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Francescantonio Oliva

Publications and source records attributed to Francescantonio Oliva.

At least 19 recordsLinked to original sources
arXiv2026

Optimal insulation and concentration breaking for nonlinear Robin boundary value problems

We consider an optimal insulation problem for a bounded domain in $\mathbb{R}^N$ driven by the $p$-Laplace operator ($p>1$). We model the convective heat transfer between the body and the environment, which corresponds, before insulation, to a nonlinear Robin boundary value problem. Assuming the body is surrounded by a thin layer of insulating material of size $\varepsilon^{\frac{1}{p-1}}$, we compute the $Γ$-limit of the governing energy functional as $\varepsilon \to 0^+$. Furthermore, we study the optimization of the heat content among all possible distributions of the insulating material with a fixed total mass. Finally, we highlight a concentration breaking phenomenon. Under a suitable non-degeneracy condition, if the boundary of the domain is connected or the external temperature profile is constant, the optimal insulating layer fails to cover the entire boundary whenever the total mass is sufficiently small. This is shown to be optimal: an explicit example provides that a disconnected boundary can trigger an anomalous double-phase transition, causing the insulation to fracture again even at intermediate mass regimes.

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arXiv2025

Existence and non-existence phenomena for nonlinear elliptic equations with $L^1$ data and singular reactions

We study existence and non-existence of solutions for singular elliptic boundary value problems as \begin{equation}\label{eintro}\begin{cases}\tag{1} \displaystyle -Δ_p u+ \frac{a(x)}{u^γ}=μf(x) \ &\text{ in }Ω, \newline u>0&\text{ in }Ω, \newline u = 0 \ &\text{ on } \partialΩ, \end{cases} \end{equation} where $Ω$ is a smooth bounded open subset of $\mathbb{R}^N$ ($N\ge 2$), $Δ_p u$ is the $p$-Laplacian with $p>1$, $0<γ\leq 1$, and $a\geq0$ is bounded and non-trivial. For any positive $ f\in L^{1}(Ω)$ we show that problem \eqref{eintro} is solvable for any $μ>μ_0>0$, for some $μ_0$ large enough. As a reciprocal outcome we also show that no finite energy solution exists if $0<μ<μ_{0*}$, for some small $μ_{0*}$. This paper extends the celebrated one of J. I. Diaz, J. M. Morel and L. Oswald ([16]) to the case $p\neq2$. Our result is also new for $p=2$ provided the singular term has a critical growth near zero (i.e. $γ=1$).

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arXiv2025

Optimal global $BV$ regularity for 1-Laplace type BVP's with singular lower order terms

In this paper we provide a complete characterization of the regularity properties of the solutions associated to the homogeneous Dirichlet problem \begin{equation*} \begin{cases} \displaystyle - Δ_1 u= h(u)f & \text{in } Ω, \\ \newline u=0 & \text{on } \partial Ω, \end{cases} \end{equation*} where $Ω\subset\mathbb{R}^N$ is a bounded open set with Lipschitz boundary, $f \in L^m(Ω)$ with $m\geq 1$ is a nonnegative function and $h\colon \mathbb{R}^+ \to \mathbb{R}^+$ is continuous, possibly singular at the origin and bounded at infinity. Without any growth restrictions on $h$ at zero, we prove existence of global finite energy solutions in $BV(Ω)$ under sharp conditions on the summability of $f$ and on the behaviour of $h$ at infinity. Roughly speaking, the faster $h$ goes to zero at infinity, the less regularity is required on $f$. In contrast to the $p$-Laplacian case ($p>1$), we show that the behaviour of $h$ at the origin plays essentially no role. The main result contains an extension of the celebrated one of Lazer-McKenna (\cite{lm}) to the case of the $1$-Laplacian as principal operator.

math.AP↗
arXiv2025

Global existence for a Leibenson type equation with reaction on Riemannian manifolds

We show a global existence result for a doubly nonlinear porous medium type equation of the form $$u_t = Δ_p u^m +\, u^q$$ on a complete and non-compact Riemannian manifold $M$ of infinite volume. Here, for $1 1$ and $q>m(p-1)$. In particular, under the assumptions that $M$ supports the Sobolev inequality, we prove that a solution for such a problem exists globally in time provided $q>m(p-1)+\frac pN$ and the initial datum is small enough; namely, we establish an explicit bound on the $L^\infty$ norm of the solution at all positive times, in terms of the $L^1$ norm of the data. Under the additional assumption that a Poincaré-type inequality also holds in $M$, we can establish the same result in the larger interval, i.e. $q>m(p-1)$. This result has no Euclidean counterpart, as it differs entirely from the case of a bounded Euclidean domain due to the fact that $M$ is non-compact and has infinite measure.

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arXiv2024

The Sattinger iteration method for 1-Laplace type problems and its application to concave-convex nonlinearities

In this paper we extend the classical sub-supersolution Sattinger iteration method to $1$-Laplace type boundary value problems of the form \begin{equation*} \begin{cases} \displaystyle -Δ_1 u = F(x,u) & \text{in}\;Ω,\\ \newline u=0 & \text{on}\;\partialΩ, \end{cases} \end{equation*} where $Ω$ is an open bounded domain of $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz boundary and $F(x,s)$ is a Caratheódory function. This goal is achieved through a perturbation method that overcomes structural obstructions arising from the presence of the $1$-Laplacian and by proving a weak comparison principle for these problems. As a significant application of our main result we establish existence and non-existence theorems for the so-called ``concave-convex'' problem involving the $1$-Laplacian as leading term.

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arXiv2024

Existence and regularity of solutions for the elliptic nonlinear transparent media equation

In this paper we study existence and regularity of solutions to Dirichlet problems as $$ \begin{cases} - {\rm div}\left(|u|^m\frac{D u}{|D u|}\right) = f & \text{in}\;Ω,\\ \newline u=0 & \text{on}\;\partialΩ, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz boundary, $m>0$, and $f$ belongs to the Lorentz space $L^{N,\infty}(Ω)$. In particular, we explore the regularizing effect given by the degenerate coefficient $|u|^m$ in order to get non-trivial and bounded solutions with no smallness assumptions on the size of the data.

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arXiv2024

Singular Elliptic PDEs: an extensive overview

In this survey we provide an overview of nonlinear elliptic homogeneous boundary value problems featuring singular zero-order terms with respect to the unknown variable whose prototype equation is $$ -Δu = {u^{-γ}} \ \text{in}\ Ω$$ where $Ω$ is a bounded subset of $\mathbb{R}^N$ ($N\geq 2$), and $γ>0$. We start by outlining the basic concepts and the mathematical framework needed for setting the problem. Both old and new key existence and uniqueness results are presented, alongside regularity issues depending on the regularity of the data. The presentation aims to be modern, self-contained and consistent. Some examples and open problems are also discussed.

math.AP↗
arXiv2024

Finite energy solutions for nonlinear elliptic equations with competing gradient, singular and $L^1$ terms

In this paper we deal with the following boundary value problem \begin{equation*} \begin{cases} -Δ_{p}u + g(u) | \nabla u|^{p} = h(u)f & \text{in $Ω$,} \newline u\geq 0 & \text{in $Ω$,} \newline u=0 & \text{on $\partial Ω$,} \ \end{cases} \end{equation*} in a domain $Ω\subset \mathbb{R}^{N}$ $(N \geq 2)$, where $1\leq p<N $, $g$ is a positive and continuous function on $[0,\infty)$, and $h$ is a continuous function on $[0,\infty)$ (possibly blowing up at the origin). We show how the presence of regularizing terms $h$ and $g$ allows to prove existence of finite energy solutions for nonnegative data $f$ only belonging to $L^1(Ω)$.

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arXiv2024

Some remarks on optimal insulation with Robin boundary conditions

We consider an optimal insulation problem of a given domain in $\mathbb R^N$. We study a model of heat trasfer determined by convection; this corresponds, before insulation, to a Robin boundary value problem. We deal with a prototype which involves the first eigenvalue of an elliptic differential operator. Such optimization problem, if the convection heat transfer coefficient is sufficiently large and the total amount of insulation is small enough, presents a symmetry breaking.

math.AP↗
arXiv2024

Bounded solutions for non-parametric mean curvature problems with nonlinear terms

In this paper we prove existence of nonnegative bounded solutions for the non-autonomous prescribed mean curvature problem in non-parametric form on an open bounded domain $Ω$ of $\mathbb{R}^N$. The mean curvature, that depends on the location of the solution $u$ itself, is asked to be of the form $f(x)h(u)$, where $f$ is a nonnegative function in $L^{N,\infty}(Ω)$ and $h:\mathbb{R}^+\mapsto \mathbb{R}^+$ is merely continuous and possibly unbounded near zero. As a preparatory tool for our analysis we propose a purely PDE approach to the prescribed mean curvature problem not depending on the solution, i.e. $h\equiv 1$. This part, which has its own independent interest, aims to represent a modern and up-to-date account on the subject. Uniqueness is also handled in presence of a decreasing nonlinearity. The sharpness of the results is highlighted by mean of explicit examples.

math.AP↗
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.

math.AP↗
arXiv2023

The Dirichlet problem for possibly singular elliptic equations with degenerate coercivity

We deal with existence, uniqueness and regularity of nonnegative solutions to a Dirichlet problem for equations as \begin{equation*} \displaystyle -\operatorname{div}\left(\frac{|\nabla u|^{p-2}\nabla u}{(1+u)^{θ(p-1)}}\right) = h(u)f \quad \text{in }Ω, \end{equation*} where $Ω$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge 2$), $p>1$, $θ\ge 0$, $f\geq 0$ belongs to a suitable Lebesgue space and $h$ is a continuous, nonnegative function which may blow up at zero and it is bounded at infinity.

math.AP↗
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.

math.AP↗
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}.

math.AP↗
arXiv2022

On the behaviour of the first eigenvalue of the $p$-Laplacian with Robin boundary conditions as $p$ goes to $1$

In this paper we study the $Γ$-limit, as $p\to 1$, of the functional $$ J_{p}(u)=\frac{\displaystyle\int_Ω|\nabla u|^p + β\int_{ \partial Ω} |u|^p}{\displaystyle \int_Ω|u|^p}, $$ where $Ω$ is a smooth bounded open set in $\mathbb R^{N}$, $p>1$ and $β$ is a real number. Among our results, for $β>-1$, we derive an isoperimetric inequality for \[ Λ(Ω,β)=\inf_{u \in BV(Ω), u\not \equiv 0} \frac{\displaystyle |Du|(Ω) + \min(β,1)\int_{ \partial Ω} |u|}{\displaystyle \int_Ω|u|} \] which is the limit as $p\to 1^{+}$ of $ λ(Ω,p,β)= \displaystyle \min_{u\in W^{1,p}(Ω)} J_{p}(u). $ We show that among all bounded and smooth open sets with given volume, the ball maximizes $Λ(Ω, β)$ when $β\in$ $(-1,0)$ and minimizes $Λ(Ω, β)$ when $β\in[0, \infty)$.

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arXiv2021

On some parabolic equations involving superlinear singular gradient terms

In this paper we prove existence of nonnegative solutions to parabolic Cauchy-Dirichlet problems with superlinear gradient terms which are possibly singular. The model equation is \[ u_t - Δ_pu=g(u)|\nabla u|^q+h(u)f(t,x)\qquad \text{in }(0,T)\timesΩ, \] where $Ω$ is an open bounded subset of $\mathbb{R}^N$ with $N>2$, $0<T<+\infty$, $1<p<N$, and $q<p$ is superlinear. The functions $g,\,h$ are continuous and possibly satisfying $g(0) = +\infty$ and/or $h(0)= +\infty$, with different rates. Finally, $f$ is nonnegative and it belongs to a suitable Lebesgue space. We investigate the relation among the superlinear threshold of $q$, the regularity of the initial datum and the forcing term, and the decay rates of $g,\,h$ at infinity.

math.AP↗
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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