A nonsmooth variational approach to differential problems. A case study of nonresonance under the first eigenvalue for a strongly nonlinear elliptic problem

Weighted $L^p$-estimates for elliptic equations with measurable coefficients in nonsmooth domains

We obtain a global weighted $L^p$ estimate for the gradient of the weak solutions to divergence form elliptic equations with measurable coefficients in a nonsmooth bounded domain. The coefficients are assumed to be merely measurable in one variable and to have small BMO semi-norms in the remaining variables, while the boundary of the domain is supposed to be Reifenberg flat, which goes beyond the category of domains with Lipschitz continuous boundaries. As consequence of the main result, we derive global gradient estimate for the weak solution in the framework of the Morrey spaces which implie…

Extension Theory and Krein-type Resolvent Formulas for Nonsmooth Boundary Value Problems

For a strongly elliptic second-order operator $A$ on a bounded domain $Ω\subset \mathbb{R}^n$ it has been known for many years how to interpret the general closed $L_2(Ω)$-realizations of $A$ as representing boundary conditions (generally nonlocal), when the domain and coefficients are smooth. The purpose of the present paper is to extend this representation to nonsmooth domains and coefficients, including the case of Hölder $C^{\frac32+\varepsilon}$-smoothness, in such a way that pseudodifferential methods are still available for resolvent constructions and ellipticity considerations. We show…

Spectral asymptotics for nonsmooth singular Green operators

Singular Green operators G appear typically as boundary correction terms in resolvents for elliptic boundary value problems on a domain Ω\subset R^n, and more generally they appear in the calculus of pseudodifferential boundary problems. In particular, the boundary term in a Krein resolvent formula is a singular Green operator. It is well-known in smooth cases that when G is of negative order -t on a bounded domain, its eigenvalues or s-numbers have the behavior (*) s_j(G) \sim c j^{-t/(n-1)} for j\to \infty, governed by the boundary dimension n-1. In some nonsmooth cases, upper estimates (**)…

Deterministic homogenization of elliptic equations with lower order terms

For a class of linear elliptic equations of general type with rapidly oscillating coefficients, we use the sigma-convergence method to prove the homogenization result and a corrector-type result. In the case of asymptotic periodic coefficients we derive the optimal convergence rates for the zero order approximation of the solution with no smoothness on the coefficients, in contrast to what has been done up to now in the literature. This follows as a result of the existence of asymptotic periodic correctors for general nonsmooth coefficients. The homogenization process is achieved through a com…

Regularity of spectral fractional Dirichlet and Neumann problems

Consider the fractional powers $(A_{\operatorname{Dir}})^a$ and $(A_{\operatorname{Neu}})^a$ of the Dirichlet and Neumann realizations of a second-order strongly elliptic differential operator $A$ on a smooth bounded subset $Ω$ of ${\Bbb R}^n$. Recalling the results on complex powers and complex interpolation of domains of elliptic boundary value problems by Seeley in the 1970's, we demonstrate how they imply regularity properties in full scales of $H^s_p$-Sobolev spaces and Hölder spaces, for the solutions of the associated equations. Extensions to nonsmooth situations for low values of $s$ a…

非光滑情形下分数阶 Dirichlet 实现的 Weyl 渐近

Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases

本文将分数阶强椭圆伪微分算子 Dirichlet 实现的 Weyl 特征值渐近公式从光滑情形推广到非光滑系数、Lipschitz 区域及含低阶扰动项的情形，并讨论了特征函数的正则性。