Title: The stability-compactness method and qualitative properties of nonlinear elliptic PDEs

Speaker: Henri Berestycki (University of Maryland, College Park & EHESS, Paris)

Date, Time, Location: Wed. October 16th @ 4:30 PM in 520 Math Hall

Abstract: Nonlinear elliptic equations describe the stationary states of numerous systems in physics, biology and medicine. Qualitative properties such as monotonicity, symmetry, stability or uniqueness are essential features of their study. In this talk, I will present a new general framework to approach this kind of questions, focusing on uniqueness. It rests on decomposing the domain into one region with a certain compactness feature and another supporting a form of spectral stability. This approach has proved to be unexpectedly versatile and in fact encompasses past works on the subject such as the general moving plane method originating in the study of minimal surfaces by Alexandrov. The results concern positive solutions of nonlinear elliptic PDEs in general unbounded domains with Dirichlet boundary conditions and for various types of reaction terms. This represents a series of joint works with Cole Graham and with Cole Graham and Jun-Cheng Wei.

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