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Leandro S. Tavares

Publications and source records attributed to Leandro S. Tavares.

10 recordsLinked to original sources
arXiv2025

A minimum problem associated with scalar Ginzburg-Landau equation and free boundary

Let $N>2$, $p\in \left(\frac{2N}{N+2},+\infty\right)$, and $Ω$ be an open bounded domain in $\mathbb{R}^N$. We consider the minimum problem $$ \mathcal{J} (u) := \displaystyle\int_{Ω} \left(\frac{1}{p}| \nabla u| ^p+λ_1\left(1-(u^+)^2\right)^2+λ_2u^+\right)\text{d}x\rightarrow \text{min} $$ over a certain class $\mathcal{K}$, where $λ_1\geq 0$ and $ λ_2\in \mathbb{R}$ are constants, and $u^+:=\max\{u,0\}$. The corresponding Euler-Lagrange equation is related to the Ginzburg-Landau equation and involves a subcritical exponent when $λ_1>0$. For $λ_1\geq 0$ and $ λ_2\in \mathbb{R}$, we prove the existence, non-negativity, and uniform boundedness of minimizers of $\mathcal{J} (u) $. Then, we show that any minimizer is locally $C^{1,α}$-continuous with some $α\in (0,1)$ and admits the optimal growth $\frac{p}{p-1}$ near the free boundary. Finally, under the additional assumption that $λ_2>0$, we establish non-degeneracy for minimizers near the free boundary and show that there exists at least one minimizer for which the corresponding free boundary has finite ($N-1$)-dimensional Hausdorff measure.

math.AP↗
arXiv2024

A multiphase eigenvalue problem on a stratified Lie group

We consider a multiphase spectral problem on a stratified Lie group. We prove the existence of an eigenfunction of $(2,q)$-eigenvalue problem on a bounded domain. Furthermore, we also establish a Pohozaev-like identity corresponding to the problem on the Heisenberg group.

math.AP↗
arXiv2023

Existence of solutions for a singular double phase problem involving a $ψ$-Hilfer Fractional operator via Nehari Manifold

In this present paper, we investigate a new class of singular double phase $p$-Laplacian equation problems with a $ψ$-Hilfer fractional operator combined from a parametric term. Motivated by the fibering method using the Nehari manifold, we discuss the existence of at least two weak solutions to such problems when the parameter is small enough. Before attacking the main contribution, we discuss some results involving the energy functional and the Nehari manifold.

math.GM↗
arXiv2021

A weighted fractional problem involving a singular nonlinearity and a $L^1$ data

In this article, we show the existence of a unique entropy solution to the following problem: \begin{equation} \begin{split} (-Δ)_{p,α}^su&= f(x)h(u)+g(x) ~\text{in}~Ω,\\ u&>0~\text{in}~Ω,\\ u&= 0~\text{in}~\mathbb{R}^N\setminusΩ,\nonumber \end{split} \end{equation} where the domain $Ω\subset \mathbb{R}^N$ is bounded and contains the origin, $ α\in[0,\frac{N-ps}{2})$, $s\in (0,1)$, $2-\frac{s}{N} 1$ and $h$ is a general singular function with singularity at 0. Further, the fractional $p$-Laplacian with weight $α$ is given by $$(-Δ)_{p,α}^su(x)=\text{P. V.}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}\frac{dy}{|x|^α|y|^α},~\forall x\in \mathbb{R}^N.$$

math.AP↗
arXiv2019

Basic results of fractional Orlicz-Sobolev space and applications to non-local problems

In this paper, we study the interplay between Orlicz-Sobolev spaces $L^{M}$ and $W^{1,M}$ and fractional Sobolev spaces $W^{s,p}$. More precisely, we give some qualitative properties of the new fractional Orlicz-Sobolev space $W^{s,M}$, where $s\in (0,1)$ and $M$ is an $N-$function. We also study a related non-local operator, which is a fractional version of the nonhomogeneous $M$-Laplace operator. As an application, we prove existence of weak solution for a non-local problem involving the new fractional $M-$Laplacian operator.

math.AP↗
arXiv2019

Mild and strong solutions for Hilfer evolution equation

In this paper, we investigate the existence and uniqueness of mild and strong solutions of fractional semilinear evolution equations in the Hilfer sense, by means of Banach fixed point theorem and the Gronwall inequality.

math.CA↗
arXiv2018

A minimum problem with free boundary and subcritical growth in Orlicz spaces

The aim of this paper is to study the heterogeneous optimization problem \begin{align*} \mathcal {J}(u)=\int_Ω(G(|\nabla u|)+qF(u^+)+hu+λ_{+}χ_{\{u>0\}} )\text{d}x\rightarrow\text{min}, \end{align*} in the class of functions $ W^{1,G}(Ω)$ with $ u-φ\in W^{1,G}_{0}(Ω)$, for a given function $φ$, where $W^{1,G}(Ω)$ is the class of weakly differentiable functions with $\int_ΩG(|\nabla u|)\text{d}x<\infty$. The functions $G$ and $F$ satisfy structural conditions of Lieberman's type that allow for a different behavior at $0$ and at $\infty$. {}{Moreover, $F$ allows for a subcritical growth.} Given functions $q,h$ and constant $λ_+\geq 0$, we address several regularity results for minimizers of $\mathcal {J}(u)$, including local $C^{1,α}-$, and local Log-Lipschitz continuities for minimizers of $\mathcal {J}(u)$ with $λ_+=0$, and {}{$λ_+\geq 0$} respectively. We also establish growth rate near the free boundary for each non-negative minimizer of $\mathcal {J}(u)$ with $λ_+=0$, and $λ_+>0$ respectively. Furthermore, under additional assumption that $F\in C^1([0,+\infty); [0,+\infty))$, local Lipschitz regularity is carried out for non-negative minimizers of $\mathcal {J}(u)$ with $λ_{+}>0$.

math.AP↗