[Seminar] Geometric PDE and Applied Analysis Seminar (12/1)

Geometric PDE and Applied Analysis Seminar (December 1, 2022)

Talk 1: 15:00-16:00

Title: ABP maximum principle with upper contact sets for fully nonlinear elliptic PDEs

Speaker: Prof. Shigeaki Koike (Waseda University)

Abstract: The Aleksandrov-Bakelmann-Pucci’s (ABP for short) maximum principle is the key tool to investigate regularity theory particularly for fully nonlinear uniformly elliptic PDEs. ABP maximum principle for classical subsolutions is to estimate the maximum by the Lⁿ norm of the inhomogeneous term over the upper contact set. It is not known that the ABP maximum principle with upper contact sets holds for weak solutions. In this talk, we present recent results for L^p-viscosity solutions on this issue, and explain its importance.

Talk 2: 16:00-17:00

Title: A perturbation theory of overdetermined problems

Speaker: Prof. Michiaki Onodera (Tokyo Institute of Technology)

Abstract: I will present a general perturbative result on overdetermined problems, such as Bernoulli's free boundary problem and Serrin's problem. A typical overdetermined problem comprises the Dirichlet problem with an additional Neumann boundary condition, so that only a particular domain allows the solvability of the problem. Unlike the case where a perturbed nonlinear problem is locally solved under the non-degeneracy of the linearized operator at a solution, our problems have a regularity deficit called a loss of derivatives, so that a simple iterative scheme in general fails. Our resolution to this issue is an introduction of a "parabolic" implicit function theorem, and it turns out that an additional monotonicity condition on the linearized operator guarantees the existence, uniqueness and some qualitative behavior of solutions to a perturbed overdetermined problem.

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