arXiv subjects

Francesco Petitta

Publications and source records attributed to Francesco Petitta.

At least 37 records · Page 2Linked to original sources

arXiv2024

Existence, non-existence and degeneracy of limit solutions to $p-$Laplace problems involving Hardy potentials as $p\to1^+$

In this paper we analyze the asymptotic behaviour as $p\to 1^+$ of solutions $u_p$ to $$ \left\{ \begin{array}{rclr} -Δ_p u_p&=&\fracλ{|x|^p}|u_p|^{p-2}u_p+f&\quad \mbox{ in } Ω,\\ u_p&=&0 &\quad \mbox{ on }\partialΩ, \end{array}\right. $$ where $Ω$ is a bounded open subset of $\mathbb{R}^N$ with Lipschitz boundary, $λ\in\mathbb{R}^+$, and $f$ is a nonnegative datum in $L^{N,\infty}(Ω)$. Under sharp smallness assumptions on the data $λ$ and $f$ we prove that $u_p$ converges to a suitable solution to the homogeneous Dirichlet problem $$\left\{ \begin{array}{rclr}- Δ_{1} u &=& \fracλ{|x|}{\rm Sgn}(u)+f & \text{in}\, Ω,\\ u&=&0 & \text{on}\ \partial Ω,\end{array}\right. $$ where $Δ_{1} u ={\rm div}\left(\frac{D u}{|Du|}\right)$ is the $1$-Laplace operator. The main assumptions are further discussed through explicit examples in order to show their optimality.

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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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arXiv2021

Nonlinear diffusion in transparent media

We consider a prototypical nonlinear parabolic equation whose flux has three distinguished features: it is nonlinear with respect to both the unknown and its gradient, it is homogeneous, and it depends only on the direction of the gradient. For such equation, we obtain existence and uniqueness of entropy solutions to the Dirichlet problem, the homogeneous Neumann problem, and the Cauchy problem. Qualitative properties of solutions, such as finite speed of propagation and the occurrence of waiting-time phenomena, with sharp bounds, are shown. We also discuss the formation of jump discontinuities both at the boundary of the solutions' support and in the bulk.

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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

1-Laplacian type problems with strongly singular nonlinearities and gradient terms

We show optimal existence, nonexistence and regularity results for nonnegative solutions to Dirichlet problems as $$ \begin{cases} \displaystyle -Δ_1 u = g(u)|D u|+h(u)f & \text{in}\;Ω,\\ u=0 & \text{on}\;\partialΩ, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N$, $f\geq 0$ belongs to $L^N(Ω)$, and $g$ and $h$ are continuous functions that may blow up at zero. As a noteworthy fact we show how a non-trivial interaction mechanism between the two nonlinearities $g$ and $h$ produces remarkable regularizing effects on the solutions. The sharpness of our main results is discussed through the use of appropriate explicit examples.

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arXiv2019

A nonlinear parabolic problem with singular terms and nonregular data

We study existence of nonnegative solutions to a nonlinear parabolic boundary value problem with a general singular lower order term and a nonnegative measure as nonhomogeneous datum, of the form $$ \begin{cases} \displaystyle u_t - Δ_p u = h(u)f+μ& \text{in}\ Ω\times (0,T),\\ u=0 &\text{on}\ \partialΩ\times (0,T),\\ u=u_0 &\text{in}\ Ω\times \{0\}, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge2$), $u_0$ is a nonnegative integrable function, $Δ_p$ is the $p$-laplace operator, $μ$ is a nonnegative bounded Radon measure on $Ω\times (0,T)$ and $f$ is a nonnegative function of $L^1(Ω\times (0,T))$. The term $h$ is a positive continuous function possibly blowing up at the origin. Furthermore, we show uniqueness of finite energy solutions in presence of a nonincreasing $h$.

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arXiv2018

The Dirichlet problem for singular elliptic equations with general nonlinearities

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{cases} \displaystyle - Δ_{1} u = h(u)f & \text{in}\, Ω,\newline u\geq 0& \text{in}\ Ω, \newline u=0 & \text{on}\ \partial Ω, \end{cases} $$ where, $Δ_{1} $ is the $1$-laplace operator, $Ω$ is a bounded open subset of $\mathbb{R}^N$ with Lipschitz boundary, $h(s)$ is a continuous function which may become singular at $s=0^{+}$, and $f$ is a nonnegative datum in $L^{N,\infty}(Ω)$ with suitable small norm. Uniqueness of solutions is also shown provided $h$ is decreasing and $f>0$. As a by-product of our method a general theory for the same problem involving the $p$-laplacian as principal part, which is missed in the literature, is established. The main assumptions we use are also further discussed in order to show their optimality.

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arXiv2017

Nonlinear diffusion in transparent media: the resolvent equation

We consider the partial differential equation $$ u-f={\rm div}\left(u^m\frac{\nabla u}{|\nabla u|}\right) $$ with $f$ nonnegative and bounded and $m\in\mathbb{R}$. We prove existence and uniqueness of solutions for both the Dirichlet problem (with bounded and nonnegative {boundary datum}) and the homogeneous Neumann problem. Solutions, which a priori belong to a space of truncated bounded variation functions, are shown to have zero jump part with respect to the ${\mathcal H}^{N-1}$ Haussdorff measure. Results and proofs extend to more general nonlinearities.

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arXiv2017

A Lazer-McKenna type problem with measures

In this paper we are concerned with a general singular Dirichlet boundary value problem whose model is the following $$ \begin{cases} -Δu = \fracμ{u^γ} & \text{in}\ Ω, u=0 &\text{on}\ \partialΩ, u>0 &\text{on}\ Ω\,. \end{cases} $$ Here $μ$ is a nonnegative bounded Radon measure on a bounded open set $Ω\subset\mathbb{R}^N$, and $γ>0$.

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arXiv2017

On the notion of renormalized solution to nonlinear parabolic equations with general measure data

Here we introduce a new notion of renormalized solution to nonlinear parabolic problems with general measure data whose model is $$ \begin{cases} u_t-Δ_{p} u =μ& \text{in}\ (0,T)\timesΩ, u=u_0 & \text{on}\ \{0\} \times Ω, u=0 &\text{on}\ (0,T)\times\partialΩ, \end{cases} $$ for any, possibly singular, nonnegative bounded measure $μ$. We prove existence of such a solutions and we discuss their main properties.

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arXiv2017

Large solutions for the elliptic $1$-Laplacian with absorption

In this paper we give a general condition on the absorption term of the 1-Laplace elliptic equation for the existence of suitable large solutions. This condition can be considered as the correspondent Keller-Osserman condition for the $p$-Laplacian, in the case $p=1$. We also provide conditions that guarantee uniqueness for solutions to such problems.

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arXiv2017

On singular elliptic equations with measure sources

We prove existence of solutions for a class of singular elliptic problems with a general measure as source term whose model is $$\begin{cases} -Δu = \frac{f(x)}{u^γ} +μ& \text{in}\ Ω, u=0 &\text{on}\ \partialΩ, u>0 &\text{on}\ Ω, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N$. Here $γ> 0$, $f$ is a nonnegative function on $Ω$, and $μ$ is a nonnegative bounded Radon measure on $Ω$.

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arXiv2017

Some remarks on the duality method for Integro-Differential equations with measure data

We deal with existence, uniqueness, and regularity for solutions of the boundary value problem $$ \begin{cases} {\mathcal L}^s u = μ&\quad \text{in $Ω$}, u(x)=0 \quad &\text{on} \ \ \mathbb{R}^N\backslashΩ, \end{cases} $$ where $Ω$ is a bounded domain of $\mathbb{R}^N$, $μ$ is a bounded radon measure on $Ω$, and ${\mathcal L}^s$ is a nonlocal operator of fractional order $s$ whose kernel $K$ is comparable with the one of the factional laplacian.

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arXiv2017

Optimal waiting time bounds for some flux-saturated diffusion equations

We consider the Cauchy problem for two prototypes of flux-saturated diffusion equations. In arbitrary space dimension, we give an optimal condition on the growth of the initial datum which discriminates between occurrence or nonoccurrence of a waiting time phenomenon. We also prove optimal upper bounds on the waiting time. Our argument is based on the introduction of suitable families of subsolutions and on a comparison result for a general class of flux-saturated diffusion equations.

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arXiv2016

Finite and Infinite energy solutions of singular elliptic problems: Existence and Uniqueness

We establish existence and uniqueness of solution for the homogeneous Dirichlet problem associated to a fairly general class of elliptic equations modeled by $$ -Δu= h(u){f} \ \ \text{in}\,\ Ω, $$ where $f$ is an irregular datum, possibly a measure, and $h$ is a continuous function that may blow up at zero. We also provide regularity results on both the solution and the lower order term depending on the regularity of the data, and we discuss their optimality.

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arXiv2015

Existence of solutions for degenerate parabolic equations with singular terms

In this paper we deal with parabolic problems whose simplest model is $$ \begin{cases} u'- Δ_{p} u + B\frac{|\nabla u|^p}{u} = 0 & \text{in} (0,T) \times Ω,\newline u(0,x)= u_0 (x) &\text{in}\ Ω, \newline u(t,x)=0 &\text{on}\ (0,T) \times \partialΩ, \end{cases} $$ where $T>0$, $N\geq 2$, $p>1$, $B > 0$, and $u_{0}$ is a positive function in $L^{\infty}(Ω)$ bounded away from zero.

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arXiv2014

Local estimates for parabolic equations with nonlinear gradient terms

In this paper we deal with local estimates for parabolic problems in $\mathbb{R}^N$ with absorbing first order terms, whose model is $\{ {l} u_t- Δu +u |\nabla u|^q = f(t,x) \quad &{in}\, (0,T) \times \mathbb{R}^N\,, \\[1.5 ex] u(0,x)= u_0 (x) &box{in}\, \mathbb{R}^N.$ where $T>0$, $N\geq 2$, $1<q\leq 2$, $f(t,x)\in L^1( 0,T; L^1_{\rm loc} (\mathbb{R}^N))$ and $u_0\in L^1_{\rm loc} (\mathbb{R}^N)$.

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arXiv2014

A nonexistence result for nonlinear parabolic equations with singular measures as data

In this paper we prove a nonexistence result for nonlinear parabolic problems with zero lower order term whose model is $$ \begin{cases} u_{t}- Δ_p u+|u|^{q-1}u=λ& \text{in}\ (0,T)\timesΩ u(0,x)=0 & \text{in}\ Ω,\\ u(t,x)=0 & \text{on}\ (0,T)\times\partialΩ, \end{cases} $$ where $Δ_p ={\rm div }(|\nabla u|^{p-2}\nabla u)$ is the usual $p$-laplace operator, $λ$ is measure concentrated on a set of zero parabolic $r$-capacity ($1<p<r$), and $q$ is large enough.

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