Dorina Mitrea and Marius Mitrea

Poisson Problems with Half-Dirichlet Boundary Conditions for Dirac Operators on Nonsmooth Manifolds

abstract:
We consider Dirichlet and Poisson type problems for the Maxwell-Dirac operator ${\mathbbD}_k=d+\delta+k\,dt\cdot$ on a Lipschitz subdomain $\Omega$ of a smooth, Riemannian manifold ${\mathcal M}$. The emphasis is on solutions of finite $L^p$ energy, i.e. sections $u$ satisfying $\iint_\Omega\left[|u|^p+|du|^p+|\delta u|^p\right]\,d\mbox{Vol} <+\infty$. In this context, we prove well-posedness for $p$ near $2$. Our approach relies heavily on the analysis of the spectra of Cauchy type operators naturally associated with the problems at hand.

Mathematics Subject Classification: 35F15, 31C12, 42B20, 58J32, 42B35, 58J05, 42B25, 78A25.

Key words and phrases: Dirac operators, Poisson problem, Dirichlet problem, Cauchy operators, Maxwell's equations, Lipschitz domains.