Perturbations of local maxima and comparison principles for boundary-degenerate linear differential equations

TL;DR

This paper establishes strong and weak maximum principles for boundary-degenerate elliptic and parabolic PDEs, allowing for perturbations that convert boundary maxima into interior maxima, extending previous results to less regular function spaces.

Contribution

It introduces novel maximum principles for boundary-degenerate operators in Sobolev spaces, relaxing regularity assumptions and handling degeneracies along parts of the boundary.

Findings

Maximum principles hold for functions in Sobolev spaces with boundary degeneracy.

Perturbation method transforms boundary maxima into interior maxima.

Results extend prior work by relaxing regularity and degeneracy assumptions.

Abstract

We develop strong and weak maximum principles for boundary-degenerate elliptic and parabolic linear second-order partial differential operators, $Au := -\mathrm{tr}(aD^2u)-<b, Du> + cu$, with partial Dirichlet boundary conditions. The coefficient, $a(x)$, is assumed to vanish along a non-empty open subset, $\partial_0\mathscr{O}$, called the \emph{degenerate boundary portion}, of the boundary, $\partial\mathscr{O}$, of the domain $\mathscr{O}\subset\mathbb{R}^d$, while $a(x)$ is non-zero at any point of the \emph{non-degenerate boundary portion}, $\partial_1\mathscr{O} := \partial\mathscr{O}\setminus\overline{\partial_0\mathscr{O}}$. If an $A$-subharmonic function, $u$ in $C^2(\mathscr{O})$ or $W^{2,d}_{\mathrm{loc}}(\mathscr{O})$, is $C^1$ up to $\partial_0\mathscr{O}$ and has a strict local maximum at a point in $\partial_0\mathscr{O}$, we show that $u$ can be perturbed, by the addition of a suitable function $w\in C^2(\mathscr{O})\cap C^1(\mathbb{R}^d)$, to a strictly $A$-subharmonic function $v=u+w$ having a local maximum in the interior of $\mathscr{O}$. Consequently, we obtain strong and weak maximum principles for $A$-subharmonic functions in $C^2(\mathscr{O})$ and $W^{2,d}_{\mathrm{loc}}(\mathscr{O})$ which are $C^1$ up to $\partial_0\mathscr{O}$. Only the non-degenerate boundary portion, $\partial_1\mathscr{O}$, is required for boundary comparisons. Our results extend those in Daskalopoulos and Hamilton (1998), Epstein and Mazzeo [arXiv:1110.0032], and the author [arXiv:1204.6613, 1306.5197], where $\mathrm{tr}(aD^2u)$ is in addition assumed to be continuous up to and vanish along $\partial_0\mathscr{O}$ in order to yield comparable maximum principles for $A$-subharmonic functions in $C^2(\mathscr{O})$, while the results developed here for $A$-subharmonic functions in $W^{2,d}_{\mathrm{loc}}(\mathscr{O})$ are entirely new.