Equivalence of viscosity and weak solutions for the -Laplacian

  • Mikko Parviainen

    Aalto University, School of Science and Technology, Institute of Mathematics, P.O. Box 11000, FI-00076 Aalto, Finland
  • Teemu Lukkari

    Department of Mathematical Sciences, NTNU, NO-7491 Trondheim, Norway
  • Petri Juutinen

    Department of Mathematics and Statistics, P.O. Box 35, FIN-40014 University of Jyväskylä, Finland


We consider different notions of solutions to the -Laplace equation

with . We show by proving a comparison principle that viscosity supersolutions and -superharmonic functions of nonlinear potential theory coincide. This implies that weak and viscosity solutions are the same class of functions, and that viscosity solutions to Dirichlet problems are unique. As an application, we prove a Radó type removability theorem.

Cite this article

Mikko Parviainen, Teemu Lukkari, Petri Juutinen, Equivalence of viscosity and weak solutions for the -Laplacian. Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 6, pp. 1471–1487

DOI 10.1016/J.ANIHPC.2010.09.004