On the regularity theory for mixed local and nonlocal quasilinear elliptic equations

Prashanta Garain*, Juha Kinnunen

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

44 Citations (Scopus)


We consider a combination of local and nonlocal p-Laplace equations and discuss several regularity properties of weak solutions. More precisely, we establish local boundedness of weak subsolutions, local Hölder continuity of weak solutions, Harnack inequality for weak solutions and weak Harnack inequality for weak supersolutions. We also discuss lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. Our approach is purely analytic and it is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. The main results apply to sign changing solutions and capture both local and nonlocal features of the equation.

Original languageEnglish
Pages (from-to)5393-5423
Number of pages31
JournalTransactions of the American Mathematical Society
Issue number8
Early online date17 Mar 2022
Publication statusPublished - 1 Aug 2022
MoE publication typeA1 Journal article-refereed


  • Regularity
  • mixed local and nonlocal p-Laplace equation
  • local bound-edness
  • Holder continuity
  • Harnack inequality
  • weak Harnack inequality
  • lower semicontinuity
  • energy estimates
  • De Giorgi-Nash-Moser theory
  • expansion of positivity


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