Singularly perturbed fully nonlinear parabolic problems and their asymptotic free boundaries

Abstract

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.

Porosity of the free boundary for p-parabolic type equations in non-divergence form

“…First of all we show Lipschitz continuity of the viscosity solution u ε with respect to x using a Bernstein type argument. The strategy to show Lipschitz regularity is based on the works [11,16] but it turns out that the result is not true for p = 1 since the constant L (see Proposition 3.1) blows up for p → 1. Finally, we will show that bound on the gradients implies limitation in the seminorm Lip (1, 1/2).…”

“…When we consider free boundary problems, optimal regularity results and sharp non-degeneracy are crucial for further analysis of the set ∂{u > 0}. In this direction, just recently we have the work [16].…”

“…Equation (1.1) is the degenerate fully nonlinear version of the parabolic problem studied by H. Shahgholian & G. Ricarte, J.M. Urbano and R. Teymurazyan in [18] and [16] respectively. They proved optimal regularity and non-degeneracy estimates for the solution.…”

“…In analogy with [16], what we want to prove is that, at each time level, the ndimensional Lebesgue measure of the free boundary is zero because it is porous. We prove this by obtaining a non-degeneracy result and by controlling the growth rate of the solution near the free boundary.…”

“…We prove this by obtaining a non-degeneracy result and by controlling the growth rate of the solution near the free boundary. As in [16], the solution to (1.1) is derived from an approximating family of functions, which are solutions to some Dirichlet problems. More precisely, we consider the following singular perturbation problem…”