Title: Improved Regularity for a Nonlocal Dead-Core Problem

Abstract: We obtain sharp regularity results for solutions of a nonlocal dead-core problem at branching points. Our approach, which does not rely on the maximum principle, introduces a new strategy for analyzing two-phase problems in the local framework - an area that remains largely unexplored.

Title: Improved regularity estimates for Hénon-type equations driven by the ∞-Laplacian

Abstract: In this work, we establish sharp and improved regularity estimates for viscosity solutions of Hénon-type equations with possibly singular weights and strong absorption governed by the infinity Laplacian under suitable assumptions on the data.  In this setting, we derive an explicit regularity exponent that depends only on universal parameters. Additionally, we prove non-degeneracy properties, providing further geometric insights into the nature of these solutions. Our regularity estimates not only improve but also extend, to some extent, the previously obtained results for zero-obstacle and dead-core problems driven by the $\infty$-Laplacian. As an application of our findings, we also address some Liouville-type results for this class of equations.

Title: Regularity for Degenerate Evolution Problems in Orlicz Spaces

Abstract: We investigate quantitative regularity estimates within the theory of degenerate parabolic partial differential equations, specifically in the context of Orlicz-Sobolev spaces. Employing a geometric tangential analysis tailored to Orlicz-Sobolev structures and intrinsic scalings, we derive precise interior H\" older regularity estimates for bounded weak solutions.

Title: Improved regularity estimates for weighted quasilinear elliptic models

Abstract: In this Lecture, we obtain sharp and improved regularity estimates for weak solutions of weighted quasilinear elliptic models of Hardy-Hénon-type, featuring an explicit regularity exponent depending only on universal parameters. We also establish higher regularity estimates and non-degeneracy properties in some specific scenarios, providing further geometric insights into such solutions. Our regularity estimates both enhance and, to some extent, extend the results arising from the C^p' conjecture for the p-Laplacian with a bounded source term. Finally, our results are noteworthy, even in the simplest model case governed by the p-Laplacian with regular coefficients, under suitable assumptions on the data, with possibly singular weight, which includes the Matukuma and Batt–Faltenbacher–Horst's equations as toy models.

This is a joint work with Disson dos Prazeres (UFS-Brazil), Gleydson C. Ricarte (UFC-Brazil), and Ginaldo S. Sá (Unicamp-Brazil).

Title: Universal Regularity for Transmission Problems in Fully Nonlinear Elliptic Models

Abstract: This manuscript focuses on borderline, gradient, and higher regularity estimates of solutions to fully nonlinear elliptic transmission problems for suitable source functions f, continuous data g, and a fixed interface \psi. Specifically, under the standard uniform ellipticity condition on the governing operators and suitable Hölder continuity assumptions on g and \psi, we obtain optimal C^{1,\alpha} regularity provided the forcing term lies in the Lebesgue space L^p for p>n. Furthermore, we establish C^{0,log-Lip} regularity in the borderline case f \in L^n. Finally, in the critical borderline case, i.e., when the forcing terms belong to the BMO space and under asymptotic concavity conditions for the governing operators with suitable Hölder continuity assumptions on the data, we obtain higher log-Lipschitz-type estimates, specifically, u \in C_{loc}^{1, log-Lip}. 

Title: Optimal Boundary Regularity for viscosity solutions of fully nonlinear elliptic equations

Abstract: We establish optimal boundary regularity results for viscosity solutions to second order fully nonlinear uniformly elliptic equations in the form F(D^2u(x),Du(x),x)= f(x) in \Omega. In particular, we obtain sharp estimates for borderline cases f\in L^n(\Omega) and f\in BMO(\Omega). For source functions in BMO(\Omega), we obtain C^{1,Log-Lip} interior regularity for flat solutions of non-convex elliptic equations. As a consequence, we obtain C^{1,Log-Lip} estimates near the boundary; which again is an optimal estimate.