Robin boundary value problems on non-smooth domains
by
Daniel Daners
Brigham Young University

We discuss L_p-L_q-smoothing properties of the resolvent of elliptic equations subject to homogeneous Robin (third) boundary conditions which are independent of the geometry of the underlying domain. These properties derive from an almost forgotten inequality due to Maz'ja. As the estimates obtained are essentially domain independent weak solutions can be defined for any bounded domain. The ideas generalize to other boundary conditions, and also to parabolic problems, where they lead to heat kernel estimates with good control over the domain dependence.

The theory should be useful wherever weak dependence of estimates on the domain is required such as for instance in domain perturbation, shape optimization or homogenization problems.

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Date received: March 31, 1999