Topics of stable solutions to elliptic equations

Takahashi, Futoshi

doi:10.1090/suga/457

Topics of stable solutions to elliptic equations
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by Futoshi Takahashi
Translated by: the author

Sugaku Expositions 34 (2021), 35-59

DOI: https://doi.org/10.1090/suga/457

Published electronically: April 28, 2021

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Abstract:

Stable solutions (sometimes called “semistable” solutions) of nonlinear elliptic partial differential equations with variational structures are defined by using the second order information of the associate variational energy functionals. A class of stable solutions includes a rather wide range of analytically and geometrically interesting solutions, such as energy minimizing solutions, minimal and extremal solutions to the classical Liouville-Gelfand type nonlinear elliptic eigenvalue problems, and monotone solutions. They are natural in the variational viewpoint, and the study of stable solutions has now become one of the most active fields in nonlinear analysis.

In this expository article, we introduce many new results that have been proved recently by many mathematicians’ groups in the world and overview (a part of) the actively ongoing studies of qualitative properties of stable solutions. We are mainly concerned with the regularity theory of extremal solutions to the Liouville-Gelfand type nonlinear elliptic eigenvalue problems and show that the regularity of extremal solutions is closely related to global information of the problem, such as the dimension and/or geometry of the domain.

In some cases, extremal solutions arise from the superposition of stable minimal solutions and unstable solutions of mountain-pass type; thus those are “degenerate”. In this sense, the study of qualitative properties of extremal solutions is interesting in itself. Also good understanding of extremal solutions is important to obtain the global information of the solution set. The reader will come to know the method of mathematical analysis of more general stable solutions through the study of the regularity theory of extremal solutions.

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