Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic p -Laplace equations

Agnid Banerjee, Prashanta Garain, Juha Kinnunen

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional p-Laplace equations which includes the fractional parabolic p-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results extend some recent results in the local parabolic case, and in the nonlocal elliptic case, to the nonlocal parabolic case. We prove the required energy estimates and measure theoretic De Giorgi type lemmas in the fractional setting.

Original languageEnglish
Article number2250032
JournalCommunications in Contemporary Mathematics
Volume25
Issue number8
DOIs
StatePublished - 1 Oct 2023

Keywords

  • De Giorgi's method
  • Doubly nonlinear parabolic equation
  • energy estimates
  • fractional p -Laplace equation
  • porous medium equation

ASJC Scopus subject areas

  • Applied Mathematics
  • General Mathematics

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