arXiv subjects

Sergio Segura de León

Publications and source records attributed to Sergio Segura de León.

11 recordsLinked to original sources

arXiv2026

The inhomogeneous Total Variation Flow with $L^1$-data

This paper is devoted to the study of the Dirichlet problem for the parabolic equation driven by the $1$--Laplacian operator under minimal integrability assumptions. Specifically, we consider \begin{equation*} u'-\Div(Du/|D u|)=f\qquad\text{ in } (0,+\infty)\timesΩ\,, \end{equation*} where $Ω\subset\R^N$ is a bounded open set with Lipschitz boundary, $u_0\in L^1(Ω)$ is the initial datum, and $f\in L_{loc}^1(0,+\infty; L^1(Ω))$ is the source term. We establish the existence and uniqueness of entropy solutions in this low-regularity setting. Our approach relies on an approximation scheme and an entropy formulation adapted to the \mbox{$1$--Laplacian} structure. Additional results include comparison between solutions, further regularity when data have higher integrability and an analysis of the long-time decay of solutions in the homogeneous case.

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arXiv2025

Stationary equation of the relativistic heat diffusion in transparent media having $L^1$--data

Our objective is to prove existence of a solution to the Dirichlet problem for an equation arising in the theory of radiation hydrodynamics to deal with the radiating energy in transparent media. We study its stationary equation with $L^1$--datum in a bounded domain. This problem was addressed in [11] for regular data (data belonging to $L^N(Ω)$) and a bounded solution was obtained. In our framework, the proof of existence is far from trivial since the solution sought cannot be bounded. Consequently, the Anzellotti theory of pairings does not apply and we have to use new developments to introduce the meaning of solution. We also study the regularity of solutions when data belong to $L^p(Ω)$, with $1<p<N$. Our result is coherent with the regularity found in [11].

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arXiv2025

Quasi-linear parabolic equations having a superlinear gradient term which depends on the solution

In this paper, we study existence and regularity for solutions to parabolic equations having a superlinear lower order term depending on both the solution and its gradient. Two different situations are analyzed. On the one hand, we assume that the initial datum belongs to an Orlicz space of exponentially summable functions. On the other, data in an appropriate Lebesgue space satisfying a smallness condition are considered. Our results are coherent with those of previous papers in similar frameworks.

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arXiv2023

The role of absorption terms in Dirichlet problems for the prescribed mean curvature equation

In this paper we study existence and uniqueness of solutions to Dirichlet problems as $$ \begin{cases} g(u) -{\rm div}\left(\frac{D u}{\sqrt{1+|D u|^2}}\right) = f & \text{in}\;Ω,\\ \newline u=0 & \text{on}\;\partialΩ, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz boundary, $g:\mathbb{R}\to\mathbb{R}$ is a continuous function and $f$ belongs to some Lebesgue spaces. In particular, under suitable saturation and sign assumptions, we explore the regularizing effect given by the absorption term $g(u)$ in order to get a solutions for data $f$ merely belonging to $L^1(Ω)$ and with no smallness assumptions on the norm. We also prove a sharp boundedness result for data in $L^{N}(Ω)$ as well as uniqueness if $g$ is increasing.

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arXiv2023

On a nonlinear Robin problem with an absorption term on the boundary and $L^1$ data

We deal with existence and uniqueness of nonnegative solutions to \begin{equation*} \left\{ \begin{array}{l} -Δu = f(x) \text{ in }Ω, \frac{\partial u}{\partial ν} + λ(x) u = \frac{g(x)}{u^η} \text{ on } \partialΩ, \end{array} \right. \end{equation*} where $η\ge 0$ and $f,λ$ and $g$ are nonnegative integrable functions. The set $Ω\subset\mathbb{R}^N (N> 2)$ is open and bounded with smooth boundary and $ν$ denotes its unit outward normal vector. More generally, we handle equations driven by monotone operators of $p$-Laplacian type jointly with nonlinear boundary conditions. We prove existence of an entropy solution and check that this solution is unique under natural assumptions. Among other features, we study the regularizing effect given to the solution by both the absorption and the nonlinear boundary term.

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arXiv2022

Existence and uniqueness for the inhomogeneous 1-Laplace evolution equation revisited

In this paper we deal with an inhomogeneous parabolic Dirichlet problem involving the 1-Laplacian operator. We show the existence of a unique solution when data belong to $L^1(0,T;L^2(Ω))$ for every $T>0$. As a consequence, global existence and uniqueness for data in $L^1_{loc}(0,+\infty;L^2(Ω))$ is obtained. Our analysis retrieves the results of \cite{SW} in a correct and complete way.

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arXiv2022

Behaviour of solutions to $p$-Laplacian with Robin boundary conditions as $p$ goes to $1$

We study the asymptotic behaviour, as $p\to 1^{+}$, of the solutions of the following inhomogeneous Robin boundary value problem: \begin{equation} \label{pbabstract} \tag{P} \left\{\begin{array}{ll} \displaystyle -Δ_p u_p = f & \text{in }Ω, \displaystyle |\nabla u_p|^{p-2}\nabla u_p\cdot ν+λ|u_p|^{p-2}u_p = g& \text{on } \partialΩ, \end{array}\right. \end{equation} where $Ω$ is a bounded domain in $\mathbb R^{N}$ with sufficiently smooth boundary, $ν$ is its unit outward normal vector and $Δ_p v$ is the $p$-Laplacian operator with $p>1$. The data $f\in L^{N,\infty}(Ω)$ (which denotes the Marcinkiewicz space) and $λ,g$ are bounded functions defined on $\partialΩ$ with $λ\ge0$. We find the threshold below which the family of $p$--solutions goes to 0 and above which this family blows up. As a second interest we deal with the $1$-Laplacian problem formally arising by taking $p\to 1^+$ in \eqref{pbabstract}.

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arXiv2020

Gelfand type problems involving the 1-Laplacian operator

In this paper, the theory of Gelfand problems is adapted to the 1--Laplacian setting. Concretely, we deal with the following problem \begin{equation*} \left\{\begin{array}{cc} -Δ_1u=λf(u) &\hbox{in }Ω\,;\\[2mm] u=0 &\hbox{on }\partialΩ\,; \end{array} \right. \end{equation*} where $Ω\subset\mathbb{R}^N$ ($N\ge1$) is a domain, $λ\geq 0$ and $f\>:\>[0,+\infty[\to]0,+\infty[$ is any continuous increasing and unbounded function with $f(0)>0$. It is proved the existence of a threshold $λ^*=\frac{h(Ω)}{f(0)}$ (being $h(Ω)$ the Cheeger constant of $Ω$) such that there exists no solution when $λ>λ^*$ and the trivial function is always a solution when $λ\leλ^*$. The radial case is analyzed in more detail showing the existence of multiple solutions (even singular) as well as the behaviour of solutions to problems involving the $p$--Laplacian as $p$ tends to 1, which allows us to identify proper solutions through an extra condition.

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arXiv2020

The Dirichlet problem for the $1$-Laplacian with a general singular term and $L^1$-data

We study the Dirichlet problem for an elliptic equation involving the $1$-Laplace operator and a reaction term, namely: $$ \left\{\begin{array}{ll} \displaystyle -Δ_1 u =h(u)f(x)&\hbox{in }Ω\,,\\ u=0&\hbox{on }\partialΩ\,, \end{array}\right. $$ where $ Ω\subset \mathbb{R}^N$ is an open bounded set having Lipschitz boundary, $f\in L^1(Ω)$ is nonnegative, and $h$ is a continuous real function that may possibly blow up at zero. We investigate optimal ranges for the data in order to obtain existence, nonexistence and (whenever expected) uniqueness of nonnegative solutions.

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arXiv2019

Regularizing effects concerning elliptic equations with a superlinear gradient term

We consider the homogeneous Dirichlet problem for an elliptic equation driven by a linear operator with discontinuous coefficients and having a subquadratic gradient term. This gradient term behaves as $g(u)|\nabla u|^q$, where $1<q<2$ and $g(s)$ is a continuous function. Data belong to $L^m(Ω)$ with $1\le m <\frac{N}{2}$ as well as measure data instead of $L^1$-data, so that unbounded solutions are expected. Our aim is, given $1\le m<\frac N2$ and $1<q<2$, to find the suitable behaviour of $g$ close to infinity which leads to existence for our problem. We show that the presence of $g$ has a regularizing effect in the existence and summability of the solution. Moreover, our results adjust with continuity with known results when either $g(s)$ is constant or $q=2$.

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arXiv2017

Parabolic equations with natural growth approximated by nonlocal equations

In this paper we study several aspects related with solutions of nonlocal problems whose prototype is $$ u_t =\displaystyle \int_{\mathbb{R}^N} J(x-y) \big( u(y,t) -u(x,t) \big) \mathcal G\big( u(y,t) -u(x,t) \big) dy \qquad \mbox{ in } \, Ω\times (0,T)\,, $$ being $ u (x,t)=0 \mbox{ in } (\mathbb{R}^N\setminus Ω)\times (0,T)\,$ and $ u(x,0)=u_0 (x) \mbox{ in } Ω$. We take, as the most important instance, $\mathcal G (s) \sim 1+ \fracμ{2} \frac{s}{1+μ^2 s^2 }$ with $μ\in \mathbb{R}$ as well as $u_0 \in L^1 (Ω)$, $J$ is a smooth symmetric function with compact support and $Ω$ is either a bounded smooth subset of $\mathbb{R}^N$, with nonlocal Dirichlet boundary condition, or $\mathbb{R}^N$ itself. The results deal with existence, uniqueness, comparison principle and asymptotic behavior. Moreover we prove that if the kernel rescales in a suitable way, the unique solution of the above problem converges to a solution of the deterministic Kardar-Parisi-Zhang equation.

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