%0 Journal Article
%T Perturbations of local maxima and comparison principles for boundary-degenerate linear differential equations
%A Paul M. N. Feehan
%J Mathematics
%D 2013
%I arXiv
%X We develop strong and weak maximum principles for boundary-degenerate elliptic and parabolic linear second-order partial differential operators, $Au := -\tr(aD^2u)-<b, Du> + cu$, with \emph{partial Dirichlet boundary conditions}. The coefficient, $a(x)$, is assumed to vanish along a non-empty open subset, $\partial_0\sO$, called the \emph{degenerate boundary portion}, of the boundary, $\partial\sO$, of the domain $\sO\subset\RR^d$, while $a(x)$ is non-zero at any point of the \emph{non-degenerate boundary portion}, $\partial_1\sO := \partial\sO\less\bar{\partial_0\sO}$. If an $A$-subharmonic function, $u$ in $C^2(\sO)$ or $W^{2,d}_{\loc}(\sO)$, is $C^1$ up to $\partial_0\sO$ and has a strict local maximum at a point in $\partial_0\sO$, we show that $u$ can be perturbed, by the addition of a suitable function $w\in C^2(\sO)\cap C^1(\RR^d)$, to a strictly $A$-subharmonic function $v=u+w$ having a local maximum in the interior of $\sO$. Consequently, we obtain strong and weak maximum principles for $A$-subharmonic functions in $C^2(\sO)$ and $W^{2,d}_{\loc}(\sO)$ which are $C^1$ up to $\partial_0\sO$. Only the non-degenerate boundary portion, $\partial_1\sO$, 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 $\tr(aD^2u)$ is in addition assumed to be continuous up to and vanish along $\partial_0\sO$ in order to yield comparable maximum principles for $A$-subharmonic functions in $C^2(\sO)$, while the results developed here for $A$-subharmonic functions in $W^{2,d}_{\loc}(\sO)$ are entirely new. Finally, we obtain analogues of all the preceding results for parabolic linear second-order partial differential operators.
%U http://arxiv.org/abs/1305.5098v2