arXiv subjects

João Vítor da Silva

Publications and source records attributed to João Vítor da Silva.

6 recordsLinked to original sources

arXiv2025

Regularity properties for $p-$dead core problems and their asymptotic limit as $p \to \infty$

We study regularity issues and the limiting behavior as $p\to\infty$ of nonnegative solutions for elliptic equations of $p-$Laplacian type ($2 \leq p< \infty$) with a strong absorption: $$ -Δ_p u(x) + λ_0(x) u_{+}^q(x) = 0 \quad \text{ in } \quad Ω\subset \mathbb{R}^N, $$ where $λ_0>0$ is a bounded function, $Ω$ is a bounded domain and $0\leq q 0\} \cap Ω$ where the sharp regularity exponent is given explicitly by $γ= \frac{1}{1-\ell}$. Finally, some weak geometric and measure theoretical properties as non-degeneracy, uniform positive density, porosity and convergence of the free boundaries are proved.

math.AP↗

arXiv2025

Sharp regularity estimates for quasi-linear elliptic dead core problems and applications

In this manuscript we study geometric regularity estimates for quasi-linear elliptic equations of $p$-Laplace type ($1 < p< \infty$) with strong absorption condition: $$ -\text{div}\,(Φ(x, u, \nabla u)) + λ_0(x) u_{+}^q(x) = 0 \quad \text{in} \quad Ω\subset \mathbb{R}^N, $$ where $Φ: Ω\times \mathbb{R}_{+} \times \mathbb{R}^N \to \mathbb{R}^N$ is a vector field with an appropriate $p$-structure, $λ_0$ is a non-negative and bounded function and $0\leq q 0\} \cap Ω$, where the regularity exponent is given explicitly by $γ= \frac{p}{p-1-q} \gg 1$. Some weak geometric and measure theoretical properties as non-degeneracy, uniform positive density and porosity of free boundary are proved. As an application, a Liouville-type result for entire solutions is established provided that their growth at infinity can be controlled in an appropriate manner. Finally, we obtain finiteness of $(N-1)$-Hausdorff measure of free boundary for a particular class of dead core problems. The approach employed in this article is novel even to dead core problems governed by the $p$-Laplace operator $-Δ_p u + λ_0 u^qχ_{\{u>0\}} = 0$ for any $λ_0>0$. \newline \newline \noindent \textbf{Keywords:} Quasi-linear elliptic operators of $p$-Laplace type, improved regularity estimates, Free boundary problems of dead core type, Liouville type results, Hausdorff measure estimates.

math.AP↗

arXiv2021

Free boundary regularity for a class of one-phase problems with non-homogeneous degeneracy

We consider a one-phase free boundary problem governed by doubly degenerate fully non-linear elliptic PDEs with non-zero right hand side, which should be understood as an analog (non-variational) of certain double phase functionals in the theory of non-autonomous integrals. By way of brief elucidating example, such non-linear problems in force appear in the mathematical theory of combustion, as well as in the study of some flame propagation problems. In such an environment we prove that solutions are Lipschitz continuous and they fulfil a non-degeneracy property. Furthermore, we address the Caffarelli's classification scheme: Flat and Lipschitz free boundaries are locally $C^{1, β}$ for some $0 < β(universal) < 1$.

math.AP↗

arXiv2018

Sharp regularity estimates for quasilinear evolution equations

We establish sharp geometric $C^{1+α}$ regularity estimates for bounded weak solutions of evolution equations of $p$-Laplacian type. Our approach is based on geometric tangential methods, and makes use of a systematic oscillation mechanism combined with an adjusted intrinsic scaling argument.

math.AP↗

arXiv2017

A limiting free boundary problem with gradient constraint and Tug-of-War games

In this manuscript we deal with regularity issues and the asymptotic behaviour (as $p \to \infty$) of solutions for elliptic free boundary problems of $p-$Laplacian type ($2 \leq p< \infty$): \begin{equation*} -Δ_p u(x) + λ_0(x)χ_{\{u>0\}}(x) = 0 \quad \mbox{in} \quad Ω\subset \mathbb{R}^N, \end{equation*} with a prescribed Dirichlet boundary data, where $λ_0>0$ is a bounded function and $Ω$ is a regular domain. First, we prove the convergence as $p\to \infty$ of any family of solutions $(u_p)_{p\geq 2}$, as well as we obtain the corresponding limit operator (in non-divergence form) ruling the limit equation, $$ \left\{ \begin{array}{rcrcl} \max\left\{-Δ_{\infty} u_{\infty}, \,\, -|\nabla u_{\infty}| + χ_{\{u_{\infty}>0\}}\right\} & = & 0 & \text{in} & Ω\cap \{u_{\infty} \geq 0\} \\ u_{\infty} & = & g & \text{on} & \partial Ω. \end{array} \right. $$ Next, we obtain uniqueness for solutions to this limit problem together with a number of weak geometric and measure theoretical properties as non-degeneracy, uniform positive density, porosity and convergence of the free boundaries. Finally, we show that any solution to the limit operator is a limit of value functions for a specific Tug-of-War game.

math.AP↗

arXiv2016

Cavity type problems ruled by infinity Laplacian operator

We study a singularly perturbed problem related to infinity Laplacian operator with prescribed boundary values in a region. We prove that solutions are locally (uniformly) Lipschitz continuous, they grow as a linear function, are strongly non-degenerate and have porous level surfaces. Moreover, for some restricted cases we show the finiteness of the (n-1)-dimensional Hausdorff measure of level sets. The analysis of the asymptotic limits is carried out as well.

math.AP↗