Related papers: Incorporating local boundary conditions into nonlo…

In this paper, having introduced a convergence of a series on the root vectors in the Abel-Lidskii sense, we present a valuable application to the evolution equations. The main issue of the paper is an approach allowing us to principally…

The nonlocal models of peridynamics have successfully predicted fractures and deformations for a variety of materials. In contrast to local mechanics, peridynamic boundary conditions must be defined on a finite volume region outside the…

In this paper we survey some results on the Dirichlet problem \[\left\{ \begin{array}{rcll} L u &=&f&\textrm{in }\Omega \\ u&=&g&\textrm{in }\mathbb R^n\backslash\Omega \end{array}\right.\] for nonlocal operators of the form…

We present an existence and uniqueness result for weak solutions of Dirichlet boundary value problems governed by a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to $L^1(\Omega)$ and to…

We consider the wave operator on static, Lorentzian manifolds with timelike boundary and we discuss the existence of advanced and retarded fundamental solutions in terms of boundary conditions. By means of spectral calculus we prove that…

In this paper, we study two local--nonlocal settings for parabolic--elliptic evolution systems. In our problems we have a disjoint partition of the spacial domain $\Omega$ as $\Omega=A\cup B$ and we first consider a local parabolic equation…

Results on the peridynamics equilibrium and evolution equations over the space of periodic vector-distributions in multi-spatial dimensions are presented. The associated operator considered is the linear state-based peridynamic operator for…

In this paper we are interested in integro-differential elliptic and parabolic equations involving nonlocal operators with order less than one, and a gradient term whose coercivity growth makes it the leading term in the equation. We obtain…

In this paper, with a fixed $p\in (1,+\infty)$ and a bounded domain $\Omega \subset \mathbb{R}^N$ whose boundary $\partial\Omega$ fulfills the $C^1$ regularity, we study a boundary value problem involving a nonlocal operator assigning to…

We derive the long time asymptotic of solutions to an evolutive Hamilton-Jacobi-Bellman equation in a bounded smooth domain, in connection with ergodic problems recently studied in \cite{bcr}. Our main assumption is an appropriate…

We study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary. This problem may be seen as a generalization of the usual Neumann problem for the heat equation. First, we prove existence,…

We propose a reformulation of the boundary integral equations for the Helmholtz equation in a domain in terms of incoming and outgoing boundary waves. We obtain transfer operator descriptions which are exact and thus incorporate features…

In this paper second order elliptic boundary value problems on bounded domains $\Omega\subset\dR^n$ with boundary conditions on $\partial\Omega$ depending nonlinearly on the spectral parameter are investigated in an operator theoretic…

Within the framework of Hilbert spaces, we solve nonlocal problems in bounded domains with prescribed conditions on the complement of the domain. Our main focus is on the inhomogeneous Neumann problem in a rather general setting. We also…

In this work, we address a parabolic problem featuring a potentially doubly nonlinear term, governed by a combination of local and nonlocal operators (see Problem P1 below). We first establish the local existence of weak energy solutions…

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 evolution equations of the form \begin{equation*}\label{Abstract equation} \dot u(t)+ A(t)u(t)=0,\ \ t\in[0,T],\ \ u(0)=u_0, \end{equation*} where $A(t),\ t\in [0,T],$ are associated with a non-autonomous sesquilinear form…

In this paper, we consider non-autonomous Ornstein-Uhlenbeck operators in smooth exterior domains $\Omega\subset \R^d$ subject to Dirichlet boundary conditions. Under suitable assumptions on the coefficients, the solution of the…

In this article we present the first results on domain decomposition methods for nonlocal operators. We present a nonlocal variational formulation for these operators and establish the well-posedness of associated boundary value problems,…

We consider a non-relativistic particle in a one-dimensional box with all possible quantum boundary conditions that make the kinetic-energy operator selfadjoint. We determine the Wigner functions of the corresponding eigenfunctions and…