in the viscosity sense for suitable data p, q ∈ (0,∞), a, g, where ζε one behaves singularly of order O(ε^−1 ) near ϵ-level surfaces. In such a context, we establish existence of certain solutions. We also prove that solutions are locally (uniformly) Lipschitz continuous, and they grow in a linear fashion. Moreover, solutions and their free boundaries possess a sort of measure-theoretic and weak geometric properties. Particularly, for a restricted class of non-linearities, we prove the finiteness of the (N − 1)-dimensional Hausdorff measure of level sets. At the end, we address a complete and in-deep analysis concerning the asymptotic limit as ε → 0+, which is related to one-phase solutions of inhomogeneous non-linear free boundary problems in flame propagation and combustion theory. This is a joint-work with J.V. da Silva (IMECC-UNICAMP) and G.C Ricarte (UFC).

We briefly discuss topics related to the regularity of solutions to fully nonlinear elliptic equations in this talk. Especially, we focus on regularity whenever some kind of geometric control of the solution only on one side is present. In our case, this will be some kind of convexity. If time permits, we will address some open questions related to the subject. This is joint work with Alessio Figalli (ETH-Zurich) and J. Ederson Braga (UFC).

I will discuss a foundational question pertaining to the theory of diffusion processes, which, put in layman's terms, can be stated as follows: how much degeneracy in a given diffusion system is too much? More precisely, we are interested in models with a diffusion agent, represented by a 2nd order (fully nonlinear) elliptic operator, and a law of degeneracy, accounting for diffusion resistance of the model. A classical theorem proven independently by Caffarelli and Trundinger in the late 80’s assures that solutions of non-degenerate models are C^{1, α} and a recent theorem proven by Imbert and Silvestre yields Holder continuity of solutions “independently” of the law of degeneracy. We want to understand the (difficult) borderline case regarding (universal) C^1 regularity.

for m > 1 and p > 2 and f ∈ L q,r. More precisely, we show that solutions are locally of class C^{0,β}, where β depends explicitly only on the optimal Holder exponent for solutions of the homogeneous case, the integrability of f in space and time, and nonlinearity terms p and m. The family of equations (0.1) generalizes two well-known cases: the porous media equation, case p = 2, and the p-Laplacian equation, case m = 1. For the very particular case m = 1 and p = 2 we recover the standard heat equation u_t = ∆u. The main motivation for the study of this class of nonlinear evolution equations is their physical relevance, for example, in the study of non-Newtonian fluids, plasma physics, ground water problems, image-analysis, motion of viscous fluids and in the modeling of an ideal gas flowing isoentropically in a inhomogeneous porous medium. Technically, the equation (0.1) exhibits a double nonlinear dependence, on both the solution u and its gradient Du making its diffusion properties degenerate along the parabolic zero set ∂{u(x, t) ̸= 0} as well as along the set of critical points ∂{Du(x, t) ̸= 0}. The main difficulty is obtain the desired estimates near points where the doubly degeneracy occurs. Inspired in recent works, we do a careful tangential analysis with a subtle use of a parabolic intrinsic scaling related to this setting.

Singular and degenerate partial differential equations are unavoidable in the modelling of several phenomena, like phase transitions and chemotaxis, and are also used in machine learning in the context of semi-supervised learning. They encompass a crucial issue in the analysis of pdes, namely wether we can still derive analytical estimates when the crucial algebraic assumption of ellipticity collapses. We provide a broad overview of qualitative versus quantitative regularity estimates for solutions of these equations, introducing the method of intrinsic scaling and deriving sharp estimates by means of geometric tangential analysis. We discuss, in particular, recent results concerning the Stefan problem, the parabolic p-Poisson equation, the porous medium equation and Trudinger’s equation.

In this talk, we will discuss new universal bounds for the Hessian integrability exponent of viscosity super-solutions of fully nonlinear, uniformly elliptic equations. Such estimates yield a quantitative improvement on the decay of this exponent with respect to the dimension. In particular we solve, in the negative, the Armstrong-Silvestre-Smart Conjecture on the optimal exponent for the Hessian integrability. This is a joint work with Eduardo Teixeira.