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Session S18 - Recent progress in non-linear PDEs and their applications

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Fully non-linear singularly perturbed models with non-homogeneous degeneracy

Elzon Cezar Bezerra Junior

Universidade Federal do Ceará , Brasil   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

This work is devoted to study non-variational, non-linear singularly perturbed elliptic models enjoying a double degeneracy character with prescribed boundary value in a domain. In its simplest form, for each $\varepsilon>0$ fixed, we seek a non-negative function $u^{\epsilon}$ satisfying $$ \left\{ \begin{array}{rclcl} \left[|\nabla u^{\varepsilon}|^p + \mathfrak{a}(x)|\nabla u^{\varepsilon}|^q \right] \Delta u^{\varepsilon} & = & \zeta_{\varepsilon}(x, u^{\varepsilon}) & \mbox{in} & \Omega,\\ u^{\varepsilon}(x) & = & g(x) & \mbox{on} & \partial \Omega, \end{array} \right. $$ in the viscosity sense for suitable data $p, q \in (0, \infty)$, $\mathfrak{a}$, $g$, where $\zeta_{\varepsilon}$ one behaves singularly of order $\mbox{O} \left(\epsilon^{-1} \right)$ near $\epsilon$-level surfaces. In such a context, we establish existence of certain solutions. We also prove that solutions are locally (uniformly) Lipschitz continuous, and they grow in a linear fashion. Moreover, solutions and their free boundaries possess a sort of measure-theoretic and weak geometric properties. Particularly, for a restricted class of non-linearities, we prove the finiteness of the $(N-1)$-dimensional Hausdorff measure of level sets. At the end, we address a complete and in-deep analysis concerning the asymptotic limit as $\varepsilon \to 0^{+}$, which is related to one-phase solutions of inhomogeneous non-linear free boundary problems in flame propagation and combustion theory.

Keywords: Singular perturbation methods, doubly degenerate fully non-linear operators, geometric regularity theory.

Joint work with João Vitor da Silva and Gleydson Chaves Ricarte.

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