Infinity Laplacian equation with strong absorptions

Abstract

We study regularity properties of solutions to reaction-diffusion equations ruled by the infinity laplacian operator. We focus our analysis in models presenting plateaus, i.e. regions where a non-negative solution vanishes identically. We obtain sharp geometric regularity estimates for solutions along the boundary of plateaus sets. In particular we show that the (n − ε)-Hausdorff measure of the plateaus boundary is finite, for a universal number ε > 0.

Geometric regularity estimates for fully nonlinear elliptic equations with free boundaries

“…In our first result, we establish the precise asymptotic behavior at which nonnegative viscosity solutions leave their dead‐core sets. This is an important piece of information in several free boundary problems (see [4, 5, 32, 35] and [38]), and it plays a pivotal role in establishing many weak geometric properties (see Section 5 for more details). Theorem Let u be a nonnegative, bounded viscosity solution to (1.2) in B 1 and let be a point in the closure of the non‐coincidence set.…”

“…Despite of the fact that there is a huge amount of literature on dead‐core problems in divergence form and theirs qualitative features, quantitative counterparts in non‐variational elliptic models like (1.2) are far less studied due to the rigidity of the structure of such operators (see [4, 20] and [38] as enlightening examples). Therefore, the treatment of such free boundary problems requires the development of new ideas and modern techniques.…”

“…It is worth highlighting that our results generalize and/or recover and/or complement previous ones (see [38] (dead‐core problems with uniformly elliptic structure), [36, Sect. 4] ( p ‐dead core problems) and [40] (quenching type problems), see also [4]) by allowing deal with more general classes of operators. Moreover, different from the former results, we suggest different approaches and techniques to address such regularity issues, which enable us to establish a number of significant applications for our findings.…”

Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients

“…The study of free boundary in discontinuous media is also a very active field of research in itself, see in particular [T16] for related problems involving a fully nonlinear dead-core problems, [ALT16] for dead-core problems driven by the infinity Laplacian, and [PT16] for cavity problems in rough media. See also [BT14] for a case in which the coefficients belong to the space of vanishing mean oscillation.…”

A free boundary optimization problem for the ∞-Laplacian

“…which, despite being too degenerate to realistically represent a physical diffusion process, has been previously used in the context of free boundary problems, for example in [5], where a dead-core problem is considered. We stress that one of the major difficulties when dealing with ∆ ∞ relies precisely on the fact that it diffuses only in the direction of the gradient, which changes point-by-point and depends on the solution itself.…”