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We construct a solution operator for $\overline{\partial}$ equation that gains $\frac{1}{2}$ derivative in the fractional Sobolev space $H^{s,p}$ on bounded strictly pseudoconvex domains in $\mathbb{C}^n$ with $C^2$ boundary, for all $1 < p…
In this paper we study the obstacle problems for the Navier (spectral) fractional Laplacian $\left(-\Delta_\Omega\right)^{\!s}$ of order $s\in(0,1)$, in a bounded domain $\Omega\subset\mathbb R^n$.
We define fractional derivatives $\pppa$ in Sobolev spaces based on $L^p(0,T)$ by an operator theory, and characterize the domain of $\pppa$ in subspaces of the Sobolev-Slobodecki spaces $W^{\alpha,p}(0,T)$. Moreover we define $\pppa u$ for…
By virtue of a suitable approximation argument, we prove a Pohozaev identity for nonlinear nonlocal problems on $\mathbb{R}^N$ involving the fractional $p-$Laplacian operator. Furthermore we provide an application of the identity to show…
We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave-convex term. We characterize completely the range of parameters for which solutions of the…
In this paper, we are interested in studying the multiplicity, uniqueness, and nonexistence of solutions for a class of singular elliptic eigenvalue problem for the Dirichlet fractional $(p,q)$-Laplacian. The nonlinearity considered…
In this paper, we study the local boundedness of local weak solutions to the following parabolic equation associated with fractional $p$-Laplacian type operators $$ \partial_t…
We qualify a relevant range of fractional powers of the so-called Hamiltonian of point interaction in three dimensions, namely the singular perturbation of the negative Laplacian with a contact interaction supported at the origin. In…
In this paper we study the fractional p(., .)-Laplacian and we introduce the corresponding nonlocal conormal derivative for this operator. We prove basic properties of the corresponding function space and we establish a nonlocal version of…
We study semilinear problems in bounded $C^{1,1}$ domains for non-local operators with a boundary condition. The operators cover and extend the case of the spectral fractional Laplacian. We also study harmonic functions with respect to the…
In this work we study the homogenization for eigenvalues of the fractional $p-$Laplace in a bounded domain both with Dirichlet and Neumann conditions. We obtain the convergence of eigenvalues and the explicit order of the convergence rates.
In this paper, we develop some properties of the $a_{x,y}(.)$-Neumann derivative for the fractional $a_{x,y}(.)$-Laplacian operator. Therefore we prove the basic proprieties of the correspondent function spaces. In the second part of this…
In this article, we consider mixed local and nonlocal Sobolev $(q,p)$-inequalities with extremal in the case $0<q<1<p<\infty$. We prove that the extremal of such inequalities is unique up to a multiplicative constant that is associated with…
For fractional derivatives and time-fractional differential equations, we construct a framework on the basis of the operator theory in fractional Sobolev spaces. Our framework provides a feasible extension of the classical Caputo and the…
In this paper, we are concerned with the asymptotic behavior of weak solutions to certain elliptic and parabolic problems involving the fractional $p$-Laplacian in cylindrical domains that become unbounded in one direction. The nonlocal…
In this article, we derive the existence of positive solutions of a semi-linear, non-local elliptic PDE, involving a singular perturbation of the fractional laplacian, coming from the fractional Hardy-Sobolev-Maz'ya inequality, derived in…
We establish new Euclidean Sobolev logarithmic inequalities in the framework of fractional Sobolev spaces and their weighted version. Our approach relies on a interpolation inequality, which can be viewed as a fractional…
Sharp affine fractional $L^p$ Sobolev inequalities for functions on $\mathbb R^n$ are established. The new inequalities are stronger than (and directly imply) the sharp fractional $L^p$ Sobolev inequalities. They are fractional versions of…
In this paper, we prove a version of the logarithmic Sobolev inequality of fractional order on noncommutative $n$-tori for any dimension $n\geq 2$.
The aim of this paper is to study a class of nonlocal fractional Laplacian equations depending on two real parameters. More precisely, by using an appropriate analytical context on fractional Sobolev spaces due to Servadei and Valdinoci, we…