Maximum principles for boundary-degenerate linear parabolic differential operators

Abstract

We develop weak and strong maximum principles for boundary-degenerate, linear, parabolic, second-order partial differential operators, $Lu := -u_t-\tr(aD^2u)-\langle b, Du\rangle + cu$, with \emph{partial} Dirichlet boundary conditions. The coefficient, $a(t,x)$, is assumed to vanish along a non-empty open subset, $\mydirac_0!\sQ$, called the \emph{degenerate boundary portion}, of the parabolic boundary, $\mydirac!\sQ$, of the domain $\sQ\subset\RR^{d+1}$, while $a(t,x)$ may be non-zero at points in the \emph{non-degenerate boundary portion}, $\mydirac_1!\sQ := \mydirac!\sQ\less\bar{\mydirac_0!\sQ}$. Points in $\mydirac_0!\sQ$ play the same role as those in the interior of the domain, $\sQ$, and only the non-degenerate boundary portion, $\mydirac_1!\sQ$, is required for boundary comparisons. We also develop comparison principles and a priori maximum principle estimates for solutions to boundary value and obstacle problems defined by boundary-degenerate parabolic operators, again where only the non-degenerate boundary portion, $\mydirac_1!\sQ$, is required for boundary comparisons. Our results complement those in our previous articles [arXiv1204.6613, arXiv:1305.5098].

34 pages, 2 figures. This article is the parabolic analogue of arXiv:1204.6613 and restates background material (definitions, notation, spaces) from arXiv:1305.5098

References in corpus (3)

• A Schauder approach to degenerate-parabolic partial differential equations with unbounded coefficients
• C^{1,1} regularity for degenerate elliptic obstacle problems
• Existence, uniqueness, and global regularity for degenerate elliptic obstacle problems in mathematical finance

Cited by in corpus (1)

• On the martingale problem for degenerate-parabolic partial differential operators with unbounded coefficients and a mimicking theorem for Ito processes