Some decomposition theorems and their application to non‐linear potential theory and Hodge theory

Abstract

Abstract The paper considers Dirichlet (or Neumann type) boundary value problems of generalized potential theory on Lipschitz manifolds with boundary. Here ϵ denotes a permissible non‐linearity. The existence theory is developed in the framework of monotone operators. The approach covers a variety of applications including fluid dynamics and electro‐ and magneto‐statics. Only fairly weak regularity assumptions are required (e.g. Lipschitz boundary, L ∞ ‐coefficients). As a by‐product we obtain a non‐linear Hodge theorem generalizing a result by L. M. Sibner and R. J. Sibner (‘A non‐linear Hodge‐DeRham theorem’, Acta Math. , 125 , 57–73 (1970)).