Browsing by Subject "Free boundary problems"
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Item Regularity estimates for some free boundary problems of obstacle-type(2016-05) Jain, Rohit, Ph. D.; Caffarelli, Luis A.; Figalli, Alessio; Chen, Thomas; Vasseur, Alexis; Roquejoffre, Jean-MichelWe study regularity estimates for solutions to implicit constraint obstacle problems and penalized boundary obstacle problems. We first prove regularity estimates for the solution and the free boundary in the classical stochastic impulse control problem. We show that the free boundary partial {u < Mu}, where Mu is the implicit constraint obstacle, can be decomposed into a union of regular points, singular points, and degenerate points with corresponding regularity and measure theoretic estimates. We then turn to generalizing our analysis to the fully nonlinear problem with obstacles admitting a general modulus of semi-convexity omega(r). We prove that solutions to the fully nonlinear stochastic impulse control problem are C^{omega(r)} up to C^{1,1}. Finally we turn our attention to study both nonuniform and uniform estimates for the penalized boundary obstacle problem, Delta^{1/2}u^{epsilon} =beta_{epsilon} (u^{epsilon}). We obtain sharp estimates for the solution in both the nonuniform and uniform theory.Item Regularity for solutions of nonlocal fully nonlinear parabolic equations and free boundaries on two dimensional cones(2013-05) Chang Lara, Hector Andres; Caffarelli, Luis A.On the first part, we consider nonlinear operators I depending on a family of nonlocal linear operators [mathematical equations]. We study the solutions of the Dirichlet initial and boundary value problems [mathematical equations]. We do not assume even symmetry for the kernels. The odd part bring some sort of nonlocal drift term, which in principle competes against the regularization of the solution. Existence and uniqueness is established for viscosity solutions. Several Hölder estimates are established for u and its derivatives under special assumptions. Moreover, the estimates remain uniform as the order of the equation approaches the second order case. This allows to consider our results as an extension of the classical theory of second order fully nonlinear equations. On the second part, we study two phase problems posed over a two dimensional cone generated by a smooth curve [mathematical symbol] on the unit sphere. We show that when [mathematical equation] the free boundary avoids the vertex of the cone. When [mathematical equation]we provide examples of minimizers such that the vertex belongs to the free boundary.Item Regularization in phase transitions with Gibbs-Thomson law(2010-12) Guillen, Nestor Daniel; Caffarelli, Luis A.; Gamba, Irene; Souganidis, Panagiotis; de La Llave, Rafael; Vasseur, Alexis; Engquist, BjornWe study the regularity of weak solutions for the Stefan and Hele- Shaw problems with Gibbs-Thomson law under special conditions. The main result says that whenever the free boundary is Lipschitz in space and time it becomes (instantaneously) C[superscript 2,alpha] in space and its mean curvature is Hölder continuous. Additionally, a similar model related to the Signorini problem is introduced, in this case it is shown that for large times weak solutions converge to a stationary configuration.