Browsing by Subject "Regularity"

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    Problems in non linear PDE : equilibrium configurations in periodic media and non local diffusion
    (2012-08) Davila, Gonzalo, 1982-; Caffarelli, Luis A.; Arapostathis, Aristotle; de la Llave, Rafael; Gamba, Irene; Knopf, Dan; Pavlovic, Natasa
    We study three different problems in non linear PDE. The first problem relates to finding equilibrium configurations in periodic media, more precisely, given an Area-Dirichlet functional J, which is periodic under integer translations and given three planes in R[superscript d], we proof there exists at least one minimizer such that it’s positive part, negative part and zero set remain at a uniform bounded distance of each plane. The second and third problem are related to non local diffusion, in the elliptic non symmetric case and parabolic case. In both cases we are interested in proving interior regularity for solutions of the aforementioned equations.
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    The double obstacle problem and the two membranes problem
    (2018-05) Duque, Luis Felipe; Caffarelli, Luis A.; Vasseur, Alexis; Arapostathis, Aristotle; Sirbu, Mihai
    In the first part of this dissertation, we study the existence, regularity and the free boundary of the double obstacle problem in different formulations that involve linear, elliptic, parabolic and fully nonlinear equations. The second part focuses on the two membranes problem for fully nonlinear elliptic operators, here we prove the existence of solutions and then we prove the optimal regularity when the operators involved are the Pucci operators. Finally, we give an example that shows that no regularity for the free boundary is to be expected to hold in general.