Boundary-degenerate elliptic operators and Hölder continuity for solutions to variational equations and inequalities

Paul M.N. Feehan; Camelia A. Pop

doi:10.1016/j.anihpc.2016.07.005

Boundary-degenerate elliptic operators and Hölder continuity for solutions to variational equations and inequalities

Paul M.N. Feehan, Camelia A. Pop

Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

We prove local supremum bounds, a Harnack inequality, Hölder continuity up to the boundary, and a strong maximum principle for solutions to a variational equation defined by an elliptic operator which becomes degenerate along a portion of the domain boundary and where no boundary condition is prescribed, regardless of the sign of the Fichera function. In addition, we prove Hölder continuity up to the boundary for solutions to variational inequalities defined by this boundary-degenerate elliptic operator.

Original language	English (US)
Pages (from-to)	1075-1129
Number of pages	55
Journal	Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume	34
Issue number	5
DOIs	https://doi.org/10.1016/j.anihpc.2016.07.005
State	Published - May 29 2017

Bibliographical note

Funding Information:
PF was partially supported by NSF grant DMS-1059206.

Publisher Copyright:
© 2016 Elsevier Masson SAS

Keywords

Degenerate diffusion process
Degenerate elliptic differential operator
Harnack inequality
Hölder continuity
Variational inequality
Weighted Sobolev space

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AU - Pop, Camelia A.

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N2 - We prove local supremum bounds, a Harnack inequality, Hölder continuity up to the boundary, and a strong maximum principle for solutions to a variational equation defined by an elliptic operator which becomes degenerate along a portion of the domain boundary and where no boundary condition is prescribed, regardless of the sign of the Fichera function. In addition, we prove Hölder continuity up to the boundary for solutions to variational inequalities defined by this boundary-degenerate elliptic operator.

AB - We prove local supremum bounds, a Harnack inequality, Hölder continuity up to the boundary, and a strong maximum principle for solutions to a variational equation defined by an elliptic operator which becomes degenerate along a portion of the domain boundary and where no boundary condition is prescribed, regardless of the sign of the Fichera function. In addition, we prove Hölder continuity up to the boundary for solutions to variational inequalities defined by this boundary-degenerate elliptic operator.

KW - Degenerate diffusion process

KW - Degenerate elliptic differential operator

KW - Harnack inequality

KW - Hölder continuity

KW - Variational inequality

KW - Weighted Sobolev space

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Boundary-degenerate elliptic operators and Hölder continuity for solutions to variational equations and inequalities

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