相关论文: Nonlocal operators of order near zero

We study Dirichlet problems for fractional Laplace equations of the form $(-\Delta)^{\frac{\alpha}{2}} u = f(x,u)$ in $\mathbb{R}^{n}$ for $0<\alpha<n$ where the nonlinearity $f(x,u) = \sum_{i=1}^{M} \sigma_{i} u^{q_i} + \omega$ involves…

We consider a broad class of nonlinear integro-differential equations with a kernel whose differentiability order is described by a general function $\phi$. This class includes not only the fractional $p$-Laplace equations, but also…

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…

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 establish optimal regularity estimates and smoothness of free boundaries for nonlocal obstacle problems governed by a very general class of integro-differential operators with possibly singular kernels. More precisely, in…

This paper deals with fractional Sobolev spaces on a compact Riemannian manifold. We prove a Sobolev inequality in the critical range with an optimal constant for these fractional Sobolev spaces. We use this result to study the existence of…

We study the Dirichlet problem for the stationary Schr\"odinger fractional Laplacian equation $(-\Delta)^s u + V u = f$ posed in bounded domain $ \Omega \subset \mathbb R^n$ with zero outside conditions. We consider general nonnegative…

In this work, we establish some abstract results on the perspective of the fractional Musielak-Sobolev spaces, such as: uniform convexity, Radon-Riesz property with respect to the modular function, $(S_{+})$-property, Brezis-Lieb type Lemma…

In this paper we consider a family of non local functionals of convolution-type depending on a small parameter $\varepsilon>0$ and $\Gamma$-converging to local functionals defined on Sobolev spaces as $\varepsilon\to 0$. We study the…

In this work, we consider the nonlocal obstacle problem with a given obstacle $\psi$ in a bounded Lipschitz domain $\Omega$ in $\mathbb{R}^{d}$, such that $\mathbb{K}_\psi^s=\{v\in H^s_0(\Omega):v\geq\psi \text{ a.e. in…

In this paper we present a preliminary study on the Dirichlet-to-Neumann operator with respect to a second order elliptic operator with measurable coefficients, including first order terms, namely, the operator on $L^2(\partial\Omega)$…

We will present versions of the Rellich-Kondrachov theorem for pseudo-differential operators acting on localizable Hardy spaces. One of the techniques includes boundedness properties for pseudodifferential operators with symbols in the…

We establish a Sobolev-type inequality in Lorentz spaces for $\mathcal{L}$-superharmonic functions \[ \|u\|_{L^{\frac{nq}{n-\alpha q},t}(\mathbb{R}^n)} \leq c \left\| \frac{u(x) - u(y)}{|x-y|^{\frac{n}{q}+\alpha}}…

For indices p and q, 1 < p <= q < infini and a linear operator L satisfying some weak-type boundedness conditions on suitable function spaces, we give in the Dunkl setting sufficient conditions on nonnegative pairs of weight functions to…

The objective of this paper is twofold. First, we conduct a careful study of various functional inequalities involving the fractional Laplacian operators, including nonlocal Sobolev-Poincar\'e, Nash, Super Poincar\'e and logarithmic Sobolev…

We prove existence and uniqueness of mild solutions to Sobolev type fractional nonlocal dynamic equations in Banach spaces. The Sobolev nonlocal condition is considered in terms of a Riemann-Liouville fractional derivative. A Lagrange…

Let $D$ be a bounded $C^2$-domain. Consider the following Dirichlet initial-boundary problem of nonlocal operators with a drift: $$ \partial_t u={\mathscr L}^{(\alpha)}_\kappa u+b\cdot \nabla u+f\ \mathrm{in}\ \mathbb R_+\times D,\ \…

We consider a family of self-adjoint Ornstein--Uhlenbeck operators $L_{\alpha} $ in an infinite dimensional Hilbert space H having the same gaussian invariant measure $\mu$ for all $\alpha \in [0,1]$. We study the Dirichlet problem for the…

We study integral functionals defined on scalar Sobolev spaces of the form $$E[f]:u\mapsto \int_\Omega f(x,u(x),\nabla u(x)) d x,$$ with an emphasis on the non-convex case, and the difficulties it involves to prevent the Lavrentiev…

We study nonuniform Sobolev spaces, i.e., spaces of functions whose partial derivatives lie in possibly different Lebesgue spaces. Although standard proofs do not apply, we show that nonuniform Sobolev spaces share similar properties as the…