Browsing by Subject "Free boundary"
Now showing 1 - 3 of 3
- Results Per Page
- Sort Options
Item Fully nonlinear equations with applications to grad equations in plasma physics ; Interaction between a one phase free boundary problem and an obstacle problem ; Optimal trace Sobolev inequalities(2022-08-05) Tomasetti, Ignacio; Caffarelli, Luis A.; Maggi, Francesco, 1978-; Patrizi, Stefania; Stinga, Pablo RIn this thesis we address three different problems. First, we prove existence and regularity for a fully nonlinear and nonlocal equation which arises in plasma physics. This is a generalization of Grad Equations which model the behavior of plasma confined in a toroidal vessel called TOKAMAK. We prove existence of a W [superscript 2,p]-viscosity solution and regularity up to C [superscript 1, alpha] [Ω overlined] for any α<1. Then we elaborate in how to improve this regularity near the boundary. The main ingredient to study is the nonlocality due to the presence of the measure of the superlevel sets in the equations. Second, we address a problem which models a reaction-diffusion process. Existence is proved, and the solution solves a one phase free boundary problem in the lower half space [doublestruck R] [superscript 3 underscored]. Its trace in [doublestruck R superscript 2] solves an obstacle problem for a given obstacle. We study the exchange between the diffusion in the horizontal plane and the lower half space. Third, we work with a family of variational problems with critical volume and trace constraints. This arises from the study of "best p-Sobolev inequalities" for n≥2 and p [is an element of](1,n). We extend the analysis from [MV05] and [MN17] for an open set Ω [is a proper subset of] [doublestruck R superscript n]. We prove existence of minimizers for Ω bounded and n>2p, and existence of generalized minimizers for n>p. We also establish rigidity results for the comparison theorem "balls have the worst best Sobolev inequalities" from [MV05].Item Optimal regularity and nondegeneracy for minimizers of an energy related to the fractional Laplacian(2011-08) Yang, Ray; Caffarelli, Luis A.; Arapostathis, Ari; Beckner, William; Tsai, Richard; Vasseur, AlexisWe study the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian through the extension technique of Caffarelli and Silvestre. Specifically, we show that minimizers of the energy [mathematical equation] where [mathematical equations] with 0 < [gamma] < 1, with free behavior on the set {y=0}, are Holder continuous with exponent [Beta] = 2[sigma]/2-[gamma]. These minimizers exhibit a free boundary: along {y = 0}, they divide into a zero set {u = 0} and a positivity set where {u > 0}; we call the interface between these sets the free boundary. The regularity is optimal, due to the non-degeneracy property of the minimizers: in any ball of radius r centered at the free boundary, the minimizer grows (in the supremum sense) like r[Beta]. This work is related to, but addresses a different problem from, recent work of Caffarelli, Roquejoffre, and Sire.Item The Allen-Cahn Euclidean isoperimetric problem and the Grad-Mercier equation in plasma physics(2023-08-07) Restrepo Montoya, Daniel Eduardo; Caffarelli, Luis A.; Maggi, Francesco, 1978-; Delgadino, Matias; Bonforte, MatteoThis 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.