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

摘要

AbstractThe paper considers Dirichlet (or Neumann type) boundary value problems of generalized potential theorydocumentclass{article}pagestyle{empty}begin{document}$$ {rm d}alpha = f,;delta varepsilon left(alpha right) = g,{rm in},M, $$end{document}documentclass{article}pagestyle{empty}begin{document}$$ alpha = 0,{rm on},partial M $$end{document}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 Ma