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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.

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 $$ -\Delta u = {u^{-\gamma}} \ \text{in}\ \Omega $$ where $\Omega$ is a bounded subset of $\mathbb{R}^N$ ($N\geq 2$), and $\gamma>0$. 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.

In this paper we study existence and regularity of solutions to Dirichlet problems as $$ \begin{cases} - {\rm div}\left(|u|^m\frac{D u}{|D u|}\right) = f & \text{in}\;\Omega,\\ \newline u=0 & \text{on}\;\partial\Omega, \end{cases} $$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz boundary, $m>0$, and $f$ belongs to the Lorentz space $L^{N,\infty}(\Omega)$. 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.

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 - \Delta_1 u= h(u)f & \text{in } \Omega, \\ \newline u=0 & \text{on } \partial \Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N$ is a bounded open set with Lipschitz boundary, $f \in L^m(\Omega)$ 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 $BV(\Omega)$ 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.

In this paper we deal with the following boundary value problem \begin{equation*} \begin{cases} -\Delta_{p}u + g(u) | \nabla u|^{p} = h(u)f & \text{in $\Omega$,} \newline u\geq 0 & \text{in $\Omega$,} \newline u=0 & \text{on $\partial \Omega$,} \ \end{cases} \end{equation*} in a domain $\Omega \subset \mathbb{R}^{N}$ $(N \geq 2)$, where $1\leq p

In this paper we study existence and uniqueness of solutions to Dirichlet problems as $$ \begin{cases} g(u) -{\rm div}\left(\frac{D u}{\sqrt{1+|D u|^2}}\right) = f & \text{in}\;\Omega,\\ \newline u=0 & \text{on}\;\partial\Omega, \end{cases} $$ where $\Omega$ 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 $L^1(\Omega)$ and with no smallness assumptions on the norm. We also prove a sharp boundedness result for data in $L^{N}(\Omega)$ as well as uniqueness if $g$ is increasing.

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.

We deal with existence and uniqueness of nonnegative solutions to \begin{equation*} \left\{ \begin{array}{l} -\Delta u = f(x) \text{ in }\Omega, \frac{\partial u}{\partial \nu} + \lambda(x) u = \frac{g(x)}{u^\eta} \text{ on } \partial\Omega, \end{array} \right. \end{equation*} where $\eta\ge 0$ and $f,\lambda$ and $g$ are nonnegative integrable functions. The set $\Omega\subset\mathbb{R}^N (N> 2)$ is open and bounded with smooth boundary and $\nu$ 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.

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 -\Delta_p u_p = f & \text{in }\Omega, \displaystyle |\nabla u_p|^{p-2}\nabla u_p\cdot \nu +\lambda |u_p|^{p-2}u_p = g& \text{on } \partial\Omega, \end{array}\right. \end{equation} where $\Omega$ is a bounded domain in $\mathbb R^{N}$ with sufficiently smooth boundary, $\nu$ is its unit outward normal vector and $\Delta_p v$ is the $p$-Laplacian operator with $p>1$. The data $f\in L^{N,\infty}(\Omega)$ (which denotes the Marcinkiewicz space) and $\lambda,g$ are bounded functions defined on $\partial\Omega$ with $\lambda\ge0$. 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}.