Universität Augsburg
Institut für Mathematik

Let \(\Omega\) be an arbitrary bounded open subset of \(\mathbb{R}^d\), and let \(\mathcal{A}: H^1_\mathrm{loc}(\Omega)\to\mathcal{D}'(\Omega)\) be a second-order elliptic operator in divergence form with bounded measurable coefficients

where all the lower-order coefficients may be complex-valued. Throughout we assume that \(u\in H^1_0(\Omega)\) and \(\mathcal{A} u = 0\) implies \(u=0\).

We study the corresponding Dirichlet problem for continuous boundary data. So given \(\varphi\in C(\partial\Omega)\), we seek an \(\mathcal{A}\)-harmonic \(u\in H^1_\mathrm{loc}(\Omega)\) such that \(u\) attains \(\varphi\) at the boundary in a suitable sense. Our main result establishes that there exists a bounded solution operator \(T: C(\partial\Omega)\to C_\mathrm{b}(\Omega)\) such that \(T\varphi\) is the unique \(\mathcal{A}\)-harmonic function satisfying \(\Phi-T\varphi\in H^1_0(\Omega)\) for some \(\Phi\in C(\overline{\Omega})\) with \(\Phi|_{\partial\Omega}=\varphi\). This yields a Perron-type generalised solution solely under the uniqueness assumption stated above, but without a divergence condition, positivity or the maximum principle.

In fact, with general complex coefficients \(c_k\) a reduction to a setting with the maximum principle is not possible. Instead, we reduce to the contractive case. The first key step is to identify a sufficiently general condition on the coefficients that ensures contractivity of \(T\). Remarkably, the Auscher–Barthélemy–Bénilan–Ouhabaz condition, which characterises the \(L_\infty\)-contractivity of the \(C_0\)-semigroup generated by the realisation of \(-\mathcal{A}\) in \(L_2(\Omega)\), turns out to be sufficient. We will focus on this contractivity criterion and explain its role for the proof of the main result.

The talk is based on recent joint work with Wolfgang Arendt and Tom ter Elst.