Fully nonlinear singularly perturbed equations and asymptotic free boundaries

Abstract

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.

Singularly perturbed fully nonlinear parabolic problems and their asymptotic free boundaries

“…However, when F is concave or convex, then the limit exists. In fact, from the dominated convergence theorem one gets a stronger assertion (see [18,Proposition 6.1] for details): if F satisfies (A1), then…”

“…This alternative approach opens an avenue leading also to non-variational free boundary problems. Recently, the singular perturbation problem F (x, D 2 u ε ) = β ε (u ε ), which is the elliptic counterpart of (E ε ), was studied in [18]; the authors obtain Lipschitz estimates and study the limiting free boundary problem. Our aim in this paper is to extend these results to the parabolic case.…”

“…We do not impose a free boundary condition and thus the limiting problem is not understood as overdetermined. Unlike the elliptic case (see, for example, [18]), one can not apply the Harnack inequality in order to prove the (uniform) regularity of solutions. The reason is that we can only compare functions on parabolic boundaries, not on the top of a cylinder; we are thus unable to pass from one level to another.…”

“…We overcome this difficulty by using a Bernstein type argument (see the proof of Proposition 4.1). For the same reason, the study of the free boundary of the limiting problem requires a totally different approach: in the elliptic case, using a covering argument, one can prove the finiteness of the (n − 1)-dimensional Hausdorff measure of the free boundary (see [18]).…”

Cavity type problems ruled by infinity Laplacian operator

“…We state the following theorem independently of the (E ε ) context, since it may be of independent interest. For the proof we refer to [15] (see also [1]). Theorem 2.1.…”