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

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.