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

2013/05/22 by Paul M. N. Feehan, Feehan, Paul M. N. · 1 citation
Computer Science · Mathematics · #35B51 #35J70 #35J86 #35K65 #35K85 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Primary 35B50 #Spectral Theory in Mathematical Physics #math.AP #msc:35B50 #msc:35B51 #msc:35D40 #msc:35J70 #msc:35J86 #msc:35K65 #msc:35K85 #secondary 35D40

paper · pdf · doi:10.48550/arxiv.1305.5098

55 pages, 1 figure, incorporating final galley proof corrections. Includes summary of background material from its companion articles arXiv:1204.6613 and 1306.5197. To appear in Transactions of the American Mathematical Society

openalex publication_date 2013/05/22 · arxiv created 2020/04/23 · arxiv updated 2020/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We develop strong and weak maximum principles for boundary-degenerate elliptic and parabolic linear second-order partial differential operators, Au := -tr(aD²u)-<b, Du> + cu, with partial Dirichlet boundary conditions. The coefficient, a(x), is assumed to vanish along a non-empty open subset, ∂₀\mathscrO, called the degenerate boundary portion, of the boundary, ∂\mathscrO, of the domain \mathscrO⊂ℝ^d, while a(x) is non-zero at any point of the non-degenerate boundary portion, ∂₁\mathscrO := ∂\mathscrO∖∂₀\mathscrO. If an A-subharmonic function, u in C²(\mathscrO) or W^2,d_loc(\mathscrO), is C¹ up to ∂₀\mathscrO and has a strict local maximum at a point in ∂₀\mathscrO, we show that u can be perturbed, by the addition of a suitable function w∈ C²(\mathscrO)∩ C¹(ℝ^d), to a strictly A-subharmonic function v=u+w having a local maximum in the interior of \mathscrO. Consequently, we obtain strong and weak maximum principles for A-subharmonic functions in C²(\mathscrO) and W^2,d_loc(\mathscrO) which are C¹ up to ∂₀\mathscrO. Only the non-degenerate boundary portion, ∂₁\mathscrO, 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(aD²u) is in addition assumed to be continuous up to and vanish along ∂₀\mathscrO in order to yield comparable maximum principles for A-subharmonic functions in C²(\mathscrO), while the results developed here for A-subharmonic functions in W^2,d_loc(\mathscrO) are entirely new.

Citations

• Degenerate Diffusion Operators Arising in Population Biology
• Maximum principles for boundary-degenerate second-order linear elliptic differential operators

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• Maximum principles for boundary-degenerate linear parabolic differential operators

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