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, $Au := -\mathrm {tr}(aD^2u)-\langle b, Du\rangle + cu$, with partial Dirichlet boundary conditions. The coefficient, $a(x)$, is assumed to vanish along a nonempty open subset, $\partial _0\mathscr {O}$, called the degenerate boundary portion, of the boundary, $\partial \mathscr {O}$, of the domain $\mathscr {O}\subset \mathbb {R}^d$, while $a(x)$ is nonzero at any point of the nondegenerate 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 $\partial _0\mathscr {O}$. Points in $\partial _0\mathscr {O}$ play the same role as those in the interior of the domain, $\mathscr {O}$, and only the nondegenerate boundary portion, $\partial _1\mathscr {O}$, is required for boundary comparisons. Moreover, we obtain a comparison principle for a solution and supersolution in $W^{2,d}_{\mathrm {loc}}(\mathscr {O})$ to a unilateral obstacle problem defined by $A$, again where only the nondegenerate boundary portion, $\partial _1\mathscr {O}$, is required for boundary comparisons. Our results extend those of Daskalopoulos and Hamilton, Epstein and Mazzeo, and Feehan, 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. Finally, we obtain analogues of all the preceding results for parabolic linear second-order partial differential operators, $Lu := -u_t - \mathrm {tr}(aD^2u)-\langle b, Du\rangle + cu$.