PERTURBATIONS of LOCAL MAXIMA and COMPARISON PRINCIPLES for BOUNDARY-DEGENERATE LINEAR DIFFERENTIAL EQUATIONS

Paul M.N. Feehan

doi:https://doi.org/10.1090/tran/7246

PERTURBATIONS of LOCAL MAXIMA and COMPARISON PRINCIPLES for BOUNDARY-DEGENERATE LINEAR DIFFERENTIAL EQUATIONS

Paul M.N. Feehan

SAS - Mathematics

Rutgers, The State University

Research output: Contribution to journal › Article › peer-review

Abstract

We develop strong and weak maximum principles for boundarydegenerate 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 nonempty open subset, ∂0O, called the degenerate boundary portion, of the boundary, ∂O, of the domain O ⊂ Rd, while a(x) is nonzero at any point of the nondegenerate boundary portion, ∂1O := ∂O \ ∂0O. If an A-subharmonic function, u in C2(O) or W2,d loc (O), is C1 up to ∂0O and has a strict local maximum at a point in ∂0O, we show that u can be perturbed, by the addition of a suitable function w ∈ C2(O) ∩ C1(Rd), to a strictly A-subharmonic function v = u + w having a local maximum in the interior of O. Consequently, we obtain strong and weak maximum principles for A-subharmonic functions in C2(O) and W2,d loc (O) which are C1 up to ∂0O. Points in ∂0O play the same role as those in the interior of the domain, O, and only the nondegenerate boundary portion, ∂1O, is required for boundary comparisons. Moreover, we obtain a comparison principle for a solution and supersolution in W2,d loc (O) to a unilateral obstacle problem defined by A, again where only the nondegenerate boundary portion, ∂1O, is required for boundary comparisons. Our results extend those of Daskalopoulos and Hamilton, Epstein and Mazzeo, and Feehan, where tr(aD2u) is in addition assumed to be continuous up to and vanish along ∂0O in order to yield comparable maximum principles for A-subharmonic functions in C2(O), while the results developed here for A-subharmonic functions in W2,d loc (O) are entirely new. Finally, we obtain analogues of all the preceding results for parabolic linear second-order partial differential operators, Lu :=-ut-tr(aD2u)-_b,Du_+ cu.

Original language	English (US)
Pages (from-to)	5275-5332
Number of pages	58
Journal	Transactions of the American Mathematical Society
Volume	373
Issue number	8
DOIs	https://doi.org/10.1090/tran/7246
State	Published - Aug 2020

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

Keywords

Boundary-degenerate elliptic differential operators
Boundary-degenerate parabolic differential operators
Comparison principles
Maximum principles
Obstacle problems
Viscosity solutions

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https://doi.org/10.1090/tran/7246

Cite this

@article{0d9a3b3d0c2c425daad198f247d852e9,

title = "PERTURBATIONS of LOCAL MAXIMA and COMPARISON PRINCIPLES for BOUNDARY-DEGENERATE LINEAR DIFFERENTIAL EQUATIONS",

abstract = "We develop strong and weak maximum principles for boundarydegenerate 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 nonempty open subset, ∂0O, called the degenerate boundary portion, of the boundary, ∂O, of the domain O ⊂ Rd, while a(x) is nonzero at any point of the nondegenerate boundary portion, ∂1O := ∂O \ ∂0O. If an A-subharmonic function, u in C2(O) or W2,d loc (O), is C1 up to ∂0O and has a strict local maximum at a point in ∂0O, we show that u can be perturbed, by the addition of a suitable function w ∈ C2(O) ∩ C1(Rd), to a strictly A-subharmonic function v = u + w having a local maximum in the interior of O. Consequently, we obtain strong and weak maximum principles for A-subharmonic functions in C2(O) and W2,d loc (O) which are C1 up to ∂0O. Points in ∂0O play the same role as those in the interior of the domain, O, and only the nondegenerate boundary portion, ∂1O, is required for boundary comparisons. Moreover, we obtain a comparison principle for a solution and supersolution in W2,d loc (O) to a unilateral obstacle problem defined by A, again where only the nondegenerate boundary portion, ∂1O, is required for boundary comparisons. Our results extend those of Daskalopoulos and Hamilton, Epstein and Mazzeo, and Feehan, where tr(aD2u) is in addition assumed to be continuous up to and vanish along ∂0O in order to yield comparable maximum principles for A-subharmonic functions in C2(O), while the results developed here for A-subharmonic functions in W2,d loc (O) are entirely new. Finally, we obtain analogues of all the preceding results for parabolic linear second-order partial differential operators, Lu :=-ut-tr(aD2u)-_b,Du_+ cu.",

keywords = "Boundary-degenerate elliptic differential operators, Boundary-degenerate parabolic differential operators, Comparison principles, Maximum principles, Obstacle problems, Viscosity solutions",

author = "Feehan, {Paul M.N.}",

note = "Funding Information: Received by the editors June 18, 2015, and, in revised form, March 2, 2017. 2010 Mathematics Subject Classification. Primary 35B50, 35B51, 35J70, 35K65; Secondary 35D40, 35J86, 35K85. Key words and phrases. Comparison principles, boundary-degenerate elliptic differential operators, boundary-degenerate parabolic differential operators, maximum principles, obstacle problems, viscosity solutions. The author was partially supported by NSF grant DMS-1237722 and a visiting faculty appointment at the Department of Mathematics at Columbia University. Publisher Copyright: {\textcopyright} 2020 American Mathematical Society. All rights reserved.",

year = "2020",

month = aug,

doi = "https://doi.org/10.1090/tran/7246",

language = "English (US)",

volume = "373",

pages = "5275--5332",

journal = "Transactions of the American Mathematical Society",

issn = "0002-9947",

publisher = "American Mathematical Society",

number = "8",

}

TY - JOUR

T1 - PERTURBATIONS of LOCAL MAXIMA and COMPARISON PRINCIPLES for BOUNDARY-DEGENERATE LINEAR DIFFERENTIAL EQUATIONS

AU - Feehan, Paul M.N.

N1 - Funding Information: Received by the editors June 18, 2015, and, in revised form, March 2, 2017. 2010 Mathematics Subject Classification. Primary 35B50, 35B51, 35J70, 35K65; Secondary 35D40, 35J86, 35K85. Key words and phrases. Comparison principles, boundary-degenerate elliptic differential operators, boundary-degenerate parabolic differential operators, maximum principles, obstacle problems, viscosity solutions. The author was partially supported by NSF grant DMS-1237722 and a visiting faculty appointment at the Department of Mathematics at Columbia University. Publisher Copyright: © 2020 American Mathematical Society. All rights reserved.

PY - 2020/8

Y1 - 2020/8

N2 - We develop strong and weak maximum principles for boundarydegenerate 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 nonempty open subset, ∂0O, called the degenerate boundary portion, of the boundary, ∂O, of the domain O ⊂ Rd, while a(x) is nonzero at any point of the nondegenerate boundary portion, ∂1O := ∂O \ ∂0O. If an A-subharmonic function, u in C2(O) or W2,d loc (O), is C1 up to ∂0O and has a strict local maximum at a point in ∂0O, we show that u can be perturbed, by the addition of a suitable function w ∈ C2(O) ∩ C1(Rd), to a strictly A-subharmonic function v = u + w having a local maximum in the interior of O. Consequently, we obtain strong and weak maximum principles for A-subharmonic functions in C2(O) and W2,d loc (O) which are C1 up to ∂0O. Points in ∂0O play the same role as those in the interior of the domain, O, and only the nondegenerate boundary portion, ∂1O, is required for boundary comparisons. Moreover, we obtain a comparison principle for a solution and supersolution in W2,d loc (O) to a unilateral obstacle problem defined by A, again where only the nondegenerate boundary portion, ∂1O, is required for boundary comparisons. Our results extend those of Daskalopoulos and Hamilton, Epstein and Mazzeo, and Feehan, where tr(aD2u) is in addition assumed to be continuous up to and vanish along ∂0O in order to yield comparable maximum principles for A-subharmonic functions in C2(O), while the results developed here for A-subharmonic functions in W2,d loc (O) are entirely new. Finally, we obtain analogues of all the preceding results for parabolic linear second-order partial differential operators, Lu :=-ut-tr(aD2u)-_b,Du_+ cu.

AB - We develop strong and weak maximum principles for boundarydegenerate 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 nonempty open subset, ∂0O, called the degenerate boundary portion, of the boundary, ∂O, of the domain O ⊂ Rd, while a(x) is nonzero at any point of the nondegenerate boundary portion, ∂1O := ∂O \ ∂0O. If an A-subharmonic function, u in C2(O) or W2,d loc (O), is C1 up to ∂0O and has a strict local maximum at a point in ∂0O, we show that u can be perturbed, by the addition of a suitable function w ∈ C2(O) ∩ C1(Rd), to a strictly A-subharmonic function v = u + w having a local maximum in the interior of O. Consequently, we obtain strong and weak maximum principles for A-subharmonic functions in C2(O) and W2,d loc (O) which are C1 up to ∂0O. Points in ∂0O play the same role as those in the interior of the domain, O, and only the nondegenerate boundary portion, ∂1O, is required for boundary comparisons. Moreover, we obtain a comparison principle for a solution and supersolution in W2,d loc (O) to a unilateral obstacle problem defined by A, again where only the nondegenerate boundary portion, ∂1O, is required for boundary comparisons. Our results extend those of Daskalopoulos and Hamilton, Epstein and Mazzeo, and Feehan, where tr(aD2u) is in addition assumed to be continuous up to and vanish along ∂0O in order to yield comparable maximum principles for A-subharmonic functions in C2(O), while the results developed here for A-subharmonic functions in W2,d loc (O) are entirely new. Finally, we obtain analogues of all the preceding results for parabolic linear second-order partial differential operators, Lu :=-ut-tr(aD2u)-_b,Du_+ cu.

KW - Boundary-degenerate elliptic differential operators

KW - Boundary-degenerate parabolic differential operators

KW - Comparison principles

KW - Maximum principles

KW - Obstacle problems

KW - Viscosity solutions

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U2 - https://doi.org/10.1090/tran/7246

DO - https://doi.org/10.1090/tran/7246

M3 - Article

VL - 373

SP - 5275

EP - 5332

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 8

ER -

PERTURBATIONS of LOCAL MAXIMA and COMPARISON PRINCIPLES for BOUNDARY-DEGENERATE LINEAR DIFFERENTIAL EQUATIONS

Abstract

ASJC Scopus subject areas

Keywords

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