Related papers: The Dirichlet problem for nonlocal L\'evy-type ope…
In this paper, we study Dirichlet problem for non-local operator on bounded domains in ${\mathbb R}^d$ $$ {\cal L}u = {\rm div}(A(x) \nabla (x)) + b(x) \cdot \nabla u(x) + \int_{{\mathbb R}^d} (u(y)-u(x) ) J(x, dy) , $$ where…
We study a Dirichlet problem in the entire space for some nonlocal degenerate elliptic operators with internal nonlinearities. With very mild assumptions on the boundary datum, we prove existence and uniqueness of the solution in the…
We consider the Dirichlet problem on general, possibly nonsmooth bounded domain, for elliptic linear equation with uniformly elliptic divergence form operator. We investigate carefully the relationship between weak, soft and the…
In this paper, we investigate the existence of weak solution for a Kirchhoff type problem driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions {\small$$…
In this work, we study the Dirichlet problem associated with a strongly coupled system of nonlocal equations. The system of equations comes from a linearization of a model of peridynamics, a nonlocal model of elasticity. It is a nonlocal…
We investigate a connection between solvability of the Dirichlet problem for an infinitely degenerate elliptic operator and the validity of an Orlicz-Sobolev inequality in the associated subunit metric space. For subelliptic operators it is…
In this paper we study the maximum principle, the existence of eigenvalue and the existence of solution for the Dirichlet problem for operators which are fully-nonlinear, elliptic but presenting some singularity or degeneracy which are…
We study the problem of existence, uniqueness and regularity of probabilistic solutions of the Cauchy problem for nonlinear stochastic partial differential equations involving operators corresponding to regular (nonsymmetric) Dirichlet…
We investigate the characterization of generators $\mathcal{L}$ of L\'evy processes satisfying the Liouville theorem: Bounded functions $u$ solving $\mathcal{L}[u]=0$ are constant. These operators are degenerate elliptic of the form…
We show continuity in generalized weighted Morrey spaces of sub-linear integral operators generated by some classical integral operators and commutators. The obtained estimates are used to study global regularity of the solution of the…
We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space $\mathbb{R}^d_+$, where the coefficients are the product of $x_d^\alpha, \alpha \in (-\infty, 1),$ and a bounded uniformly elliptic…
We prove an optimal extension and trace theorem for Sobolev spaces of nonlocal operators. The extension is given by a suitable Poisson integral and solves the corresponding nonlocal Dirichlet problem. We give a Douglas-type formula for the…
In this article we consider direct and inverse problems for $\alpha$-stable, elliptic nonlocal operators whose kernels are possibly only supported on cones and which satisfy the structural condition of \emph{directional antilocality} as…
In this paper, we study the solvability of a Cauchy- Dirichlet problem for nonlinear parabolic equation with non standard growths and nonlocal terms. We show the existence of weak solutions of the considered problem under more general…
We develop a Widder-type theory for nonlocal heat equations involving quite general L\'evy operators. Thus, we consider nonnegative solutions and look for conditions on the operator that ensure: (i) uniqueness of nonnegative classical and…
We study the Dirichlet problem in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order, with bounded, complex-valued coefficients. Our main result gives a sharp condition…
For nonlinear operators of fractional $p$-Laplace type, we consider two types of solutions to the nonlocal Dirichlet problem: Sobolev solutions based on fractional Sobolev spaces and Perron solutions based on superharmonic functions. These…
In this paper we study an equation driven by a nonlocal anisotropic operator with homogeneous Dirichlet boundary conditions. We find at least three non trivial solutions: one positive, one negative and one of unknown sign, using variational…
We derive a criterium for the almost sure finiteness of perpetual integrals of \LL processes for a class of real functions including all continuous functions and for general one-dimensional L\'evy processes that drifts to plus infinity.…
We present a new approach to absolute continuity of laws of Poisson functionals. The theoretical framework is that of local Dirichlet forms as a tool to study probability spaces. The method gives rise to a new explicit calculus that we show…