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

Research output: Contribution to journalArticlepeer-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 languageEnglish (US)
Pages (from-to)5275-5332
Number of pages58
JournalTransactions of the American Mathematical Society
Volume373
Issue number8
DOIs
StatePublished - 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

Fingerprint

Dive into the research topics of 'PERTURBATIONS of LOCAL MAXIMA and COMPARISON PRINCIPLES for BOUNDARY-DEGENERATE LINEAR DIFFERENTIAL EQUATIONS'. Together they form a unique fingerprint.

Cite this