The Maxwell Compactness Property in Bounded Weak Lipschitz Domains with Mixed Boundary Conditions

Research output: Contribution to journalResearch articleContributedpeer-review


  • Sebastian Bauer - , University of Duisburg-Essen (Author)
  • Dirk Pauly - , Institute of Analysis, University of Duisburg-Essen (Author)
  • Michael Schomburg - , University of Duisburg-Essen (Author)


Let Omega subset of R-3 be a bounded weak Lipschitz domain with boundary Gamma :- partial derivative Omega divided into two weak Lipschitz submanifolds Gamma(tau) and Gamma(nu) and let epsilon denote an L-infinity-matrix field inducing an inner product in L-2(Omega). The main result of this contribution is the so called Maxwell compactness property, i.e., the Hilbert space {E is an element of L-2(Omega) : rot E is an element of L-2(Omega), div epsilon E is an element of L-2(Omega), nu X E vertical bar Gamma(tau) = 0,nu.epsilon E vertical bar Gamma(nu) = 0} is compactly embedded into L-2(Omega). We will also prove some canonical applications, such as Maxwell estimates, Helmholtz decompositions and a static solution theory. Furthermore, a Fredholm alternative for the underlying time-harmonic Maxwell problem and all corresponding and related results for exterior domains formulated in weighted Sobolev spaces are straight forward.


Original languageEnglish
Pages (from-to)2912-2943
Number of pages32
JournalSIAM journal on mathematical analysis
Issue number4
Publication statusPublished - 2016

External IDs

ORCID /0000-0003-4155-7297/work/145224239
WOS 000385023400021



  • Helmholtz decomposition, Maxwell compactness property, Maxwell estimate, Electromagneto static, Mixed boundary conditions, Vector potentials, weak Lipschitz domain