Unilateral Problems for Quasilinear Operators with Fractional Riesz Gradients

Abstract

In this work, we develop the classical theory of monotone and pseudomonotone operators in the class of convex constrained Dirichlet-type problems involving fractional Riesz gradients in bounded and in unbounded domains $\Omega\subset\mathbb{R}^d$. We consider the problem of finding $u\in K^s$, such that, \begin{equation*} \int_{\mathbb{R}^d}{\boldsymbol{a}(x,u,D^s u)\cdot D^s(v-u)}\,dx+\int_\Omega{b(x,u,D^s u)(v-u)}\,dx\geq 0 %\langle F, v-u\rangle \end{equation*} for all $v\in K^s$. Here $K^s\subset\Lambda^{s,p}_0(\Omega)$ is a non-empty, closed and convex set of a fractional Sobolev type space $\Lambda^{s,p}_0(\Omega)$ with $0\leq s\leq 1$ and $1<p<\infty$, and $D^s$ denotes the distributional Riesz fractional gradient, with two limit cases: $D^1=D$ representing the classical gradient in the classical Sobolev space $\Lambda^{1,p}_0(\Omega)=W^{1,p}_0(\Omega)$, and $-D^0=R$ denotes the vector-valued Riesz transform within $\Lambda^{0,p}_0(\Omega)=\{u\in L^p(\mathbb{R}^d):\, u=0 \mbox{ a.e. in } \mathbb{R}^d\setminus\Omega\}$. We discuss the existence and uniqueness of solutions in this novel framework and we obtain new results on the continuous dependence, with respect to the fractional parameter $s$, of variational solutions corresponding to several classical assumptions on the structural functions $\boldsymbol{a}$ and $b$ adapted to the fractional framework. We introduce an extension of the Mosco convergence for convex sets $K^s$ with respect to the parameter $s$, including the limit cases $s=1$ and $s=0$, to prove weak or strong convergences of the solutions $u_s$ and their fractional gradients $D^su$, according to different cases. Several applications are illustrated with examples of unilateral problems, including quasi-variational inequalities with constraints of obstacle type $u\geq \psi$ and $s$-gradient type $|D^su|\leq g$.