Correction to: On Nonlocal Variational and Quasi-Variational Inequalities with Fractional Gradient

1 Correction to: Appl Math Optim https://doi.org/10.1007/s00245-019-09610-0

¹We correct the norm of the usual fractional Sobolev space

$$\begin{aligned} H_0^\sigma (\Omega ),\quad \text {by setting its norm }\Vert u\Vert _{H_0^\sigma (\Omega )}=\Vert D^\sigma u\Vert _{L^2(\mathbb {R}^N)^N},\quad 0<\sigma <1, \end{aligned}$$

where the $L^2$ norm of the distributional Riesz fractional gradient of u, extended by zero in $\mathbb {R}^N\setminus \Omega$,

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

must be taken in the whole $R^N$ and not only in $\Omega$, the bounded open set where the problem is considered.

Consequently, all the integrals involving the $D^\sigma$ are then taken also in the whole $\mathbb {R}^N$ and the condition (2.1) should be read as follows: $A=A(x):\mathbb {R}^N\rightarrow \mathbb {R}^{N\times N}$ is a bounded and measurable matrix, not necessarily symmetric, such that, for some $a_*, a^*>0$ and for a.e. $x\in \mathbb {R}^N$ and all $\xi ,\eta \in \mathbb {R}^N$,

$$\begin{aligned} a_*|\xi |^2\le A(x)\xi \cdot \xi \quad \text {and}\quad A(x)\xi \cdot \eta \le a^*|\xi ||\eta |. \end{aligned}$$

We also need to correct the definitions of

$$\begin{aligned} L^\infty _\nu ({\mathbb {R}^N})= & {} \big \{v\in L^\infty ({\mathbb {R}^N}):v(x)\ge \nu >0 \text{ a.e. } x\in {\mathbb {R}^N}\big \},\\ \Upsilon _\infty ^\sigma (\Omega )= & {} \big \{\upsilon \in H^\sigma _0(\Omega ): D^\sigma \upsilon \in L^\infty (\mathbb {R}^N)^N\big \}\\ \mathbb {K}_g^\sigma= & {} \big \{v\in H^\sigma _0(\Omega ):|D^\sigma v|\le g\text { a.e. in }\mathbb {R}^N\big \}. \end{aligned}$$

and to justify the following inclusions.

Proposition

For any $g\in L^\infty _\nu ({\mathbb {R}^N})$,

$$\begin{aligned} \mathbb {K}_g^\sigma \subset \Upsilon ^\sigma _\infty (\Omega )\subset \mathscr {C}^{0,\beta }(\overline{\Omega })\subset L^\infty (\Omega ), \end{aligned}$$

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

$$\begin{aligned} \Vert u\Vert _{L^\infty (\Omega )}\le \kappa \Vert u\Vert _{H^\sigma _0(\Omega )} \end{aligned}$$

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

Proof

For $u\in \Upsilon _\infty ^\sigma (\Omega )$, as $H^\sigma _0(\Omega )\subset L^{2^*}(\Omega )$ and recalling that $u=0$ on $\mathbb {R}^N\setminus \Omega$, with the estimate

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

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 $2^{**}\ge 2$ is any finite number if $2^*\sigma \ge N$, by using Sobolev imbeddings. Iterating with a bootstrap argument, we obtain $u\in L^{\sigma ,q}(\mathbb {R}^N)^N$, for any $q>\tfrac{N}{\sigma }$, and then by Morrey estimates also $u\in \mathscr {C}^{0,\beta }(\overline{\Omega })$, with $\beta =\sigma -\tfrac{N}{q}$. $\square$

With this estimate, all the proofs of the paper remain essentially the same with simple adaptations. For completion, we restate the main theorems on variational inequalities and the reader may check the details in the corrected pre-print available in https://arxiv.org/pdf/1903.02646.pdf.

Theorem 2.1

Assume that $f_i\in L^1(\Omega )$ and $g_i\in L^\infty (\mathbb {R}^N)$, $g_i\ge 0$, for $i=1,2$. Then there exists a unique solution $u_i$ to

$$\begin{aligned} u_i\in \mathbb {K}_{g_i}^\sigma :\qquad \int _{\mathbb {R}^N} AD^\sigma u_i\cdot D^\sigma (v-u_i)\ge \int _\Omega f_i(v-u_i),\quad \forall v\in \mathbb {K}_{g_i}^\sigma ,\qquad \qquad \end{aligned}$$

such that

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

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 some $C_1=\kappa /a_{*}>0$, we have

$$\begin{aligned} \Vert u_1-u_2\Vert _{H^\sigma _0(\Omega )}\le C_1\Vert f_1-f_2\Vert _{L^1(\Omega )}. \end{aligned}$$

Moreover, if in addition $f_i\in L^{2^{\#}}(\Omega )$, $i=1,2$, 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 $2^*=q$, $2^{\#}=q'=\frac{q}{q-1}$ when $\sigma =\frac{1}{2}$ and $2^*=\infty$, $2^\#=1$ when $\sigma >\frac{1}{2}$, and $g_1=g_2$, then $L^{2^\#}(\Omega )\ni f\mapsto u\in H^\sigma _0(\Omega )$ is Lipschitz continuous:

$$\begin{aligned} \Vert u_1-u_2\Vert _{H^\sigma _0(\Omega )}\le C_\#\Vert f_1-f_2\Vert _{L^{2^\#}(\Omega )}, \end{aligned}$$

for $C_\#=C_*/{a_*}>0$, where $C_*$ is the constant of the Sobolev embedding $H^\sigma _0(\Omega )\hookrightarrow L^{2^*}(\Omega )$.

Theorem 3.1

If $g\in L^\infty _\nu ({\mathbb {R}^N})$ and $f\in L^{2^\#}(\Omega )$, then the Lagrange multipliers problem

$$\begin{aligned} \varvec{\langle }\lambda D^\sigma u,&D^\sigma v\varvec{\rangle }_{\big (L^\infty (\mathbb {R}^N)^N\big )^{\varvec'}\times L^\infty (\mathbb {R}^N)^N}+\int _{\mathbb {R}^N} AD^\sigma u\cdot D^\sigma v=\int _{\Omega }f v,\qquad \forall v\in \Upsilon _\infty ^\sigma (\Omega ),\\&|D^\sigma u|\le g\ \text { a.e. in }{\mathbb {R}^N},\ \lambda \ge 0 \text{ and } \lambda (|D^\sigma u|-g)=0\ \text{ in } L^\infty (\mathbb {R}^N)^{\varvec'} \end{aligned}$$

has a solution

$$\begin{aligned} (\lambda , u)\in L^{\infty }({\mathbb {R}^N})^{\varvec'}\times \Upsilon _\infty ^\sigma (\Omega ). \end{aligned}$$

Moreover, u solves the variational inequality of Theorem 2.1.

In the last section on quasi-variational inequalities, where the nonlinear map for the threshold $g=G[u]$, depending on the solution u, has now image in $L^\infty _\nu (\mathbb {R}^N)$, the results are also almost unchanged, with the exception of the Theorem 4.3, which is improved by the estimate of the above proposition, essentially with the same proof (see https://arxiv.org/pdf/1903.02646.pdf). It reads now in the following form.

Theorem 4.3

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

$$\begin{aligned} G: \mathscr {C}^0(\overline{\Omega })\rightarrow L^\infty _\nu (\mathbb {R}^N)\qquad \text { is a continuous operator.} \end{aligned}$$

Then there exists a solution of the quasi-variational inequality

$$\begin{aligned}&u\in \mathbb {K}_{G[u]}^\sigma =\big \{v\in H^\sigma _0(\Omega ):|D^\sigma v|\le G[u]\text { a.e. in }\mathbb {R}^N \big \}\\&\int _{\mathbb {R}^N} AD^\sigma u\cdot D^\sigma (v-u)\ge \int _{\Omega }f(v-u),\qquad \forall v\in \mathbb {K}_{G[u]}^\sigma . \end{aligned}$$