arXiv subjects

Daniela Giachetti

Publications and source records attributed to Daniela Giachetti.

8 recordsLinked to original sources

arXiv2026

New definitions of a solution of a one-dimensional elliptic equation with a singular first order divergence term

In this paper, which is a continuation of our recent paper [1], we give two new definitions of a weak solution of the one-dimensional, elliptic, nonlinear singular problem which is formally written as $$ \begin{cases} \displaystyle -\frac{d}{dx}\left(a(x) \frac{d u}{dx}\right) = - \frac{d \phi (u) }{dx}- \frac{d g(x) }{dx}& \text{in}\;(0,L),\\ \newline u(0)=u(L)=0\,, & \end{cases} $$ where $\phi:\mathbb{R}\mapsto\mathbb{R}\cup\{+\infty\}$ is a singular function whose model is given by $\displaystyle\phi(s)=\phi_\gamma (s)=\frac{1}{|s|^{\gamma}}$ with $\gamma>0$. In these two definitions the solutions belong respectively to the space $u\in W_0^{1,1}(0,L)$ with $\phi(u)\in L^1(0,L)$, and to the space $u\in W_0^{1,m}(0,L)$ with $\phi(u)\in L^m(0,L)$ for $m>1$. In these frameworks we state and prove results of non-existence, existence, and non-isolation of the solutions.

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arXiv2024

Unexpected phenomena in a one dimensional elliptic equation with a singular first order divergence term

We study existence of a weak solution for one-dimensional problems as \begin{equation}\label{intro}\tag{1} \begin{cases} \displaystyle -\frac{d}{dx}\left(a(x) \frac{d u}{dx}\right) = - \frac{d \phi (u) }{dx}- \frac{d g(x) }{dx}& \text{in}\;(0,L), u(0)=u(L)=0\,, & \end{cases} \end{equation} where $a$ is a positive bounded function, $g\in L^2(0,L)$, and $\phi:\mathbb{R}\mapsto \mathbb{R}\cup \{+\infty\}$ is continuous as a function with values in $\mathbb{R}\cup \{+\infty\}$. Some relevant qualitative and quantitative facts concerning such problems and their solutions are described. In particular a precise characterization of the behaviour of suitable approximating solution is provided. Of particular (and independent) interest is the study of an associated ODE for which, we prove existence, uniqueness and comparison results. As a consequence of our arguments, a delicate stability result as well a quite unexpected multiplicity result is shown for problems as in \eqref{intro}.

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arXiv2023

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 $\Omega$ 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 $L^{N,\infty}(\Omega)$ 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.

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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 $$ \begin{cases} \displaystyle -\Delta_1 u = g(u)|D u|+h(u)f & \text{in}\;\Omega,\\ u=0 & \text{on}\;\partial\Omega, \end{cases} $$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N$, $f\geq 0$ belongs to $L^N(\Omega)$, 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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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 $$\begin{cases} \displaystyle - \Delta_{1} u = h(u)f & \text{in}\, \Omega,\newline u\geq 0& \text{in}\ \Omega, \newline u=0 & \text{on}\ \partial \Omega, \end{cases} $$ where, $\Delta_{1} $ is the $1$-laplace operator, $\Omega$ 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 $L^{N,\infty}(\Omega)$ 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

Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at $u=0$ in a domain with many small holes

We perform the homogenization of the semilinear elliptic problem \begin{equation*} \begin{cases} u^\varepsilon \geq 0 & \mbox{in} \; \Omega^\varepsilon,\\ \displaystyle - div \,A(x) D u^\varepsilon = F(x,u^\varepsilon) & \mbox{in} \; \Omega^\varepsilon,\\ u^\varepsilon = 0 & \mbox{on} \; \partial \Omega^\varepsilon.\\ \end{cases} \end{equation*} In this problem $F(x,s)$ is a Carath\'eodory function such that $0 \leq F(x,s) \leq h(x)/\Gamma(s)$ a.e. $x\in\Omega$ for every $s > 0$, with $h$ in some $L^r(\Omega)$ and $\Gamma$ a $C^1([0, +\infty[)$ function such that $\Gamma(0) = 0$ and $\Gamma'(s) > 0$ for every $s > 0$. On the other hand the open sets $\Omega^\varepsilon$ are obtained by removing many small holes from a fixed open set $\Omega$ in such a way that a "strange term" $\mu u^0$ appears in the limit equation in the case where the function $F(x,s)$ depends only on $x$. We already treated this problem in the case of a "mild singularity", namely in the case where the function $F(x,s)$ satisfies $0 \leq F(x,s) \leq h(x) (\frac 1s + 1)$. In this case the solution $u^\varepsilon$ to the problem belongs to $H^1_0 (\Omega^\varepsilon)$ and its definition is a "natural" and rather usual one. In the general case where $F(x,s)$ exhibits a "strong singularity" at $u = 0$, which is the purpose of the present paper, the solution $u^\varepsilon$ to the problem only belongs to $H_{\tiny loc}^1(\Omega^\varepsilon)$ but in general does not belongs to $H^1_0 (\Omega^\varepsilon)$ any more, even if $u^\varepsilon$ vanishes on $\partial\Omega^\varepsilon$ in some sense. Therefore we introduced a new notion of solution (in the spirit of the solutions defined by transposition) for problems with a strong singularity. This definition allowed us to obtain existence, stability and uniqueness results.

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arXiv2016

Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at $ u = 0 $

In this paper we consider a semilinear elliptic equation with a strong singularity at $u=0$, namely $ \displaystyle u\geq 0 \mbox{ in } \Omega$, $ \displaystyle - div \,A(x) D u = F(x,u) \mbox{ in} \; \Omega$, $u = 0 \mbox{ on} \; \partial \Omega$, with $F(x,s)$ a Carath\'eodory function such that $$ 0\leq F(x,s)\leq \frac{h(x)}{\Gamma(s)}\,\,\mbox{ a.e. } x\in\Omega,\, \forall s>0, $$ with $h$ in some $L^r(\Omega)$ and $\Gamma$ a $C^1([0,+\infty[)$ function such that $\Gamma(0)=0$ and $\Gamma'(s)>0$ for every $s>0$. We introduce a notion of solution to this problem in the spirit of the solutions defined by transposition. This definition allows us to prove the existence and the stability of this solution, as well as its uniqueness when $F(x,s)$ is nonincreasing in $s$.

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arXiv2015

A semilinear elliptic equation with a mild singularity at $u=0$: existence and homogenization

In this paper we consider semilinear elliptic equations with singularities, whose prototype is the following \begin{equation*} \begin{cases} \displaystyle - div \,A(x) D u = f(x)g(u)+l(x)& \mbox{in} \; \Omega,\\ u = 0 & \mbox{on} \; \partial \Omega,\\ \end{cases} \end{equation*} where $\Omega$ is an open bounded set of $\mathbb{R}^N,\, N\geq 1$, $A\in L^\infty(\Omega)^{N\times N}$ is a coercive matrix, $g:[0,+\infty)\rightarrow [0,+\infty]$ is continuous, and $0\leq g(s)\leq {{1}\over{s^\gamma}}+1$ $\forall s>0$, with $0<\gamma\leq 1$ and $f,l \in L^r(\Omega)$, $r={{2N}\over{N+2}}$ if $N\geq 3$, $r>1$ if $N=2$, $r=1$ if $N=1$, $f(x), l(x)\geq 0$ a.e. $x \in \Omega$. We prove the existence of at least one nonnegative solution and a stability result; moreover uniqueness is also proved if $g(s)$ is nonincreasing or "almost nonincreasing". Finally, we study the homogenization of these equations posed in a sequence of domains $\Omega^\epsilon$ obtained by removing many small holes from a fixed domain $\Omega$.

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