Related papers: Generalized solutions of nonlocal elliptic problem…

This paper investigates realisations of elliptic differential operators of general order on manifolds with boundary following the approach of B\"ar-Ballmann to first order elliptic operators. The space of possible boundary values of…

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 study linear boundary-value problems for systems of first-order ordinary differential equations with the most general boundary conditions in the normed spaces of continuously differentiable functions on a finite closed interval. The…

This paper investigates the mathematical properties and numerical approximation of a class of nonlocal elliptic partial differential equations of the form \begin{equation*} -\Delta u + \lambda \, G(u) = f, \end{equation*} where $\Delta$…

The main topic is the development of a Fredholm theory in a new class of spaces called M-polyfolds. In the subsequent Volume II the theory will be generalized to an even larger class of spaces called polyfolds, which can also incorporate…

We study nonlocal elliptic and parabolic equations on $C^{1,\tau}$ open sets in weighted Sobolev spaces, where $\tau\in (0,1)$. The operators we consider are infinitesimal generators of symmetric stable L\'evy processes, whose L\'evy…

We consider a boundary value problem in a bounded domain involving a degenerate operator of the form $$L(u)=-\textrm{div} (a(x)\nabla u)$$ and a suitable nonlinearity $f$. The function $a$ vanishes on smooth 1-codimensional submanifolds of…

We consider the mapping properties of generalized Laplace-type operators ${\mathcal L} = \nabla^* \nabla + {\mathcal R}$ on the class of quasi-asymptotically conical (QAC) spaces, which provide a Riemannian generalization of the QALE…

We introduce a refined Sobolev scale on a vector bundle over a closed infinitely smooth manifold. This scale consists of inner product H\"ormander spaces parametrized with a real number and a function varying slowly at infinity in the sense…

This work concerns with the existence of solutions for the following class of nonlocal elliptic problems \begin{equation*}\label{00} \left\{ \begin{array}{l} (-\Delta)^{s}u + u = |u|^{p-2}u\;\;\mbox{in $\Omega$},\\ u \geq 0 \quad \mbox{in}…

We develop local elliptic regularity for operators having coefficients in a range of Sobolev-type function spaces (Bessel potential, Sobolev-Slobodeckij, Triebel-Lizorkin, Besov) where the coefficients have a regularity structure typical of…

We prove higher-order fractional Sobolev regularity for fully nonlinear, uniformly elliptic equations in the presence of unbounded source terms. More precisely, we show the existence of a universal number $0< \varepsilon <1$, depending only…

We study the zero exterior problem for the elliptic equation $$ \Delta^{\alpha/2}u-\lambda u=f, \quad x\in D\,; \quad u|_{D^c}=0 $$ as well as for the parabolic equation $$ u_t=\Delta^{\alpha/2}u+f, \quad t>0,\, x\in D \,; \quad…

We prove the well posedness in weighted Sobolev spaces of certain linear and nonlinear elliptic boundary value problems posed on convex domains and under singular forcing. It is assumed that the weights belong to the Muckenhoupt class $A_p$…

We consider an elliptic boundary problem over a bounded region $\Omega$ in $\mathbb{R}^n$ and acting on the generalized Sobolev space $W^{0,\chi}_p(\Omega)$ for $1 < p < \infty$. We note that similar problems for $\Omega$ either a bounded…

This paper aims to establish counterparts of fundamental regularity statements for solutions to elliptic equations in the setting of low-dimensional structures such as, for instance, glued manifolds or CW-complexes. The main result proves…

We study a class of nonlinear elliptic problems with Dirichlet conditions in the framework of the Sobolev anisotropic spaces with variable exponent, involving an anisotropic operator on an unbounded domain $\Omega\subset \>I\!\!R^{N}\>(N…

We study an asymptotic behavior of solutions to elliptic equations of the second order in a two dimensional exterior domain. Under the assumption that the solution belongs to $L^q$ with $q \in [2,\infty)$, we prove a pointwise asymptotic…

This paper considers boundary value problems for a class of singular elliptic operators which appear naturally in the study of asymptotically anti-de Sitter (aAdS) spacetimes. These problems involve a singular Bessel operator acting in the…

We consider a general class of parametrized displacement boundary value problems in incompressible nonlinear elasticity. We prove the existence of an unbounded solution branch of classical injective solutions emanating from the unforced…