Revista Matemática Iberoamericana


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Volume 35, Issue 5, 2019, pp. 1535–1558
DOI: 10.4171/rmi/1091

Published online: 2019-06-17

Singularly perturbed fully nonlinear parabolic problems and their asymptotic free boundaries

Gleydson C. Ricarte[1], Rafayel Teymurazyan[2] and José Miguel Urbano[3]

(1) Universidade Federal do Ceará, Fortaleza, Brazil
(2) Universidade de Coimbra, Portugal
(3) Universidade de Coimbra, Portugal

We study fully nonlinear singularly perturbed parabolic equations and their limits. We show that solutions are uniformly Lipschitz continuous in space and Hölder continuous in time. For the limiting free boundary problem, we analyse the behaviour of solutions near the free boundary. We show, in particular, that, at each time level, the free boundary is a porous set and, consequently, is of Lebesgue measure zero. For rotationally invariant operators, we also derive the limiting free boundary condition.

Keywords: Parabolic fully nonlinear equations, singularly perturbed problems, Lipschitz regularity, porosity of the free boundary

Ricarte Gleydson, Teymurazyan Rafayel, Urbano José Miguel: Singularly perturbed fully nonlinear parabolic problems and their asymptotic free boundaries. Rev. Mat. Iberoam. 35 (2019), 1535-1558. doi: 10.4171/rmi/1091