Maximum principles for boundary-degenerate second-order linear elliptic differential operators

TL;DR

This paper establishes maximum principles and related results for second-order linear elliptic operators that degenerate along part of the boundary, broadening the understanding of boundary behavior for such PDEs.

Contribution

It proves maximum principles, Hopf lemma, and uniqueness results for boundary-degenerate elliptic operators, including boundary conditions and variational solutions, regardless of the sign of the Fichera function.

Findings

Maximum principles hold despite boundary degeneracy.

Uniqueness of solutions for boundary value and obstacle problems.

Maximum principle estimates for solutions with weighted Sobolev spaces.

Abstract

We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the smooth subsolutions along this boundary vanishing locus ensures that these maximum principles hold irrespective of the sign of the Fichera function. Boundary conditions need only be prescribed on the complement in the domain boundary of the principal symbol vanishing locus. We obtain uniqueness and a priori maximum principle estimates for smooth solutions to boundary value and obstacle problems defined by these boundary-degenerate elliptic operators for partial Dirichlet or Neumann boundary conditions along the complement of the boundary vanishing locus. We also prove weak maximum principles and uniqueness for solutions to the corresponding variational equations and inequalities defined with the aide of weighted Sobolev spaces. The domain is allowed to be unbounded when the operator coefficients and solutions obey certain growth conditions.