⩽C\left(\delta \right).$$

(13)

Thanks to (12) and using (13), it is easy to verify

$$\sum _{N+1}^{\infty }{\mu }_i^2{y}_i^2⩽C\left(\delta \right)\left(\underset{i⩾N+1}{sup}\right({\displaystyle \frac{{\mu }_i}{{\nu }_i}}{)}^2)\underset{N\to \infty }{\to }0,$$

(14)

uniformly with respect to all the points. Of course, considering (14) suggests that introducing a slight degree of compactness should be sufficient to ensure that truncation effects do not introduce significant errors.

(A₃)

Assumption (12) is typical of the infinite-dimensional setting and can be neglected for finite-dimensional case.

2.3. Uniform Convergence and a Priori Estimation

• Before we present the main results, we would explain why uniform convergence plays a central role in our work. Sub- and supersolutions are required to be bounded uniformly continuous and weakly continuous. A very interesting problem concerns uniform convergence. In infinite dimension, the balls are no longer compact, and uniform continuity is a real estimate. In order to mathematically justify this uniform convergence limit of our equation, we must be precise. Uniform convergence is obtained by a compactness result. Compactness in infinite-dimensional spaces can be very tricky but less so in our case because we have an operator B that is compact since $\mathrm{Tr}\left(B^2\right)<\infty$; that is, it enables uniform convergence but not more than that.
  Let us clarify what we mean by the function u uniformly continuous on the bounded set. The following lemma is needed to ensure this point.
  The trick is to take $u\in BUC\left(H\right)$. Then, from the estimate

  $$\left|u\right(x)-u(y\left)\right|⩽\omega \left(\right|B(x-y)\left|\right),$$

  (15)

  this indicates that

  $$|u\left(P_{N}x\right)-u_N\left(x\right)|\underset{N\to \infty }{\to }0.$$

  (16)
  Now, assume that $u\in C^0\left(H\right)$ (continuous in the usual sense of the term, that is, strongly continuous) and take $x_k\to x$, weakly convergent as $k\to +\infty$; this implies that we are in a bounded state $|x_k|,|x|⩽R_0$ (a weakly converging sequence is bounded); then, we are in a situation where

  $$\forall \epsilon >0,\exists N,\mathrm{such} \mathrm{that}|u\left(P_N{x}_k\right)-u\left(x_k\right)|⩽\epsilon ,$$

  (17)

  and of course

  $$|u\left(P_{N}x\right)-u\left(x\right)|⩽\epsilon .$$

  (18)
  This means that the sequence $P_N{x}_k$ lives in a finite-dimensional space, and $x_k$ converges to x. As a consequence, we obtain that

  $$P_N{x}_k\underset{k\to \infty }{\to }{P}_{N}x,$$

  weakly or strongly, since we are in finite dimension. The function $u\in C^0\left(H\right)$ for strong convergence and therefore

  $$u\left(P_N{x}_k\right)\underset{k\to \infty }{\to }u\left(x\right).$$
  It thereby follows from the bounds in (17) and (18) that

  $$|u\left(x_k\right)-u\left(x\right)|⩽2\epsilon +u\left(x\right)-P_N\left(x_k\right)⟶0,$$

  and we thereby obtain the bound

  $$limsup|u\left(x_k\right)-u\left(x\right)|⩽2\epsilon .$$
  Summarizing, we have proved the following lemma, which is in effect in the proof of the main result:
  Lemma 1. 

  Let $u_N$ be the approximate solution of Equation (1). Define $u_N\left(x_N\right):=u\left(P_{N}x\right)$. Then,

  (i)

  The sequences $u_N$ converge uniformly in x on every bounded set of H to a function $u\in BUC\left(H\right)$:

  $$\left|u\right(P_{N}x)-u_N\left(x\right)|\underset{N\to \infty }{\to }0.$$

  (19)

  (ii)

  The function u depends on a finite number of variables if and only if u weakly continuous since if u is weakly continuous

  $$P_{N}x\underset{N\to \infty }{\to }x$$

  uniformly on bounded subsets.

• Estimation a priori on $Du$
  We start with a quick review of the properties of $Du$. Having u bounded enabled us to have an estimate of the gradient and to show that the gradient is compactified, that is to say $Du\in D\left(A\right)$. Let us explain what we mean by the linear module. Formally, it follows from

  $$\forall x,y\in H,\left|u\right(x)-u(y\left)\right|⩽\omega \left(B\right|x-y\left|\right),$$

  (20)

  that

  $${\displaystyle \frac{1}{h}}|u(x+h\xi )-u\left(x\right)|⩽L\left|B\xi \right|$$

  for all $h>0$ and for any vector $\xi$, meaning that it is differentiable in any case if it is for positive instances. An immediate consequence is that, for the directional derivative in the direction $\xi$, we obtain

  $$\left|\right(\nabla u\left(x\right),\xi \left)\right|⩽L\left|B\xi \right|.$$

  (21)
  Indeed, if we denote $B\xi =\eta$ and $\xi =B^{-1}\eta$, then

  $$\left|\right(B^{-1}\nabla u\left(x\right),\eta \left)\right|⩽L\left|\eta \right|,\forall \eta ,$$

  (22)

  where $\eta$ is a point in the image of H by B, which is dense in H, and then it is true whatever $\eta$ is. To make this precise, the continuity modulus means $B^{-1}\nabla u$ is bounded as follows:

  $$|B^{-1}\nabla u\left(x\right)|⩽C,$$

  (23)

  which is not surprising because it is the same as saying that $v\in \mathrm{Lip}$, where $u\left(x\right)=v\left(Bx\right)$, which means

  $$\nabla u=B\nabla v.$$

  (24)
  Therefore, if we have a $\omega$ that is linear, that means we have a v that is Lipschitzian, so $\nabla v$ is bounded and $B^{-1}\nabla u$ is bounded; it is consistent. We see that, when we start to manipulate the size of the gradients, we can examine u or v to gain insight. Therefore, $B^{-1}\nabla u$ bounded.
  Compactness of $Du$. We have compactness in gradient $Du$, which enables us to pass to the limit in $F\left(Du\right)$. Thanks to a priori estimates, we control the term $F\left(Du\right)$ given the sense of the test function in the definition of viscosity solutions.

3. The Main Results

We now turn to the issue of the uniqueness viscosity solution of Equation (1) according to the standard definition of viscosity solution that was introduced by Lions in [35].

Definition 2 

(Viscosity solution). A function $u\in C\left(H\right)$ is called a viscosity subsolution (supersolution) of Equation (1) if for every $x_0\in H$ and for every test function $\phi :H\to \mathbb{R}$ continuously differentiable with respect to x whenever $u-\phi$ has a local maximum (respectively, $u-\phi$ has a local minimum) at $x_0\in H$ and

$$u\left(x_0\right)-\Delta \phi \left(x_0\right)+F\left(D\phi \left(x_0\right)\right)⩽f\left(x_0\right)$$

(25)

(respectively,

$$u\left(x_0\right)-\Delta \phi \left(x_0\right)+F\left(D\phi \left(x_0\right)\right)⩾f\left(x_0\right)).$$

(26)

Finally, a function $u\in C\left(H\right)$ is called viscosity solution of Equation (1) if u is both a viscosity subsolution and a viscosity supersolution of Equation (1).

Hereafter, we discard the term “viscosity” and simply speak of sub- and supersolution.

In this section, we prove a comparison result for viscosity solutions that, in turn, implies the uniqueness of solution of (1) under certain conditions.

The main point here is that, to measure the regularity chains, it is appropriate to take $H_N$. One can construct a sequence of approximation solutions $u_N$ by any method that yields a consistent weak formulation. Here, we conduct this with the Galerkin method. We are looking for $u_N$, and we project the equation

$$u_N-\Delta u_N+F\left(D{u}_N\right)=f\left(x\right)\mathrm{for}x\in H,$$

(27)

indexed by $N=1,2,3,\dots$, where $u_N$ takes values in $H_N$ and where the solutions $u_N$ of (27) are uniformly continuous in x in any bounded set of H. We obtain an equation in finite dimension. We also emphasize that, because $u_N\in H$, therefore we return to the previous calculations; there is no difference regarding all the $P_N$ that are in the equation as we multiply by $P_N$ to measure the norm $L^2$, and we can remove them because $u_N\in H_N$. Therefore, they are the same estimates.

We are now ready to state the main result, which is the central research question of this paper.

Theorem 1 

(Comparison principle). Assume that assumptions (H1), (H2), (H3), (H4), (H5), and (H6) are fulfilled. Let u and $v\in BUC\left(H\right)$ be B-continuous and such that u is a viscosity subsolution and v is a viscosity supersolution of the stationary Equation (1), respectively, with f replaced by g. Then, $u⩽v$ in H.

In particular, the stationary partial differential Equation (1) admits a unique solution.

4. Proof of the Main Theorem

The proof is based on an original (and clever) idea of doubling the variables. As already explained, we prove the theorem by the approximation argument.

• The idea of the approach is as follows. Since u is B-continuous, we take a Hilbert–Schmidt operator $B^{\prime }$ in the same base as B but whose eigenvalues tend a little slower towards 0, such as $B{\left(B^{\prime }\right)}^{-1}$ being compact; therefore, the operator B stifles larger coordinates than $B^{\prime }$. Instead of transforming the equation with B, we focus on $B^{\prime }$ such that it remains $B{\left(B^{\prime }\right)}^{-1}$-continuous since the transformation with $B^{\prime }$ consumes ${\left(B^{\prime }\right)}^{-1}$ and we still have B. Consequently, if we have such a modulus of continuity with a compact operator, this leads to weakly continuous. It is worth noting that, in Theorem 1, the solutions u and v differ from $u=v\left(Bx\right)$.
  Actually, we follow the more refined approach of Ishii [1] and Ishii and Lions [30,31,46]. The main difficulty in the proof is the lack of compactness in infinite-dimensional spaces.

• Step I. Construction of appropriate regularizations of u and v.

Let $N⩾0$. If x denotes a generic point in H, we may always choose an orthonormal basis $(e_1,e_2,\dots )$ of H so that $H_N$ coincides with the vector space generated by $(e_1,e_2,\dots ,e_N)$. We denote by $x_N$, $y_N$, …points in H and by $x_N^{\prime }$, $y_N^{\prime }$ …points in $H_N^{\perp }$; finally, we indifferently write $x=(x_N,x_N^{\prime })$ or $x=x_N+x_N^{\prime }$.

With these notations, the first step toward the proof of Theorem 1 is the construction of appropriate regularizations of u and v.

We begin the proof by assuming the extra conditions. We set for $\delta \in [0,1]$ and for all $x\in H$ 

$$\begin{array}{c}\hfill u_N\left(x\right)=\underset{z_N^{\prime }\in H_N^{\perp }}{sup}\left\{u(x_N,z_N^{\prime })-{\displaystyle \frac{1}{\delta }}{|x_N^{\prime }-z_N^{\prime }|}^2\right\},\end{array}$$

(28)

$$\begin{array}{c}\hfill v_N\left(x\right)=\underset{z_N^{\prime }\in H_N^{\perp }}{inf}\left\{v(x_N,z_N^{\prime })-{\displaystyle \frac{1}{\delta }}{|x_N^{\prime }-z_N^{\prime }|}^2\right\},\end{array}$$

(29)

and we claim that, for all $x_N^{\prime }\in H_N^{\perp }$, $u_N(·,x_N)$, and $v_N(·,x_N^{\prime })$ as functions over H, are, respectively, viscosity sub- and supersolutions of

$$-\Delta u_N+F\left(D{u}_N\right)+u_N⩽f(·,x_N^{\prime })+\omega \left(\delta \right),$$

(30)

$$-\Delta v_N+F\left(D{v}_N\right)+v_N⩾g(·,x_N^{\prime })-\omega \left(\delta \right),$$

(31)

where $\omega \left(\delta \right)⩾0$ and tends towards 0 as $\delta \to 0_{+}$. This maintains clarity once the complete result has been established.

The extra assumptions (30) and (31) were used to guarantee that it has a global maximum point $({\overline{x}}_N,{\overline{z}}_N^{\prime })$.

Let us remark at this stage that we have clearly

$$\begin{array}{c}\underset{z_N^{\prime }\in H_N^{\perp }}{sup}{u}_N(x_N,z_N^{\prime })-v_N(y_N,z_N^{\prime })\hfill \\ =sup\left\{u(x_N,x_N^{\prime })-v(y_N,y_N^{\prime })-{\displaystyle \frac{1}{\delta }}{|z_N^{\prime }-x_N^{\prime }|}^2\right.\hfill \\ \left.-{\displaystyle \frac{1}{\delta }}{|z_N^{\prime }-y_N^{\prime }|}^2/x_N^{\prime },y_N^{\prime },z_N^{\prime }\in H_N^{\perp }\right\}\hfill \\ =sup\left\{u(x_N,x_N^{\prime })-v(y_N,y_N^{\prime })-{\displaystyle \frac{1}{2\delta }}{|x_N^{\prime }-y_N^{\prime }|}^2|x_N^{\prime },y_N^{\prime }\in H_N^{\perp }\right\}\hfill \end{array}$$

for all $x_N,y_N\in H_N$.

We first conclude the proof of Theorem 1 by admitting this claim and then we prove it.

Now, once inequalities (30) and (31) are established, we just have to appeal to the finite-dimensional uniqueness proof (as in [11,47,48]) to deduce for all $N>0$ that

$$\begin{array}{c}\underset{H_N}{sup}\{u_N(x_N,x_N^{\prime })-v_N(x_N,x_N^{\prime })\}\hfill \\ ⩽2\omega \left(\delta \right)+\underset{H_N}{sup}(f(·,x_N^{\prime })-g(·,x_N^{\prime }))+sup\{F\left(D{u}_N\right)-F\left(D{v}_N\right)\}.\hfill \end{array}$$

(32)

Observe, indeed, that $u_N$ and $v_N$ are bounded and, respectively, upper and lower semicontinuous on H.

Next, we remark that the right-hand side of (32) is independent of $x_N^{\prime }$ and that $x_N^{\prime }$ is an arbitrary point in $H_N^{\perp }$.

Observe that u is a subsolution with f replaced by f defined in the same way, and, for v, a supersolution, f is replaced by g, but this is enough of our proof since in any case $f-g$ are very close to each other because of the B-module of continuity. Then, we have

$${\parallel f-g\parallel }_{\infty }⩽2\omega \left(\delta \right).$$

(33)

An immediate consequence is that, by using (9), one thereby obtains from (32) that

$$\underset{H}{sup}\{u_N-v_N\}⩽\underset{H}{sup}(f-g)+2\omega \left(\delta \right).$$

(34)

We thereby obtain

$$u_N⩽v_N+sup(f-g),$$

(35)

and we conclude since

$$v_N+sup(f-g)⩽v_N+\omega \left(\delta \right).$$

(36)

Roughly speaking,

$$u_N⩽v_N+\omega \left(\delta \right).$$

(37)

This ensures that the subsequence is

$$u_N\to u,v_N\to v\mathrm{as}N\to \infty$$

(38)

uniformly in any bounded set. In (38), pointwise convergence would suffice, but it is uniform because of the B-module of uniform continuity.

At this point, we just have to recall from (29) that $u_N⩾u$ and $v_N⩽v$ to deduce

$$\underset{H}{sup}\{u-v\}⩽\underset{H}{sup}(f-g)+2\omega \left(\delta \right).$$

(39)

We may then conclude upon formally setting $N\to \infty$ and $\delta \to 0_{+}$.

Therefore, to complete the uniqueness proof for (1), we have to prove claims (30) and (31).

We now use a rather standard technique of reduction to finite-dimensional spaces to produce appropriate test functions.

We begin with incorrect proof, which, ignoring the technical difficulties of minimization in infinite-dimensional spaces, at least clearly shows the idea of the proof. In this incomplete argument, we assume that suprema in (29) are always achieved. Then, to check (30), we consider $x_N^{\prime }\in H_N^{\perp }$, $\varphi \in C^2\left(H_N\right)$, ${\overline{x}}_N$ a maximum point of $u_N(x_N,{\overline{x}}_N^{\prime })-\varphi \left(x_N\right)$ over H, and we have to show

$$-\Delta \varphi \left({\overline{x}}_N\right)+F\left(D\varphi \left({\overline{x}}_N\right)\right)+u_N({\overline{x}}_N,{\overline{x}}_N^{\prime })⩽f({\overline{x}}_N,{\overline{x}}_N^{\prime })+\omega \left(\delta \right),$$

(40)

for some $\omega \left(\delta \right)$. With the above simplification strategy, we obtain a point ${\overline{z}}_N^{\prime }\in H_N^{\perp }$ such that

$$u_N({\overline{x}}_N,{\overline{x}}_N^{\prime })=u({\overline{x}}_N,{\overline{z}}_N^{\prime })-{\displaystyle \frac{1}{\delta }}{|{\overline{x}}_N^{\prime }-{\overline{z}}_N^{\prime }|}^2.$$

(41)

Then, we have

$$\begin{array}{c}u(x_N,x_N^{\prime })-\varphi \left(x_N\right)-{\displaystyle \frac{1}{\delta }}{|x_N^{\prime }-{\overline{z}}_N^{\prime }|}^2\hfill \\ ⩽u_N(x_N,{\overline{x}}_N^{\prime })-\varphi \left(x_N\right)⩽u_N({\overline{x}}_N,{\overline{x}}_N^{\prime })-\varphi \left({\overline{x}}_N\right)\hfill \\ =u({\overline{x}}_N,{\overline{z}}_N^{\prime })-\varphi \left({\overline{x}}_N\right)-{\displaystyle \frac{1}{\delta }}{|{\overline{z}}_N^{\prime }-{\overline{x}}_N^{\prime }|}^2,\hfill \end{array}$$

for all $x_N\in H_N$, $x_N^{\prime }\in H_N^{\perp }$. In other words, $({\overline{x}}_N,{\overline{z}}_N^{\prime })$ is a global maximum point over $H_N\times H_N^{\perp }\simeq H$ of $u-\varphi \left(x_N\right)-{\displaystyle \frac{1}{\delta }}{|x_N^{\prime }-{\overline{x}}_N^{\prime }|}^2$. Therefore, the immediate consequence is that, by definition of viscosity (sub)solutions, we have

$$-\Delta \varphi \left({\overline{x}}_N\right)+F\left(D\varphi \left({\overline{x}}_N\right)\right)+u({\overline{x}}_N,{\overline{z}}_N^{\prime })⩽f({\overline{x}}_N,{\overline{z}}_N^{\prime }).$$

(42)

Then, (40) follows from (42) since $u_N({\overline{x}}_N,{\overline{x}}_N^{\prime })⩽u({\overline{x}}_N,{\overline{z}}_N^{\prime })$ (see (41)) and specifically

$${\displaystyle \frac{1}{\delta }}{|{\overline{x}}_N^{\prime }-{\overline{z}}_N^{\prime }|}^2⩽\{\underset{H}{sup}u-\underset{H}{inf}u\}.$$

(43)

Indeed, (43) follows from (41) and the fact that $u_N⩽u$ on H. It remains to show the validity of (30) and (31).

• Step II. The perturbed maximization: Showing the validity of (30) and (31).

• One of the drawbacks of this method is that it does not guarantee the existence of maxima. We have applied the finite-dimensional maximum principle. Since we work in an infinite-dimensional space, there is no reason for the maxima to be attained. It remains to show that this maximum exists.
  To this aim, it suffices to use the results of generic optimization and the fact that the function involved is B-upper-semicontinuous. We can achieve this with the help of the Bishop–Phelps Theorem [49] or the use of the perturbed optimization principle, which does not require compactness or local compactness of the space due to Stegall [50], Ekeland, and Lebourg [51,52,53,54].
  However, to avoid using a linear perturbation of the Bishop–Phelps Theorem et al., this is circumvented by the use of a “softer operator” other than the B operator [55]. A very interesting problem is the question of the robustness of this method. Indeed, the lack of compactness in infinite-dimensional Hilbert spaces is circumvented by working with suitable compact subsets, in particular with the operator C defined in ((H6)). It is worth noticing that $y\in H_N^{\perp }$ means that the N-first coordinates are set to 0. We employ the radial function defined in (45) and (46) instead of the squared norm $\sqrt{{\left|Cy\right|}^2}$ since the latter causes the Laplacian to blow up and employing B is not conclusive to obtain our estimations, and then we obtain a uniform subsolution.
  Let us select a radial function $\mu :H⟶\mathbb{R}$ that is nondecreasing and linearly growing that localizes the problem and is also a perturbation that enables one to “generate maxima” and such that $D\mu$ and $D^2\mu$ are bounded and uniformly continuous. Let

  $$\mu \left(y\right)=\sqrt{1+{C\left|y\right|}^2}.$$

  (44)
  Upon choosing $\mu \left(y\right)$, the first step towards the proof of comparison result is the construction of appropriate regularizations of u and v. Define, for $\gamma >0$, the functions

  $$u_N\left(x\right)=\underset{y\in H_N^{\perp }}{sup}\{u(x,y)-\gamma \sqrt{1+{\left|Cy\right|}^2}\},x\in H_N,$$

  (45)

  $$v_N\left(x\right)=\underset{z\in H_N^{\perp }}{inf}\{v(x,z)+\gamma \sqrt{1+{\left|Cz\right|}^2}\},x\in H_N$$

  (46)
  (i.e., the “usual” sup-convolution or inf-convolution). According to [44,55,56], $u_N\left(x\right)$ and $v_N\left(x\right)$ are $C^{1,1}$ functions on H.
  The general classical techniques of viscosity theory tell us what we should expect: the $u_N$ in (45) is a subsolution in $H_N$.

• On the one hand, for $\gamma =0$, the equation we expect is the one in which we replace u with $u_N$; obviously, the second derivatives are canceled from the $(N+1)$th direction. We claim that $u_N$ as function over $H_N$ is a viscosity subsolution of

  $$u_N-\Delta u_N+F(\nabla u_N)⩽\underset{y\in H_N^{\perp }}{sup}f(x,y).$$

  (47)
  Likewise, from the fact that $v_N$ is a supersolution, we have

  $$v_N-\Delta v_N+F(\nabla v_N)⩾\underset{y\in H_N^{\perp }}{inf}f(x,y).$$

  (48)
  On the other hand, since H is an infinite-dimensional separable Hilbert space, we do not know where the maximum is reached, so we take the worst case to establish boundedness. Thanks to the choice regarding $\mu$, instead of working with solution u of (1), we consider the equation satisfied by $u-\gamma \mu \left(y\right)$. Replacing $Du$ by the function $Du-\gamma \mu \left(y\right)$ and bearing in mind that $\mu \left(y\right)$ is Lipschitzian, thus bounded, one has

  $$F(Du\pm \gamma D\left(\sqrt{1+{C\left|y\right|}^2}\right))⩽C\gamma ,$$

  (49)

  which is a consequence of (H2). Finally, let us find an explicit bound for the function $\mu \left(y\right)$. On the other hand, direct calculation shows that

  $$D\left\{\sqrt{1+{\left|Cy\right|}^2}\right\}={\displaystyle \frac{C^2}{{(1+|Cy|}^2{)}^{3/2}}},$$

  and provides the term of the form ${\displaystyle {\left|Cy\right|}^2{(\mu \left(y\right)}^{-3/2}}$, which is dominated by $\mathrm{Tr}\left(C^2\right)<\infty$.
  Hence, one deduces the following Hilbert–Schmidt corrector

  $$\Delta \{u-\gamma \sqrt{1+{\left|Cy\right|}^2}\}⩽C\gamma .$$

  (50)
  Upon collecting the above estimates and putting all these into (47), we obtain

  $$u_N-\Delta u_N+F(\nabla u_N)⩽C\gamma +\underset{y\in H_N^{\perp }}{sup}\{f(x,y)-\gamma \sqrt{1+{\left|Cy\right|}^2}\}.$$

  (51)
  A similar argument shows that, regarding $v_N$, one obtains

  $$v_N-\Delta v_N+F(\nabla v_N)⩾-C\gamma +\underset{z\in H_N^{\perp }}{inf}\{f(x,z)-\gamma \sqrt{1+{\left|Cz\right|}^2}\}.$$

  (52)
  Computing the difference between the right-hand sides of (51) and (52) and using the uniform continuity and boundedness of $u_N$ and $v_N$, this yields the boundedness of $(u_N-v_N)$. Specifically,

  $$\begin{array}{c}(C\gamma +\underset{y\in H_N^{\perp }}{sup}\{f(x,y)-\gamma \sqrt{1+{\left|Cy\right|}^2}\})-(-C\gamma +\underset{z\in H_N^{\perp }}{inf}\{f(x,z)-\gamma \sqrt{1+{\left|Cz\right|}^2}\})\hfill \\ =2C\gamma +\underset{y\in H_N^{\perp }}{sup}\{f(x,y)-\gamma \sqrt{1+{\left|Cy\right|}^2}\}\hfill \\ -\underset{z\in H_N^{\perp }}{inf}\{f(x,z)-\gamma \sqrt{1+{\left|Cz\right|}^2}\}.\hfill \end{array}$$

  (53)
  In view of (53), we obtain by combining the equalities above

  $$\begin{array}{ccc}\hfill u_N-v_N& ⩽& 2C\gamma +\underset{y\in H_N^{\perp }}{sup}\{f(x,y)-\gamma \sqrt{1+{\left|Cy\right|}^2}\}\hfill \\ \hfill & \hfill & -\underset{z\in H_N^{\perp }}{inf}\{f(x,z)-\gamma \sqrt{1+{\left|Cz\right|}^2}\}.\hfill \end{array}$$

  (54)

• We need to show that the right-hand side of (54) approaches to zero as $N\to \infty$. If $\overline{y}$ is a point of maximum, possibly $\overline{z}$ a minimum point, it is not necessarily reached, but it suffices to find nearby points $\overline{x}$ and $\overline{y}$ such that $u_N⩽v_N$. Without loss of generality, we can assume that the maximum is strict and global on H in $\overline{y}$. We assume that suprema in (45) are always achieved. Then, to check (51), let $\phi \in C^2\left(H_N\right)$ be a test function that touches $u_N$ from above at the point $\overline{x}$. Let $\overline{y}$ be the point where the maximum is achieved in the definition of $u_N\left(\overline{x}\right)$. We have

  $$\begin{array}{ccc}\hfill \phi \left(\overline{x}\right)& =& u\left(\overline{y}\right)-\gamma \sqrt{1+C|\overline{y}{|}^2},\hfill \\ \hfill \phi \left(\overline{x}\right)& ⩾& u_N\left(x\right)⩾u(x-\gamma \sqrt{1+{C\left|y\right|}^2}).\hfill \end{array}$$
  Therefore, the function $\tilde{\phi }\left(x\right)=\phi (x-\gamma \sqrt{1+{C\left|y\right|}^2})$ is a valid test function that touches the function u from above at the point $\overline{y}$. Applying the definition of viscosity subsolution for the function u, we thereby obtain the bound

  $$-\Delta u_N\left(\overline{y}\right)+F\left(D\tilde{\phi }\left(\overline{y}\right)\right)+u_N(\overline{x},\overline{y})⩽f(\overline{x},\overline{y}).$$

  (55)
  This is the same as

  $$-\Delta u_N\left(\overline{y}\right)+F\left(D\phi \left(\overline{y}\right)\right)+u_N(\overline{x},\overline{y})⩽f(\overline{x},\overline{y}).$$

  (56)
  This completes the verification that $u_N$ is a subsolution. Naturally, there is a straightforward modification of (56) that holds that, if v is a supersolution, then $v_N$ is also a supersolution.
  But, since we do not know where the maximum is reached, $f(\overline{x},\overline{y})$ is dominated by ${\displaystyle \underset{y\in H_N^{\perp }}{sup}f(x,y)}$. Roughly speaking,

  $$-C^{\mathrm{te}}⩽f(x,0)⩽\underset{y\in H_N^{\perp }}{sup}\{f(x,y)-\gamma \sqrt{1+{\left|Cy\right|}^2}\}⩽\underset{y\in H_N^{\perp }}{sup}f(x,y)⩽C^{\mathrm{te}},$$

  (57)

  with $f(x,0)$ bounded. Thus, the supremum is bounded independently of N and $\gamma$. We now observe that

  $$-{\Delta }_{H_N}\phi +H\left({\nabla }_{H_N}\phi \right)+u_N(\overline{x},\overline{y})⩽f(\overline{x},\overline{y})⩽\underset{y\in H_N^{\perp }}{sup}f(x,y).$$

  (58)
  It is worth noting that, since the supremum is bounded and $f(x,y)$ is also bounded, and thanks to the location weight, we thereby obtain the bound

  $$\left|Cy\right|⩽{\displaystyle \frac{C_0}{\gamma }}.$$

  (59)
  Then, regarding the supremum, only the y checks bound (59). We can similarly argue that, in the infimum, only the z checks the bound

  $$\left|Cz\right|⩽{\displaystyle \frac{C_0}{\gamma }}.$$

  (60)
  Moreover, y and z belong to $H_N^{\perp }$, which entails eliminating the first N-coordinates and assuming that C transforms bounded sets into compacts; this means that $By$ and $Bz$ are in a compact $K_{\gamma }$ depending only on $\gamma$; they are in $H_N^{\perp }$. It is worth noting that B is a compact operator since $\mathrm{Tr}\left(B^2\right)<\infty$. It is a compact operator as soon as we hold that the eigenvalues tend to 0 when $N\to \infty$ and the trace is finite and we recall that, by the B-continuity modulus, the large coordinates have little importance.
  The compactness of $By$ and $Bz$ guarantees that the maximum point exists. Letting N tend to infinity, we find that

  $$\left|By\right|⩽{\omega }_{\gamma }\left(N\right)\underset{N\to \infty }{\to }0,\left|Bz\right|⩽{\omega }_{\gamma }\left(N\right)\underset{N\to \infty }{\to }0,$$

  (61)

  where the limit depends on $\gamma$ because of the $1/\gamma$ in (59) and (60).
  As a consequence, thanks to the penalization terms, the maximization points are C-bounded, so they are B-compact; therefore, we have a maximum point, and it is clear that the maximum is achieved at some point $\overline{y}$ that remains in the ball $B_{1\widehat{R}}$ for some $\widehat{R}$ independent of $\epsilon$.
  Since f has a B-modulus of continuity, this implies that

  $$|f(x,\overline{y})-f(x,\overline{z})|⩽\omega (|B(\overline{y}-\overline{z})\left|\right),$$

  (62)

  and B, $B\overline{y}$, and $B\overline{z}$ tend towards 0.
• If we apply the change in variable $y=Bx$ and set $u_N\left(x\right)\stackrel{\mathrm{def}}{=}{v}_N\left(Bx\right)$, then the functions $v_N$ are uniformly compact in x, uniformly in N, on some compact K. Then, Ascoli–Arzela lemma ensures the existence of a subsequence ${\left\{u_{n_k}\right\}}_{k=1}^{\infty }$ that is locally uniformly convergent.
  As a consequence, on account of the theorems of viscosity theory [46,57], we may pick $\gamma$ small enough to ensure that

  $$u_N-v_N⩽C\gamma +{\omega }_{\gamma }\left(N\right)\underset{N\to \infty }{\to }0.$$

  (63)
  Thus, upon combining inequalities (59), (60), (61), and (13), we have equicontinuity and with the B-module, which is compact, and we know that

  $$u_{N^{\prime }}\underset{N^{\prime }\to \infty }{\overset{\mathrm{converge}}{\to }}u,$$
  (where $u_N^{\prime }$ is a subsequence) that a priori has all the properties (20) to (24) induced by the a priori estimates. Upon formally setting $N\to \infty$ in (63), one finds that

  $$u_N\underset{N\to \infty }{\to }u,v_N\underset{N\to \infty }{\to }v.$$

  (64)
  Summing up, the previous results can be rephrased as follows:

  $$u-v⩽C\gamma +{\omega }_{\gamma }\left(N\right)\underset{N\to \infty }{\to }0$$

  (65)

  at fixed $\gamma$, which completes the proof of Theorem 1.
  We conclude by remarking that all the above apply without any difficulty to time-dependent problems. Since the results are almost identical, we do not provide any precise statement here.