| Grant number: | 23/14636-6 |
| Support Opportunities: | Research Grants - Visiting Researcher Grant - Brazil |
| Start date: | February 18, 2024 |
| End date: | January 30, 2025 |
| Field of knowledge: | Physical Sciences and Mathematics - Mathematics - Analysis |
| Principal Investigator: | Sergio Henrique Monari Soares |
| Grantee: | Sergio Henrique Monari Soares |
| Visiting researcher: | Diego Ribeiro Moreira |
| Visiting researcher institution: | Universidade Federal do Ceará (UFC). Centro de Ciências, Brazil |
| Host Institution: | Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil |
| Associated research grant: | 22/16407-1 - TESdE: Thematic on Equations and Systems of differential Equations, AP.TEM |
Abstract
In this project we plan to explore the basic and classical ingredients such as Hopf-Ole\u{\i}nik Lemma and interior and up to the boundary gradient estimates for solutions to non-uniformly elliptic equations of quasilinear type that violates the $\Delta_2$ Lieberman's condition introduced in the 90s, allowing the ellipticity to blow up. The typical operator under this condition is given by $\mathcal{L}u = div(2e^{|\nabla u|^2}) = 2e^{|\nabla u|^2}(2\Delta_{\infty}u + \Delta u) $. Our method is a geometric one based on the development of suitable barriers for these operators. This also allows us to study viscosity solutions of Bernoulli-type Free Boundary Problems governed by this type of operator. (AU)
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