In this paper we study one-phase fully nonlinear singularly perturbed elliptic problems with high energy activation potentials, ζ ε (u) with ζ ε → δ 0 · ζ . We establish uniform and optimal gradient estimates of solutions and prove that minimal solutions are non-degenerated. For problems governed by concave equations, we establish uniform weak geometric properties of approximating level surfaces. We also provide a thorough analysis of the free boundary problem obtained as a limit as the ε-parameter term goes to zero. We find the precise jumping condition of limiting solutions through the phase transition, which involves a subtle homogenization process of the governing fully nonlinear operator. In particular, for rotational invariant operators, F (D 2 u), we show the normal derivative of limiting function is constant along the interface. Smoothness properties of the free boundary are also addressed.
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.
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