On nonlocal variational and quasi-variational inequalities with fractional gradient

TL;DR

This paper extends classical variational inequalities to fractional gradient constraints, establishing existence, continuous dependence, and generalized Lagrange multipliers for these new fractional PDEs.

Contribution

It introduces a novel class of fractional variational inequalities with Riesz fractional gradients and proves existence and dependence results, extending classical theories.

Findings

Established continuous dependence on data including fractional gradient thresholds.

Proved existence of solutions to fractional quasi-variational inequalities.

Extended Lagrange multiplier theory to fractional gradient constraints.

Abstract

We extend classical results on variational inequalities with convex sets with gradient constraint to a new class of fractional partial differential equations in a bounded domain with constraint on the distributional Riesz fractional gradient, the $\sigma$-gradient ($0<\sigma<1$). We establish continuous dependence results with respect to the data, including the threshold of the fractional $\sigma$-gradient. Using these properties we give new results on the existence to a class of quasi-variational variational inequalities with fractional gradient constraint via compactness and via contraction arguments. Using the approximation of the solutions with a family of quasilinear penalisation problems we show the existence of generalised Lagrange multipliers for the $\sigma$-gradient constrained problem, extending previous results for the classical gradient case, i.e., with $\sigma=1$.

Equations

$$\big{(}D^{\sigma}u\big{)}_{j}=\frac{\partial^{\sigma}u}{\partial x_{j}^{\sigma}}=\frac{\partial\ }{\partial x_{j}}I_{1-\sigma}u,\qquad 0<\sigma<1,\quad j=1,\ldots,N,$$

$$\Big{\langle}\frac{\partial^{\sigma}u}{\partial x_{j}^{\sigma}},v\Big{\rangle}=-\Big{\langle}I_{1-\sigma}u,\frac{\partial v}{\partial x_{j}}\Big{\rangle}=-\int_{\mathbb{R}^{N}}(I_{1-\sigma}u)\frac{\partial v}{\partial x_{j}}\,dx,$$

$$I_{\alpha }u(x)=(I_{\alpha }\ast u)(x)={\gamma }_{N,\alpha }{\int }_{{\mathbb{R}}^N}\frac{u(y)}{|x-y{|}^{N-\alpha }}dy,\text{with }{\gamma }_{N,\alpha }=\frac{\Gamma (\frac{N-\alpha }{2})}{{\pi }^{\frac{N}{2}}{2}^{\alpha }\Gamma (\frac{\alpha }{2})}.$$

$$D^{\sigma }u\equiv D(I_{1-\sigma }u)=I_{1-\sigma }\ast Du,$$

$$(-\Delta {)}^{\sigma }u=-\sum _{j=1}^N\frac{{\partial }^{\sigma }}{\partial x_j^{\sigma }}\frac{{\partial }^{\sigma }}{\partial x_j^{\sigma }}u,$$

$$(-\Delta {)}^{\sigma }u\equiv C_{N,\sigma }\text{ P.V.}{\int }_{{\mathbb{R}}^N}\frac{u(x)-u(y)}{|x-y{|}^{N+2\sigma }}dy.$$

$$D^{\sigma }u(x)=R(-\Delta {)}^{\frac{\sigma }{2}}u(x)=(N+\sigma -1){\gamma }_{N,1-\sigma }{\int }_{{\mathbb{R}}^N}\frac{u(x)-u(y)}{|x-y{|}^{N+\sigma }}\frac{x-y}{|x-y|}dy,$$

$$Rf(x)={\rho }_N\text{ P.V.}{\int }_{{\mathbb{R}}^N}f(y)\frac{x-y}{|x-y{|}^{N+1}}dy.$$

$$\Vert u{\Vert }_{\sigma ,p}^p=\Vert u{\Vert }_{L^p({\mathbb{R}}^N)}^p+\Vert D^{\sigma }u{\Vert }_{L^p(R^N{)}^N}^p,0<\sigma <1,p>1,$$

$$H_0^{\sigma }(\Omega ),\text{with norm }\Vert u{\Vert }_{H_0^{\sigma }(\Omega )}=\Vert D^{\sigma }u{\Vert }_{L^2({\mathbb{R}}^N{)}^N},0<\sigma <1,$$

$$\mathbb{K}_{g}^{\sigma}=\big{\{}v\in H^{\sigma}_{0}(\Omega):|D^{\sigma}v|\leq g\text{ a.e. in }\mathbb{R}^{N}\big{\}}.$$

$$u\in {\mathbb{K}}_g^{\sigma }:{\int }_{{\mathbb{R}}^N}A{D}^{\sigma }u\cdot D^{\sigma }(v-u)\ge {\int }_{\Omega }f(v-u),\forall v\in {\mathbb{K}}_g^{\sigma }.$$

$$\displaystyle\boldsymbol{\langle}\lambda D^{\sigma}u,D^{\sigma}v\boldsymbol{\rangle}_{\big{(}L^{\infty}(\mathbb{R}^{N})^{N}\big{)}^{\boldsymbol{{}^{\prime}}}\times L^{\infty}(\mathbb{R}^{N})^{N}}+\int_{\mathbb{R}^{N}}AD^{\sigma}u\cdot D^{\sigma}v=\int_{\Omega}fv,\qquad\forall v\in\Upsilon_{\infty}^{\sigma}(\Omega),$$

$$|D^{\sigma }u|\le g\text{ a.e. in }{\mathbb{R}}^N,\lambda \ge 0\text{ and }\lambda (|D^{\sigma }u|-g)=0\text{ in }{L}^{\infty }({\mathbb{R}}^N{)}^{{}^{\prime }}$$

$$\Upsilon_{\infty}^{\sigma}(\Omega)=\big{\{}\upsilon\in H^{\sigma}_{0}(\Omega):D^{\sigma}\upsilon\in L^{\infty}(\mathbb{R}^{N})^{N}\big{\}},\qquad 0<\sigma<1,$$

$$⟨\lambda \mathbf{\alpha },\mathbf{\beta }{⟩}_{(L^{\infty }({\mathbb{R}}^N{)}^N){}^{\prime }\times L^{\infty }({\mathbb{R}}^N{)}^N}=⟨\lambda ,\mathbf{\alpha }\cdot \mathbf{\beta }{⟩}_{L^{\infty }({\mathbb{R}}^N){}^{\prime }\times L^{\infty }({\mathbb{R}}^N)}\forall \lambda \in L^{\infty }({\mathbb{R}}^N){}^{\prime }\forall \mathbf{\alpha },\mathbf{\beta }\in L^{\infty }({\mathbb{R}}^N{)}^N.$$

$$a_{\ast }|\xi {|}^2\le A(x)\xi \cdot \xi \text{and}A(x)\xi \cdot \eta \le a^{\ast }|\xi ||\eta |.$$

$$L^{\infty}_{\nu}({\mathbb{R}^{N}})=\big{\{}v\in L^{\infty}({\mathbb{R}^{N}}):v(x)\geq\nu>0\text{ a.e. }x\in{\mathbb{R}^{N}}\big{\}}.$$

$$W^{\sigma ,p}(\Omega )\hookrightarrow L^q(\Omega ),\text{ for every }q\le \frac{Np}{N-\sigma p},\text{ if }\sigma p<N,$$

$$W^{\sigma ,p}(\Omega )\hookrightarrow L^q(\Omega ),\text{ for every }q<\infty ,\text{ if }\sigma p=N,$$

$$W^{\sigma ,p}(\Omega )\hookrightarrow L^{\infty }(\Omega )\cap {\mathcal{C}}^{0,\beta }(\overline{\Omega }),\text{ for every }0<\beta \le \sigma -\frac{N}{p},\text{ if }\sigma p>N,$$

$$H^{\sigma}_{0}(\Omega)\hookrightarrow L^{2^{*}}(\Omega)\quad\text{and}\quad L^{2^{\#}}(\Omega)\hookrightarrow H^{-\sigma}(\Omega)=\big{(}H^{\sigma}_{0}(\Omega)\big{)}^{\boldsymbol{{}^{\prime}}},\ 0<\sigma<1,$$

$${\mathbb{K}}_g^{\sigma }\subset {{\rm Y}}_{\infty }^{\sigma }(\Omega )\subset {\mathcal{C}}^{0,\beta }(\overline{\Omega })\subset L^{\infty }(\Omega ),$$

$$\Vert u{\Vert }_{L^{\infty }(\Omega )}\le \kappa \Vert u{\Vert }_{H_0^{\sigma }(\Omega )},\text{for any }u\in {{\rm Y}}_{\infty }^{\sigma }(\Omega ),$$

$${\int }_{R^N}|D^{\sigma }u{|}^p\le \Vert D^{\sigma }u{\Vert }_{L^{\infty }({\mathbb{R}}^N{)}^N}^{p-2}{\int }_{{\mathbb{R}}^N}|D^{\sigma }u{|}^2,\forall 2<p<\infty ,$$

$$(2.7{)}_i{u}_i\in {\mathbb{K}}_{g_i}^{\sigma }:{\int }_{{\mathbb{R}}^N}A{D}^{\sigma }{u}_i\cdot D^{\sigma }(v-u_i)\ge {\int }_{\Omega }{f}_i(v-u_i),\forall v\in {\mathbb{K}}_{g_i}^{\sigma }.$$

$$u_i\in {\mathbb{K}}_{g_i}^{\sigma }\cap {\mathcal{C}}^{0,\beta }(\overline{\Omega }),\text{ for all }0<\beta <\sigma .$$

$$\Vert u_1-u_2{\Vert }_{H_0^{\sigma }(\Omega )}\le C_1\Vert f_1-f_2{\Vert }_{L^1(\Omega )}.$$

$$\Vert u_1-u_2{\Vert }_{H_0^{\sigma }(\Omega )}\le C_{\#}\Vert f_1-f_2{\Vert }_{L^{2^{\#}}(\Omega )},$$

$$a_{\ast }\Vert \overline{u}{\Vert }_{H_0^{\sigma }(\Omega )}^2\le {\int }_{{\mathbb{R}}^N}A{D}^{\sigma }\overline{u}\cdot D^{\sigma }\overline{u}\le \Vert \overline{f}{\Vert }_{L^{2^{\#}}(\Omega )}\Vert \overline{u}{\Vert }_{L^{2^{\ast }}(\Omega )}\le C_{\ast }\Vert \overline{f}{\Vert }_{L^{2^{\#}}(\Omega )}\Vert \overline{u}{\Vert }_{H_0^{\sigma }(\Omega )},$$

$$\Vert \overline{u}{\Vert }_{L^{\infty }(\Omega )}\le \kappa \Vert \overline{u}{\Vert }_{H_0^{\sigma }(\Omega )},$$

$$a_{\ast }\Vert \overline{u}{\Vert }_{H_0^{\sigma }(\Omega )}^2\le \Vert \overline{f}{\Vert }_{L^1(\Omega )}\Vert \overline{u}{\Vert }_{L^{\infty }(\Omega )}\le \kappa \Vert \overline{f}{\Vert }_{L^1(\Omega )}\Vert \overline{u}{\Vert }_{H_0^{\sigma }(\Omega )}.$$

Full text

1. Introduction

111This is a revised and corrected version of the authors paper “On nonlocal variational and quasi-variational inequalities with fractional gradient. Appl. Math. Optim. 80 (2019), no. 3, 835–852”.

In a series of two interesting papers [13] and [14], Shieh and Spector have considered a new class of fractional partial differential equations. Instead of using the well-known fractional Laplacian, their starting concept is the distributional Riesz fractional gradient of order $\sigma\in(0,1)$, which will be called here the $\sigma$-gradient $D^{\sigma}$, for brevity: for $u\in L^{p}(\mathbb{R}^{N})$, $1<p<\infty$, we set

[TABLE]

where $\frac{\partial\ }{\partial x_{j}}$ is taken in the distributional sense, for every $v\in\mathscr{C}^{\infty}_{0}(\mathbb{R}^{N})$,

[TABLE]

with $I_{\alpha}$ denoting the Riesz potential of order $\alpha$, $0<\alpha<1$:

[TABLE]

As it was shown in [13], $D^{\sigma}$ has nice properties for $u\in\mathscr{C}^{\infty}_{0}(\mathbb{R}^{N})$, namely

[TABLE]

[TABLE]

where the well-known fractional Laplacian may be given, for a suitable constant $C_{N,\sigma}$, by (see, for instance, [4]):

[TABLE]

It was also observed in [14] that the $\sigma$-gradient is an example of the non-local gradients considered in [9], which can be also given by

[TABLE]

in terms of the vector-valued Riesz transform (see [15], with \rho_{N}=\text{\small\Gamma\big{(}\frac{N+1}{2}\big{)}/\pi^{\frac{N+1}{2}}}):

[TABLE]

We observe that, from the properties of $D^{\sigma}$ and a result of [7] on the Riesz kernel as approximation of the identity as $\alpha\rightarrow 0$, the $\sigma$-gradient approaches the standard gradient as $\sigma\rightarrow 1$: if $Du\in L^{p}(\mathbb{R}^{N})^{N}\cap L^{q}(\mathbb{R}^{N})^{N}$, $1<q<p$, then $D^{\sigma}u\underset{\sigma\rightarrow 1}{\longrightarrow}Du$ in $L^{p}(\mathbb{R}^{N})^{N}$.

Introducing the vector space of fractional differentiable functions as the closure of $\mathscr{C}^{\infty}_{0}(\mathbb{R}^{N})$ with respect to the norm

[TABLE]

by [13, Theorem 1.7] it is exactly the Bessel potencial space $L^{\sigma,p}(\mathbb{R}^{N})\hookrightarrow W^{s,p}(\mathbb{R}^{N})$, $0\leq s<\sigma$, where $W^{s,p}(\mathbb{R}^{N})$ denotes the usual fractional Sobolev space. In [13] the solvability of the fractional partial differential equations with variable coefficients and Dirichlet data was treated in the case $p=2$, as well as the minimization of the integral functionals of the $\sigma$-gradient with $p$-growth was also considered, leading to the solvability of a fractional $p$-Laplace equation of a novel type.

In this work we are concerned with the Hilbertian case $p=2$ in a bounded domain $\Omega\subset\mathbb{R}^{N}$, with Lipschitz boundary, where the homogeneous Dirichlet problem for a general linear PDE with measurable coefficients is considered under an additional constraint on the $\sigma$-gradient. We shall consider all solutions in the usual Sobolev space

[TABLE]

where here and whenever necessary, we still denote by $u$ the extension of $u$ by zero outside of $\Omega$. By the Sobolev-Poincaré inequality, this norm is equivalent to the usual Hilbertian norm induced from $L^{\sigma,2}(\mathbb{R}^{N})=W^{\sigma,2}(\mathbb{R}^{N})=H^{\sigma}(\mathbb{R}^{N})$, $0<\sigma<1$ in the closure of the Cauchy sequences of smooth functions with compact support in $\Omega$, i.e., in $\mathscr{C}^{\infty}_{0}(\Omega)$ (see [13]).

For nonnegative functions $g\in L^{\infty}({\mathbb{R}^{N}})$, we consider the nonempty convex sets of the type

[TABLE]

Let $f\in L^{1}(\Omega)$ and $A:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N\times N}$ be a measurable, bounded and positive definite matrix. We shall consider, in Section 2, the well-posedness of the variational inequality

[TABLE]

In particular, we obtain precise estimates for the continuous dependence of the solution $u$ with respect to $f$ and $g$, and so we extend well-known results for the classical case $\sigma=1$ (see [12] and its references).

Extending the result of [2] for the gradient ($\sigma=1$) case, we prove in Section 3 the existence of generalised Lagrange multipliers for the $\sigma$-gradient constrained problem. More precisely, we show the existence of $(\lambda,u)\in L^{\infty}(\mathbb{R}^{N})^{\boldsymbol{{}^{\prime}}}\times\Upsilon_{\infty}^{\sigma}(\Omega)$ such that

[TABLE]

and, moreover, $u$ solves (1.7). Here, for each $\sigma$, we have set

[TABLE]

and

[TABLE]

Finally, in the Section 4 we consider the solvability of solutions to quasi-variational inequalities corresponding to (1.7) when the threshold $g=G[u]$ and therefore also the convex set (1.6) depend on the solution $u\in\mathbb{K}_{G[u]}^{\sigma}$. We give sufficient conditions on the nonlinear and nonlocal operator $v\mapsto G[v]$ to obtain the existence of at least one solution $u$ of (1.7) with $\mathbb{K}_{g}^{\sigma}$ replaced by $K_{G[u]}^{\sigma}$, by compactness methods, as in [6] for the case $\sigma=1$. In a special case, when $G[u](x)=\Gamma(u)\varphi(x)$ is strictly positive and separates variables with a Lipschitz functional $\Gamma:L^{2}(\Omega)\rightarrow\mathbb{R}^{+}$, we adapt an idea of [5] (see also [12]) to obtain, by a contraction principle, the existence and uniqueness of the solution of the quasi-variational inequality under the “smallness” of the product of $f$ with the Lipschitz constant of $\Gamma$ and the inverse of its positive lower bound.

2. The variational inequality with $\sigma$-gradient constraint

For some $a_{*},a^{*}>0$, let $A=A(x):\mathbb{R}^{N}\rightarrow\mathbb{R}^{N\times N}$ be a bounded and measurable matrix, not necessarily symmetric, such that, for a.e. $x\in\mathbb{R}^{N}$ and all $\xi,\eta\in\mathbb{R}^{N}$:

[TABLE]

Fixed $\nu>0$, we define

[TABLE]

Indeed, we recall (see for instance [3]) the embedding for the fractional Sobolev spaces $0<\sigma\leq 1$, $1<p<\infty$:

[TABLE]

with continuous embedding, which are also compact if also $q<\tfrac{Np}{N-\sigma p}$ in (2.3a) and $\beta<\sigma-\tfrac{N}{p}$ in (2.3c). These embedding properties extend to the the Bessel potential space $L^{\sigma,p}(\mathbb{R}^{N})$ exactly with the same values of $\sigma,p,q$ and $\beta$ (see [13, Thm. 2.2]). In particular, we have

[TABLE]

where we set $2^{*}=\frac{2N}{N-2\sigma}$ and $2^{\#}=\frac{2N}{N+2\sigma}$ when $\sigma<\frac{N}{2}$, and if $N=1$ we denote $2^{*}=q$, $2^{\#}=q^{\prime}=\frac{q}{q-1}$ when $\sigma=\frac{1}{2}$ and $2^{*}=\infty$, $2^{\#}=1$ when $\sigma>\frac{1}{2}$.

For any $g\in L^{\infty}_{\nu}({\mathbb{R}^{N}})$, the convex set $\mathbb{K}_{g}^{\sigma}$ is non-empty and closed and we have the following auxiliary result.

Proposition 2.1**.**

For any $g\in L^{\infty}_{\nu}({\mathbb{R}^{N}})$, we have the following inclusions:

[TABLE]

for all $0<\beta<\sigma$, where $\mathscr{C}^{0,\beta}(\overline{\Omega})$ is the space of Hölder continuous functions with exponent $\beta$ and the estimate

[TABLE]

holds, where $\kappa>0$ depends on $\Omega$ through the Sobolev imbeddings and on $\|D^{\sigma}u\|_{L^{\infty}(\mathbb{R}^{N})^{N}}$.

Proof.

For $u\in\Upsilon_{\infty}^{\sigma}(\Omega)$, as $H^{\sigma}_{0}(\Omega)\subset L^{2^{*}}(\Omega)$ and recalling that $u=0$ on $\mathbb{R}^{N}\setminus B_{R}$, with the estimate

[TABLE]

we obtain $u\in L^{\sigma,2^{*}}(\mathbb{R}^{N})\subset L^{2^{**}}(\mathbb{R}^{N})$, where $2^{**}=\frac{2^{*}N}{N-2^{*}\sigma}$ if $2^{*}\sigma<N$ and $2^{**}$ is any $q$ finite if $2^{*}\sigma\geq N$, by using Sobolev imbeddings (see [13, Thm. 2.2]). Iterating with a bootstrap argument, we obtain $u\in L^{\sigma,q}(\mathbb{R}^{N})^{N}\subset\mathscr{C}^{0,\beta}(\overline{\Omega})$, for any $q>\tfrac{N}{\sigma}$, with $\beta=\sigma-\tfrac{N}{q}$. ∎

Therefore, in the right hand side of the variational inequality (1.7), for $g_{i}\in L^{\infty}(\mathbb{R}^{N})$, we can take $f_{i}\in L^{1}(\Omega)$, and the first two theorems give continuous dependence results with precise estimates for two different problems with $i=1,2$:

[TABLE]

Theorem 2.1**.**

Under the assumptions (2.1), for each $f_{i}\in L^{1}(\Omega)$ and each $g_{i}\in L^{\infty}(\mathbb{R}^{N})$, $g_{i}\geq 0$, there exists a unique solution $u_{i}$ to $(2.7)_{i}$ such that

[TABLE]

When $g_{1}=g_{2}$, the solution map $L^{1}(\Omega)\ni f\mapsto u\in H^{\sigma}_{0}(\Omega)$ is Lipschitz continuous, i.e., for $C_{1}=\kappa/a_{*}>0$, where $\kappa>0$ is the constant (2.6), we have

[TABLE]

Moreover, if in addition $f_{i}\in L^{2^{\#}}(\Omega)$, $i=1,2$, with $2^{\#}$ defined in (2.4) and $g_{1}=g_{2}$, then $L^{2^{\#}}(\Omega)\ni f\mapsto u\in H^{\sigma}_{0}(\Omega)$ is Lipschitz continuous:

[TABLE]

for $C_{\#}=C_{*}/{a_{*}}>0$, where $C_{*}$ is the constant of the Sobolev embedding $H^{\sigma}_{0}(\Omega)\hookrightarrow L^{2^{*}}(\Omega)$.

Proof.

Suppose that $f_{i}\in L^{2^{\#}}(\Omega)\subset H^{-\sigma}(\Omega)$. Since the assumption (2.1) implies that $A$ defines a continuous bilinear and coercive form over $H^{\sigma}_{0}(\Omega)$, the existence and uniqueness of the solution $u_{i}\in\mathbb{K}^{\sigma}_{g_{i}}$ to $(2.7)_{i}$ is an immediate consequence of the Stampacchia Theorem (see, for instance, [11, p. 95]), and (2.8) follows from (2.5).

With our notation (1.5), the estimate (2.10) follows easily from $(2.7)_{i}$ with $g_{1}=g_{2}$ and $v=u_{j}$ ($i,j=1,2$, $i\neq j$) from

[TABLE]

where we have set $\overline{u}=u_{1}-u_{2}$ and $\overline{f}=f_{1}-f_{2}$.

By (2.6), letting $\kappa>0$ be such that

[TABLE]

we may easily conclude the estimate (2.9) with $C_{1}=\kappa/a_{*}$ for $f_{1},f_{2}\in L^{2^{\#}}(\Omega)\subset L^{1}(\Omega)$ from (1.5)i and

[TABLE]

Finally, the solvability of $(2.7)_{i}$ for $f_{i}$ only in $L^{1}(\Omega)$ can be easily obtained by taking an approximating sequence of $f_{i}^{n}\in L^{2^{\#}}(\Omega)$ such that $f_{i}^{n}\underset{n}{\rightarrow}f_{i}$ in $L^{1}(\Omega)$ and using (2.9) for that (Cauchy) sequence. The proof is complete. ∎

Remark 2.1**.**

As in [13] it is possible to extend the variational inequality with $\sigma$-gradient to arbitrary open domains $\Omega\subset\mathbb{R}^{N}$ with a generalised Dirichlet data $\varphi\in H^{\sigma}(\mathbb{R}^{N})$ such that $I_{1-\sigma}*\varphi$ is well-defined and $D^{\sigma}\varphi\in L^{\infty}(\mathbb{R}^{N})$. This would require in the definition (1.6) of $\mathbb{K}_{g}^{\sigma}$ to replace $H^{\sigma}_{0}(\Omega)$ by the space

[TABLE]

and, in addition, technical compatibility assumptions on $\varphi$ and $g$ to guarantee that the new $\mathbb{K}_{g}^{\sigma}\not=\emptyset$.

Remark 2.2**.**

It is well-known that if, in addition, $A$ is symmetric, i.e. $A=A^{T}$, the variational inequality (1.7) corresponds (and is equivalent) to the optimisation problem (see, for instance, [11])

[TABLE]

where $\mathcal{J}:\mathbb{K}_{g}^{\sigma}\rightarrow\mathbb{R}$ is the convex functional

[TABLE]

Theorem 2.2**.**

Under the framework of the previous theorem, when $f_{1}=f_{2}\in L^{1}(\Omega)$, the solution map

[TABLE]

is $\frac{1}{2}$-Hölder continuous, i.e., there exists $C_{\nu}>0$ such that

[TABLE]

Proof.

Let $\eta=\|g_{1}-g_{2}\|_{L^{\infty}(\mathbb{R}^{N})}$ and, for $i,j=1,2$, $i\neq j$, notice that

[TABLE]

if $u_{i}$ denotes the unique solution of $(2.15)_{i}$ to $g_{i}$ and $f_{i}$.

Observe that for $i=1,2,$

[TABLE]

Hence, letting $v=u_{i_{j}}$ in $(2.7)_{j}$ and using (2.1) we get (2.11) from

[TABLE]

with $C^{{}^{\prime}}_{\nu}=2\kappa^{2}\|f\|^{2}_{L^{1}(\Omega)}(a^{*}+a_{*})/a_{*}^{2}\nu>0$, by using Theorem 2.1 and the constant $\kappa$ is defined by (2.6). ∎

Remark 2.3**.**

Using the trick of the above proof, if $g_{n}\underset{n}{\rightarrow}g$ in $L^{\infty}(\mathbb{R}^{N})$ for a sequence $g_{n}\in L^{\infty}_{\nu}(\mathbb{R}^{N})$, it is clear that, for any $w\in\mathbb{K}_{g}^{\sigma}$ we can choose $w_{n}\in\mathbb{K}_{g_{n}}^{\sigma}$ such that $w_{n}\underset{n}{\rightarrow}w$ in $H^{\sigma}_{0}(\Omega)$. On the other hand, also for any sequence $w_{n}\underset{n}{\relbar\joinrel\rightharpoonup}w$ in $H^{\sigma}_{0}(\Omega)$-weak, with each $w_{n}\in\mathbb{K}_{g_{n}}^{\sigma}$, $g_{n}\underset{n}{\rightarrow}g$ in $L^{\infty}(\mathbb{R}^{N})$ implies that also $w\in\mathbb{K}_{g}^{\sigma}$. These two conditions determine that if $g_{n}\underset{n}{\rightarrow}g$ in $L^{\infty}_{\nu}(\mathbb{R}^{N})$ then the respective convex sets $\mathbb{K}_{g_{n}}^{\sigma}$ converge in the Mosco sense to $\mathbb{K}_{g}^{\sigma}$. An open question is to extend this convergence to the case $0<\sigma<1$, by dropping the strict positivity condition on $g_{n}$ and $g$, as in [1] for $\sigma=1$.

3. Existence of Lagrange multipliers

In this section we prove the existence of solution of the problem (1.8a)-(1.8b).

For $\varepsilon\in(0,1)$ and denoting $\widehat{k}_{\varepsilon}=\widehat{k}_{\varepsilon}(D^{\sigma}u^{\varepsilon})=k_{\varepsilon}(|D^{\sigma}u^{\varepsilon}|-g)$ for simplicity, we define a family of approximated quasi-linear problems

[TABLE]

where $k_{\varepsilon}:\mathbb{R}\rightarrow\mathbb{R}$ is defined by

[TABLE]

Proposition 3.1**.**

Suppose that $g\in L^{\infty}_{\nu}(\mathbb{R}^{N})$, $f\in L^{2^{\#}}(\Omega)$ and $A:{\mathbb{R}^{N}}\rightarrow\mathbb{R}^{N\times N}$ is a measurable, bounded and positive definite matrix. Then the quasi-linear problem (3.1) has a unique solution $u^{\varepsilon}\in H^{\sigma}_{0}(\Omega)$.

Proof.

The operator $B_{\varepsilon}:H^{\sigma}_{0}(\Omega)\rightarrow H^{-\sigma}(\Omega)$ defined by

[TABLE]

is bounded, strongly monotone, coercive and hemicontinuous, so problem (3.1) has a unique solution (see, for instance, [8]). ∎

Lemma 3.1**.**

If $g\in L^{\infty}_{\nu}({\mathbb{R}^{N}})$, $f\in L^{2^{\#}}(\Omega)$, $A:{\mathbb{R}^{N}}\rightarrow\mathbb{R}^{N\times N}$ is a measurable, bounded and positive definite matrix and $1\leq q<\infty$, there exist positive constants $C$ and $C_{q}$ such that, for $0<\varepsilon<1$, setting $\widehat{k}_{\varepsilon}=k_{\varepsilon}(|D^{\sigma}u^{\varepsilon}|-g)$, the solution $u^{\varepsilon}$ of the approximated problem (3.1) satisfies

[TABLE]

Proof.

Using $u^{\varepsilon}$ as test function in (3.1), we get

[TABLE]

since $A\xi\cdot\xi\geq a_{*}|\xi|^{2}$ for any $\xi\in\mathbb{R}^{N}$ by the assumptions on $A$. But $\widehat{k}_{\varepsilon}\geq 0$ and so

[TABLE]

concluding then (3.2a) and (3.2b).

Observing that the function $\varphi_{\varepsilon}=\widehat{k}_{\varepsilon}\,(t^{2}-g^{2})+g^{2}\widehat{k}_{\varepsilon}\geq\nu^{2}\widehat{k}_{\varepsilon}$ and using (3.2b), there exists a positive constant $C$ independent of $\varepsilon$ such that

[TABLE]

This implies the uniform boundedness of $\widehat{k}_{\varepsilon}$ in $L^{1}({\mathbb{R}^{N}})$ and also in $L^{\infty}({\mathbb{R}^{N}})^{\boldsymbol{{}^{\prime}}}$, i.e., (3.2c) and (3.2e) respectively.

To prove (3.2d), it is enough to notice that, for $\boldsymbol{\beta}\in L^{\infty}(\Omega)^{N}$,

[TABLE]

Because for $t-g>0$ we have $\widehat{k}_{\varepsilon}(t-g)\geq\frac{1}{m!}(t-g)^{m}$, for any $m\in\mathbb{N}$, then using (3.2c) we obtain $\|D^{\sigma}u^{\varepsilon}\|_{L^{q}(B_{R})}\leq C_{q,R}$, for any $q\in(1,\infty)$ and $R\in\mathbb{R}^{+}$, (for details see, for instance [10], replacing $Q_{T}$ by $B_{R}$). Arguing as in Proposition 2.1, we conclude the proof. ∎

Proposition 3.2**.**

For $g\in L^{\infty}_{\nu}({\mathbb{R}^{N}})$, $f\in L^{2^{\#}}(\Omega)$ and $A:{\mathbb{R}^{N}}\rightarrow\mathbb{R}^{N\times N}$ a measurable, bounded and positive definite matrix, the family $\{u^{\varepsilon}\}_{\varepsilon}$ of solutions of the approximated problems (3.1) converges weakly in $H^{\sigma}_{0}(\Omega)$ to the solution of the variational inequality (1.7).

Proof.

The uniform boundedness of $\{u^{\varepsilon}\}_{\varepsilon}$ in $H^{\sigma}_{0}(\Omega)$ implies that, at least for a subsequence,

[TABLE]

For $v\in\mathbb{K}_{g}^{\sigma}$ we have, since $\widehat{k}_{\varepsilon}>0$ when $|D^{\sigma}u_{\varepsilon}|>g\geq|D^{\sigma}v|$,

[TABLE]

and so, testing the first equation of (3.1) with $v-u^{\varepsilon}$, we get

[TABLE]

But

[TABLE]

So, utilizing the weak convergence $u^{\varepsilon}\underset{\varepsilon\rightarrow 0}{\relbar\joinrel\rightharpoonup}u$ in $H^{\sigma}_{0}(\Omega)$,

[TABLE]

Let $w\in\mathbb{K}_{g}^{\sigma}$ and setting $v=u+\theta(w-u)$, then $v\in\mathbb{K}_{g}^{\sigma}$ for any $\theta\in(0,1]$ and we get

[TABLE]

Dividing this inequality by $\theta$ and letting $\theta\rightarrow 0$, we obtain (1.7). The proof is concluded if we show that $u\in\mathbb{K}_{g}^{\sigma}$. Indeed we split $\mathbb{R}^{N}$ in three subsets

[TABLE]

and, following the steps in [10], we conclude, for arbitrary $R>0$, that

[TABLE]

because

[TABLE]

So $|D^{\sigma}u|\leq g$ a.e. in ${B_{R}}$, which means that $u\in\mathbb{K}_{g}^{\sigma}$ because $\mathbb{R}^{N}=\displaystyle\cup_{k\in\mathbb{N}}B_{k}$.

The uniqueness of solution of the variational inequality (1.7) implies that the whole sequence $\{u^{\varepsilon}\}_{\varepsilon}$ converges to $u$ in $H^{\sigma}_{0}(\Omega)$. ∎

Theorem 3.1**.**

If $g\in L^{\infty}_{\nu}({\mathbb{R}^{N}})$, $f\in L^{2^{\#}}(\Omega)$ and $A:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N\times N}$ is a measurable, bounded and positive definite matrix, then problem (1.8a)-(1.8b) has a solution

[TABLE]

Proof.

By estimates (3.2d) and (3.2e) and the Banach-Alaoglu-Bourbaki theorem we have, at least for a subsequence,

[TABLE]

and

[TABLE]

For $v\in H^{\sigma}_{0}(\Omega)$, since

[TABLE]

we obtain, letting $\varepsilon\rightarrow 0$ with $v\in\Upsilon_{\infty}^{\sigma}(\Omega)$,

[TABLE]

Taking $v=u^{\varepsilon}$ in (3.4) we get

[TABLE]

Observe first that

[TABLE]

and therefore

[TABLE]

So, from(3.6) and (3.5) with $v=u$,

[TABLE]

and then

[TABLE]

Using $\widehat{k}_{\varepsilon}(|D^{\sigma}u^{\varepsilon}|^{2}-g^{2})\geq 0$, we obtain

[TABLE]

We also have

[TABLE]

and therefore we conclude

[TABLE]

Given $v\in\mathbb{K}_{g}$, we have

[TABLE]

because, by estimate (3.2c), $\widehat{k}_{\varepsilon}$ is uniformly bounded in $L^{1}(\mathbb{R}^{N})$. So, for any $v\in\mathbb{K}_{g}$,

[TABLE]

concluding the proof of (1.8a).

Since $\displaystyle{\int_{\mathbb{R}^{N}}}\widehat{k}_{\varepsilon}v\geq 0$ for all $v\in L^{\infty}({\mathbb{R}^{N}})$ such that $v\geq 0$ then, for such $v$, we also have $\langle\lambda,v\rangle\geq 0$, which means that $\lambda\geq 0$.

For $v\in L^{\infty}(\mathbb{R}^{N})$ set $v^{+}=\max\{v,0\}$, $v^{-}=(-v)^{+}$. Since $\widehat{k}_{\varepsilon}(|D^{\sigma}u^{\varepsilon}|^{2}-g^{2})\geq 0$ then

[TABLE]

concluding that

[TABLE]

The fact that $\widehat{k}_{\varepsilon}\geq 0$ and $u\in\mathbb{K}^{\sigma}_{g}$ imply $\widehat{k}_{\varepsilon}(|D^{\sigma}u|^{2}-g^{2})v^{\pm}\leq 0$ and, therefore, integrating and letting $\varepsilon\rightarrow 0$, $\langle\lambda,(D^{\sigma}u|^{2}-g^{2})\,v^{\pm}\rangle\leq 0$, and so

[TABLE]

Writting $v=\frac{w}{|D^{\sigma}u|+g}$, for any $w\in L^{\infty}(\Omega)$, we conclude (1.8b). ∎

4. The quasi-variational inequality with $\sigma$-gradient constraint

In this section we consider a map $G$ such that

[TABLE]

is a continuous and bounded operator, where $2^{*}$ is the Sobolev exponent as in (2.4) for $0<\sigma<1$.

We recall that, whenever necessary, we still denote by $u$ the extension of $u$ by zero outside of $\Omega$.

We set

[TABLE]

and we shall consider the quasi-variational inequality

[TABLE]

Generalising a compactness argument of [6] where quasi-variational inequalities of this type were considered for the gradient case $\sigma=1$, we may give a general existence theorem.

Theorem 4.1**.**

Under the assumptions (2.1), for continuous and bounded operators $G$ satisfying (4.1) and for any $f\in L^{2^{\#}}(\Omega)$, with $2^{\#}$ as in (2.4), there exists at least one solution for the quasi-variational inequality (4.3).

Proof.

Let $u=S(f,g)$ be the unique solution of the variational inequality (1.7) with $g=G[w]$ for any $w\in L^{2^{*}}(\Omega)$. If $C_{*}>0$ denotes the Sobolev constant as in Theorem 2.1, since $f_{2}=0$ corresponds always to the solution $u_{2}=0$, we have the a priori estimate

[TABLE]

independently of $g\in L^{\infty}_{\nu}(\mathbb{R}^{N})$.

Set B_{c_{f}}=\big{\{}v\in L^{2^{*}}(\Omega):\|v\|_{L^{2^{*}}(\Omega)}\leq c_{f}\big{\}} and define the nonlinear map $T=S\circ G:L^{2^{*}}(\Omega)\ni w\mapsto u\in L^{2^{*}}(\Omega)$ where $u=S(f,G[w])\in\mathbb{K}_{G[w]}^{\sigma}\cap\mathscr{C}^{0,\beta}(\overline{\Omega})$, $0<\beta<\sigma$ by (2.8).

Clearly, (4.4) implies $T(B_{c_{f}})\subset B_{c_{f}}$ and, by the continuity of $G$ and Theorem 2.2, $T$ is also a continuous map. On the other hand, $G$ is bounded, i.e. transforms bounded sets in $L^{2^{*}}(\Omega)$ into bounded sets of $L^{\infty}_{\nu}(\mathbb{R}^{N})$ and $S\circ T$ is also a bounded operator. Therefore, by (2.8), $T(B_{c_{f}})$ is also a bounded set of $C^{0,\beta}(\overline{\Omega})$. Since the embedding $C^{0,\beta}(\overline{\Omega})\hookrightarrow L^{2^{*}}(\Omega)$ is compact, the Schauder fixed point theorem guarantees the existence of $u=Tu$, which solves (4.3). ∎

Example 4.1**.**

Consider the operator $G:L^{2^{*}}(\Omega)\rightarrow L^{\infty}_{\nu}(\mathbb{R}^{N})$ defined as follows:

[TABLE]

where $F:\mathbb{R}^{N}\times\mathbb{R}\rightarrow\mathbb{R}$ is a function bounded in $x\in\mathbb{R}^{N}$ and continuous in $w\in\mathbb{R}$, uniformly in $x\in\mathbb{R}^{N}$, satisfying, for some $\nu>0$,

[TABLE]

and for some monotone increasing function $\varphi$. We may choose

[TABLE]

where we give \vartheta\in L^{\infty}\big{(}\mathbb{R}^{N}_{x};L^{2^{\#}}(\Omega_{y})\big{)}. For $u_{n}\underset{n}{\rightarrow}u$ in $L^{2^{*}}(\Omega)$, from the estimate

[TABLE]

and by the uniform continuity of $F$, we have

[TABLE]

implying the continuity of $G$.

The boundedness of $G$ is a consequence of (4.6) and therefore $G$ satisfies the assumptions of Theorem 4.1.

Example 4.2**.**

Consider now the operator $G:H^{\sigma}_{0}(\Omega)\rightarrow L^{\infty}_{\nu}(\mathbb{R}^{N})$ given also by (4.5) with $F$ under the same assumptions as in the previous example, but now with

[TABLE]

where \Theta\in\mathscr{C}^{0}\big{(}\overline{\Omega}_{x};\mathbb{R}^{N}_{y}\big{)}. Now $G$ is not only bounded but also completely continuous, since $\Phi:H^{\sigma}_{0}(\Omega)\rightarrow\mathscr{C}^{0}(\overline{\Omega})$ is also completely continuous. Indeed, if $u_{n}\underset{n}{\relbar\joinrel\rightharpoonup}u$ in $H^{\sigma}_{0}(\Omega)$-weak, then $w_{n}=\Phi(u_{n})\underset{n}{\rightarrow}\Phi(u)=w$ in $\mathscr{C}^{0}(\overline{\Omega})$, because $\{D^{\sigma}u_{n}\}_{n}$, being bounded in $L^{2}(\mathbb{R}^{N})^{N}$ implies $\{w_{n}\}_{n}$ uniformly bounded in $\mathscr{C}^{0}(\overline{\Omega})$,

[TABLE]

and also equicontinuous in $\overline{\Omega}$ by

[TABLE]

But $G$ is not defined in the whole $L^{2^{*}}(\Omega)$ and therefore we cannot apply Theorem 4.1 to solve (4.3). Nevertheless, the solvability of (4.3) in this example is an immediate consequence of the following theorem.

Theorem 4.2**.**

Assume (2.1) and let $f\in L^{2^{\#}}(\Omega)$ as previously. If the nonlinear and nonlocal operator $G$ satisfies

[TABLE]

then there exists a solution $u$ to the quasi-variational inequality (4.3).

Proof.

Due to the estimate (4.4) and the assumption (4.9), the proof is analogous by applying the Schauder fixed point theorem to the nonlinear completely continuous map

[TABLE]

∎

Example 4.3**.**

By restricting the domain of $G$ and using the same type of Carathéodory function $F$ as in Example 4.1, we can introduce the superposition operator

[TABLE]

In order to guarantee that $G:\mathscr{C}^{0}(\overline{\Omega})\rightarrow L^{\infty}_{\nu}(\mathbb{R}^{N})$ is a continuous and bounded operator in an appropriate space to obtain a fixed point, we need to require that the function $F:\mathbb{R}^{N}\times\mathbb{R}\rightarrow\mathbb{R}$ is a bounded function in $x\in\mathbb{R}^{N}$ in each compact for the variable $u$, continuous in $u\in\mathbb{R}$ uniformly in $x\in\mathbb{R}^{N}$, and satisfying (4.6), where the function $\varphi$ is continuous and satisfies only

[TABLE]

This situation is covered by the next theorem.

Theorem 4.3**.**

Assume (2.1), let $f\in L^{1}(\Omega)$ and the functional $G$ be such that

[TABLE]

Then there exists a solution of the quasi-variational inequality (4.3).

Proof.

As before, we set $T=S\circ G:\mathscr{C}^{0}(\overline{\Omega})\rightarrow H^{\sigma}_{0}(\Omega)$ and for $w\in\mathscr{C}^{0}(\overline{\Omega})$, $u=S(f,G[w])$ solves (1.7) with $g=G[w]$.

In order to apply the Leray-Schauder principle, we set

[TABLE]

and we show that $\mathscr{S}$ is a priori bounded in $L^{\infty}(\Omega)$. For any $w\in\mathscr{S}$, $u=Tw$ solves (1.7) with $g=G[w]$. Hence we have, noting that $w=\theta u$,

[TABLE]

if $\kappa>0$ is the constant of (2.6), by Theorem 2.1, and this a priori estimate is independent of $G$.

Since, by (2.5), $T(\mathscr{C}^{0}(\overline{\Omega}))\hookrightarrow\mathscr{C}^{0,\beta}(\overline{\Omega})\hookrightarrow\mathscr{C}^{0}(\overline{\Omega})$ and this last embedding is compact, we may conclude that $T$ is a completely continuous mapping into a closed ball of $\mathscr{C}^{0}(\overline{\Omega})$ and its fixed point $u=Tu$ solves (4.3). ∎

It is clear that in general we cannot expect the uniqueness of solution to quasi-variational inequalities of the type (4.3). However, the Lipschitz continuity of the solution map $f\mapsto u$ to the variational inequality (1.7), given by Theorem 2.1, allows us to obtain, via the strict contraction Banach fixed point principle, a uniqueness result in a special case of “small” and controlled variations of the convex sets for the quasi-variational situation with separation of variables in the nonlocal constraint $G$.

We denote, for $R>0$,

[TABLE]

Theorem 4.4**.**

Let $f\in L^{2^{\#}}(\Omega)$, $\varphi\in L^{\infty}_{\nu}(\mathbb{R}^{N})$ and

[TABLE]

where $\Gamma:H^{\sigma}_{0}(\Omega)\rightarrow\mathbb{R}^{+}$ is a functional satisfying

i)* $0<\eta(R)\leq\Gamma(u)\leq E(R),\quad\forall u\in B_{R}$,*

ii)* $|\Gamma(u_{1})-\Gamma(u_{2})|\leq\gamma(R)\|u_{1}-u_{2}\|_{H^{\sigma}_{0}(\Omega)},\quad\forall u_{1},u_{2}\in B_{R},$*

for sufficiently large $R\in\mathbb{R}^{+}$, with $\eta,E$ and $\gamma$ being monotone increasing positive functions of $R$.

Then the quasi-variational inequality (4.3) has a unique solution, provided

[TABLE]

where $R_{f}\equiv C_{\#}\|f\|_{L^{2^{\#}}(\Omega)}$ with $C_{\#}=C_{*}/a_{*}$ and $C_{*}$ is the constant of the Sobolev embedding as in (4.4).

Proof.

Let $S:B_{R}\ni v\mapsto u\in H^{\sigma}_{0}(\Omega)$ be the solution map with $u=S(f,G[v])$ being the unique solution of the variational inequality (1.7) with $g=G[v]$.

The a priori estimate (4.4) implies $S(B_{R_{f}})\subset B_{R_{f}}$.

Given $v_{i}\in B_{R}$, let $u_{i}=S(v_{i})=S(f,\varphi\,\Gamma(v_{i}))$, $i=1,2$, and choose $\mu=\frac{\Gamma(v_{2})}{\Gamma(v_{1})}>1$, without loss of generality.

Setting $g=\varphi\,\Gamma(v_{1})$, we have $\mu\,g=\varphi\,\Gamma(v_{2})$ and

[TABLE]

[TABLE]

by recalling the assumptions i) and ii) and denoting $\|w\|_{\sigma}=\|w\|_{H^{\sigma}_{0}(\Omega)}$ for simplicity.

Consequently, using (4.4) and (2.10) with $f_{1}=f$ and $f_{2}=\mu\,f$, we have

[TABLE]

and the conclusion of the theorem follows immediately. ∎

Example 4.4**.**

We can take $\Gamma$ of the form

[TABLE]

with $e:\mathbb{R}^{N}\times\mathbb{R}\times\mathbb{R}^{N}\rightarrow[\eta,\infty)$, for some $\eta>0$, under a local Lipschitz condition of the type

[TABLE]

for $|v|$, $|w|$, $|\xi|$ and $|\zeta|$ less or equal to $R$.

Remark 4.1**.**

Assumptions i) and ii) have been used in Appendiz B of [5] under the implicit assumptions of smallness of the term $f$, and in [12] in a simplified and more precise form in the case of gradient type (i.e. $\sigma=1$) and for a class of general operators of p-Laplacian type.

Remark 4.2**.**

The existence of solution of the quasi-variational inequality (4.3) is obtained in this section by finding a fixed point of the map $w\mapsto S(f,G[w])=u$, under suitable assumptions. But when $u=S(f,G[w])$ is the solution of (1.7) then there exists $\lambda\in L^{\infty}(\mathbb{R}^{N}){\boldsymbol{{}^{\prime}}}$ such that $(u,\lambda)$ solves problem (1.8a)-(1.8b) with data $(f,G[w])$. In particular, when $u$ is a fixed point $u=S(f,G[u])$ it solves the quasi-variational inequality, and we immediately get existence of a solution $(\lambda,u)$ of problem (1.8a)-(1.8b) for the quasi-variational case.

Acknowledgements

The research of J. F. Rodrigues was partially done under the framework of the project PTDC/MAT-PUR/28686/2017 at CMAFcIO/ULisboa and L. Santos was partially supported by the Centre of Mathematics the University of Minho through the Strategic Project PEst UID/MAT/00013/2013.