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 $$\begin{cases} -\Delta u = \frac{\mu}{u^{\gamma}} & \text{in}\ \Omega, u=0 &\text{on}\ \partial\Omega, u>0 &\text{on}\ \Omega\,. \end{cases}$$ Here $\mu$ is a nonnegative bounded Radon measure on a bounded open set $\Omega\subset\mathbb{R}^N$, and $\gamma>0$

Flat solutions of the 1-Laplacian equation

For every $f \in L^N(\Omega)$ defined in an open bounded subset $\Omega$ of $\mathbb{R}^N$, we prove that a solution $u \in W_0^{1, 1}(\Omega)$ of the $1$-Laplacian equation ${-}\mathrm{div}{(\frac{\nabla u}{|\nabla u|})} = f$ in $\Omega$ satisfies $\nabla u = 0$ on a set of positive Lebesgue measure. The same property holds if $f \not\in L^N(\Omega)$ has small norm in the Marcinkiewicz space of weak-$L^{N}$ functions or if $u$ is a BV minimizer of the associated energy functional. The proofs rely on Stampacchia's truncation method.Comment: Dedicated to Jean Mawhin. Revised and extended version of a note written by the authors in 201

Strong maximum principle for Schr\"odinger operators with singular potential

We prove that for every $p > 1$ and for every potential $V \in L^p$, any nonnegative function satisfying $-\Delta u + V u \ge 0$ in an open connected set of $\mathbb{R}^N$ is either identically zero or its level set $\{u = 0\}$ has zero $W^{2, p}$ capacity. This gives an affirmative answer to an open problem of B\'enilan and Brezis concerning a bridge between Serrin-Stampacchia's strong maximum principle for $p > \frac{N}{2}$ and Ancona's strong maximum principle for $p = 1$. The proof is based on the construction of suitable test functions depending on the level set $\{u = 0\}$ and on the existence of solutions of the Dirichlet problem for the Schr\"odinger operator with diffuse measure data.Comment: 21 page

A semilinear problem with a W^{1,1}_0 solution

We study a degenerate elliptic equation, proving the existence of a W^{1,1}_0 distributional solution

Existence of solutions for degenerate parabolic equations with singular terms

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

The role of interplay between coefficients in the GG-convergence of some elliptic equations

We study the behavior of the solutions $u$ of the linear Dirichlet problems $- \mathrm{div} (M(x) \nabla u) + a(x) u = f(x)$ with respect to perturbations of the matrix $M(x)$ (with respect to the $G$-convergence) and with respect to perturbations of the nonnegative coefficient $a(x)$ and of the right hand side $f(x)$ satisfying the condition $|f (x)| \leq Q \, a (x)$

Renormalized solutions of elliptic equations with general measure data

We study existence and (in some case) uniqueness for elliptic equations with measure data

The maximum cardinality of minimal inversion complete sets in finite reflection groups

We compute for reflection groups of type A,B,D,F4,H3 and for dihedral groups a statistic counting the maximal cardinality of a set of elements in the group whose generalized inversions yield the full set of inversions and which are minimal with respect to this property. We also provide lower bounds for the E types that we conjecture to be the exact value of our statistic