Mini-Symposium on Operator Theory and Partial Differential Equations

13:15-14: Wolfgang Arendt

Title: Variational Solutions of the Dirichlet and the Poisson Problem

Abstract: A well known generalized solution of the Dichlet problem is due to Perron. Its behaviour close to the boundary of the given domain has been extensively studied. In this talk we introduce another notion of weak solution which we call variational solution. The boundary behaviour of this solution is defined in the sense of Sobolev. An extension theorem for continuous functions at the boundary to the entire domain allows to prove existence and uniqueness for these variational solutions. Its proof is very technical though. It turns out that for the Laplacian the variational solution and the Perron solution coincide.

Our point is that the variational solution also exists for elliptic operators with complex order coefficients where no maximum principle is possible. We will describe various other ways to characterize the variational solution, for instance by the quasi-everywhere behavior at the boundary. One point is very appealing: It is immediate to characterize when the variational solution has fine energy, a problem which puzzled Hadamard and others in the early days of potential theory.

This talk is based on recent joint work with Tom ter Elst (Auckland) and Manfred Sauter (Ulm).

14:15-15: Andrew Morris

Title: A generalised Fefferman Theorem for elliptic operators with rough coefficients

Abstract: The theory of layer potentials for complex elliptic operators $L=-\mathrm{div}A \nabla$ with bounded measurable coefficients $A$ asserts conditions on the coefficients which ensure that solutions to the Dirichlet problem for boundary data in $BMO$ are given by layer potentials satisfying a certain Carleson measure condition. In this setting, we present a converse Fatou-type result which shows that such Carleson measure control actually characterises the solutions that have a trace of bounded mean oscillation. This extends the characterisation obtained for harmonic functions by Fabes--Johnson--Neri, which built on the duality between $H^1$ and $BMO$ established by Fefferman. This is joint work with Steve Hofmann and Marius Mitrea.

15:15-16: David Rule

Title: Sparse bounds for rough pseudo differential and Fourier integral operators

Abstract: We revisit the recent work of Beltran and Cladek and show that their sparse form bounds for Hörmander symbols extend to pseudo differential operators which are merely bounded in the spatial variable. The proof is more direct in that it avoids recourse to a maximal sparse bound and also produces pointwise sparse bounds. We go on to show a sharp pointwise bound for rough Fourier integral operators which also leads to sparse bounds.