Kolloquium: Wolfgang Arendt (Universität Ulm): The Dirichlet Problem Without the Maximum Principle

The Dirichlet Problem Without the Maximum Principle – Vortragender: Wolfgang Arendt, Universität Ulm – Einladender: E. Wiedemann

Abstract: In the first part of the talk we recall known results on
the Dirichlet problem for the Laplacian. We will mention the Dirichlet
Principle, the Perron solution, Wiener’s characterization of well-posedness via
a capacity condition and several further beautiful results. The fundamental De
Giorgi-Nash Theorem allowed Stampacchia and coauthors to extend many of these
results to elliptic operators (instead of the Laplacian). For these generalizations the maximum
principle plays an important role; it is valid only if the coefficients of the
operator satisfy a certain divergence condition.

In 2019 we succeeded to prove many of the well-posedness
results without that the maximum principle holds. The only condition we need is
that 0 is not a Dirichlet eigenvalue, which is equivalent to uniqueness in the
Dirichlet problem. The Perron solution is a special challenge in this general
setting, and diverse equivalent descriptions could be given recently. The most interesting one
is even new for the Laplacian. These results are contained in the following two
articles: W. Arendt, T. ter Elst: The Dirichlet Problem without the
Maximum Principle. Ann. Inst. Fourier, Grenoble, 69 (2019) 763-782. W. Arendt, T. ter Elst, M. Sauter: The Perron Solution
for Elliptic Equations without the Maximum Principle. Math. Ann. 2024.