The Dirichlet problem for the uniformly higher-order elliptic equations in generalized weighted Sobolev-Morrey spaces (original) (raw)

Dirichlet boundary value problems for uniformly elliptic equations in modified local generalized Sobolev–Morrey spaces

Electronic Journal of Qualitative Theory of Differential Equations

In this paper, we study the boundedness of the sublinear operators, generated by Calderón-Zygmund operators in local generalized Morrey spaces. By using these results we prove the solvability of the Dirichlet boundary value problem for a polyharmonic equation in modified local generalized Sobolev-Morrey spaces. We obtain a priori estimates for the solutions of the Dirichlet boundary value problems for the uniformly elliptic equations in modified local generalized Sobolev-Morrey spaces defined on bounded smooth domains.

Elliptic Equations in Weighted Sobolev Spaces on Unbounded Domains

International Journal of Mathematics and Mathematical Sciences, 2008

We study in this paper a class of second-order linear elliptic equations in weighted Sobolev spaces on unbounded domains of R n , n ≥ 3. We obtain an a priori bound, and a regularity result from which we deduce a uniqueness theorem.

The Dirichlet problem in a class of generalized weighted Morrey spaces

Proceedings of the Institute of Mathematics and Mechanics,National Academy of Sciences of Azerbaijan

We show continuity in generalized weighted Morrey spaces M p,ϕ (w) 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 Dirichlet problem for linear uniformly elliptic operators with discontinuous data.

Boundary regularity for quasilinear elliptic equations with general Dirichlet boundary data

arXiv: Analysis of PDEs, 2018

We study global regularity for solutions of quasilinear elliptic equations of the form divA(x,u,nablau)=divF\div \A(x,u,\nabla u) = \div \F divA(x,u,nablau)=divF in rough domains Omega\OmegaOmega in Rn\R^nRn with nonhomogeneous Dirichlet boundary condition. The vector field A\AA is assumed to be continuous in uuu, and its growth in nablau\nabla unablau is like that of the ppp-Laplace operator. We establish global gradient estimates in weighted Morrey spaces for weak solutions uuu to the equation under the Reifenberg flat condition for Omega\OmegaOmega, a small BMO condition in xxx for A\AA, and an optimal condition for the Dirichlet boundary data.

Global Regularity in Generalized Morrey Spaces of Solutions to Nondivergence Elliptic Equations with VMO Coefficients

Potential Analysis, 2013

We show continuity in generalized Morrey spaces of sublinear integral operators generated by Calderón-Zygmund operator and their commutators with BMO functions. The obtained estimates are used to study global regularity of the solution of the Dirichlet problem for linear uniformly elliptic operators. Keywords Generalized Morrey spaces • Sublinear integrals • Calderón-Zygmund integrals and commutators • BMO • V MO • Elliptic equations • Dirichlet problem Mathematics Subject Classifications (2010) 35J25 • 35B40 • 42B20 • 42B35