Maximum principles for boundary-degenerate linear parabolic differential operators

2013/06/20 by Paul M. N. Feehan, Feehan, Paul M. N.
Computer Science · Engineering · Mathematics · #35B51 #35K65 #35K85 #60J60 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Primary 35B50 #Probability (math.PR) #Stability and Controllability of Differential Equations #math.AP #math.PR #msc:35B50 #msc:35B51 #msc:35D40 #msc:35K65 #msc:35K85 #msc:60J60 #secondary 35D40

paper · pdf · doi:10.48550/arxiv.1306.5197

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

openalex publication_date 2013/06/20 · arxiv created 2013/07/19 · arxiv updated 2013/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We develop weak and strong maximum principles for boundary-degenerate, linear, parabolic, second-order partial differential operators, Lu := -u_t-\tr(aD²u)-⟨ b, Du⟩ + cu, with partial Dirichlet boundary conditions. The coefficient, a(t,x), is assumed to vanish along a non-empty open subset, \mydirac₀!\sQ, called the degenerate boundary portion, of the parabolic boundary, \mydirac!\sQ, of the domain \sQ⊂\RR^d+1, while a(t,x) may be non-zero at points in the non-degenerate boundary portion, \mydirac₁!\sQ := \mydirac!\sQ\less\mydirac₀!\sQ. Points in \mydirac₀!\sQ play the same role as those in the interior of the domain, \sQ, and only the non-degenerate boundary portion, \mydirac₁!\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, \mydirac₁!\sQ, is required for boundary comparisons. Our results complement those in our previous articles [arXiv1204.6613, arXiv:1305.5098].

Citations

• Perturbations of local maxima and comparison principles for\n boundary-degenerate linear differential equations
• User’s guide to viscosity solutions of second order partial differential equations

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