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The Allen-Cahn Euclidean Isoperimetric Problem and the Grad-Mercier Equation in Plasma Physics

出版	University of Texas, 2023
URL	http://books.google.com.hk/books?id=xDto0AEACAAJ&hl=&source=gbs_api

註釋This thesis is divided in two parts. Each one of them is devoted to the study of a semilinear elliptic equation arising as a model to understand a phenomenon that lies in the interplay between physics and geometry. In the first part, we provide an approximation to the Euclidean-isoperimetric problem via the Allen-Cahn energy functional. For non-degenerate double well potentials, we prove sharp quantitative stability inequalities of quadratic type which are uniform in the length scale of the phase transitions. We also derive a rigidity theorem for critical points analogous to the classical Alexandrov's theorem for constant mean curvature boundaries. In the second part, we establish regularity and uniqueness results for Grad-Mercier type equations that arises in the context of plasma physics. We show that solutions of this problem develop naturally a dead core, which corresponds to the set where the solutions become identically equal to its maximum. We prove sharp regularity and non-degeneracy bounds for the solution when the domain exhibit certain types of symmetry, e.g., radial or axial symmetry. We also prove some initial regularity estimates for the free boundary associated with the dead core. Our approach is non-variational, only recurring to the maximum principle, which allows us to carry out our arguments for a large spectrum of elliptic (nonlinear) operators