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

Research output: Contribution to journal › Research article › Contributed › peer-review

## Contributors

## Abstract

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.

## Details

Original language | English |
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Pages (from-to) | 2912-2943 |

Number of pages | 32 |

Journal | SIAM journal on mathematical analysis |

Volume | 48 |

Issue number | 4 |

Publication status | Published - 2016 |

Peer-reviewed | Yes |

## External IDs

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

## Keywords

### ASJC Scopus subject areas

### Keywords

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