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Luigi Orsina

Publications and source records attributed to Luigi Orsina.

17 recordsLinked to original sources
arXiv2025

Quasilinear elliptic equations with singular quadratic growth terms

In this paper we deal with positive solutions for singular quasilinear problems whose model is $$ \begin{cases} -Δu + \frac{|\nabla u|^2}{(1-u)^γ}=g & \mbox{in $Ω$,}\newline \hfill u=0 \hfill & \mbox{on $\partialΩ$,} \end{cases} $$ where $Ω$ is a bounded open set of $\mathbb{R}^N$, $g\geq 0 $ is a function in some Lebesgue space, and $γ>0$. We prove both existence and nonexistence of solutions depending on the value of $γ$ and on the size of $g$.

math.AP↗
arXiv2025

Existence and nonexistence of solutions for singular quadratic quasilinear equations

We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient, whose model is $$ \begin{cases} -Δu + \frac{|\nabla u|^2}{u^γ} = f & \mbox{in } Ω,\newline \hfill u=0 \hfill & \mbox{on } \partial Ω, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N $, $γ> 0$ and $f$ is a function which is strictly positive on every compactly contained subset of $Ω$. As a consequence of our main results, we prove that the condition $γ<2$ is necessary and sufficient for the existence of solutions in $H^{1}_{0}(Ω)$ for every sufficiently regular $f$ as above.

math.AP↗
arXiv2021

An Agmon-Allegretto-Piepenbrink principle for Schroedinger operators

We prove that each Borel function $V : Ω\to [-\infty, +\infty]$ defined on an open subset $Ω\subset \mathbb{R}^{N}$ induces a decomposition $Ω= S \cup \bigcup_{i} D_{i}$ such that every function in $W^{1,2}_{0}(Ω) \cap L^{2}(Ω; V^{+} dx)$ is zero almost everywhere on $S$ and existence of nonnegative supersolutions of $-Δ+ V$ on each component $D_{i}$ yields nonnegativity of the associated quadratic form $\int_{D_{i}} (|\nabla ξ|^2+Vξ^2)$.

math.AP↗
arXiv2018

On the nonexistence of Green's function and failure of the strong maximum principle

Given any Borel function $V : Ω\to [0, +\infty]$ on a smooth bounded domain $Ω\subset \mathbb{R}^{N}$, we establish that the strong maximum principle for the Schrödinger operator $-Δ+ V$ in $Ω$ holds in each Sobolev-connected component of $Ω\setminus Z$, where $Z \subset Ω$ is the set of points which cannot carry a Green's function for $- Δ+ V$. More generally, we show that the equation $- Δu + V u = μ$ has a distributional solution in $W_{0}^{1, 1}(Ω)$ for a nonnegative finite Borel measure $μ$ if and only if $μ(Z) = 0$.

math.AP↗
arXiv2018

Hopf potentials for the Schrödinger operator

We establish the Hopf boundary point lemma for the Schrödinger operator $-Δ+ V$ involving potentials $V$ that merely belong to the space $L^{1}_{loc}(Ω)$. More precisely, we prove that among all supersolutions $u$ of $-Δ+ V$ which vanish on the boundary $\partialΩ$ and are such that $V u \in L^{1}(Ω)$, if there exists one supersolution which satisfies $\partial u/\partial n < 0$ almost everywhere on $\partialΩ$ with respect to the outward unit vector $n$, then such a property holds for every nontrivial supersolution in the same class. We rely on the existence of nontrivial solutions of the nonhomogeneous Dirichlet problem with boundary datum in $L^{\infty}(\partialΩ)$.

math.AP↗
arXiv2017

Flat solutions of the 1-Laplacian equation

For every $f \in L^N(Ω)$ defined in an open bounded subset $Ω$ of $\mathbb{R}^N$, we prove that a solution $u \in W_0^{1, 1}(Ω)$ of the $1$-Laplacian equation ${-}\mathrm{div}{(\frac{\nabla u}{|\nabla u|})} = f$ in $Ω$ satisfies $\nabla u = 0$ on a set of positive Lebesgue measure. The same property holds if $f \not\in L^N(Ω)$ 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.

math.AP↗
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 $γ>0$.

math.AP↗
arXiv2016

The role of interplay between coefficients in the $G$-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 $|f (x)| \leq Q \, a (x)$.

math.AP↗
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.

math.AP↗
arXiv2014

Strong maximum principle for Schrödinger operators with singular potential

We prove that for every $p > 1$ and for every potential $V \in L^p$, any nonnegative function satisfying $-Δu + V u \ge 0$ 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 Bénilan 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 Schrödinger operator with diffuse measure data.

math.AP↗
arXiv2014

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

We compute for reflection groups of type $A,B,D,F_4,H_3$ 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.

math.RT↗
arXiv2011

A nonlinear degenerate elliptic problem with W^{1,1}_0 solutions

We study a nonlinear equation with an elliptic operator having degenerate coercivity. We prove the existence of a unique W^{1,1}_0 distributional solution under suitable summability assumptions on the source in Lebesgue spaces. Moreover, we prove that our problem has no solution if the source is a Radon measure concentrated on a set of zero harmonic capacity.

math.AP↗
arXiv2000

ad-nilpotent $\frak b$-ideals in sl(n) having a fixed class of nilpotence: combinatorics and enumeration

We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of $sl(n+1,\Bbb C)$. We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence. We study the relationships between these results and the combinatorics of Dyck paths, based upon a remarkable bijection between ad-nilpotent ideals and Dyck paths. Finally, we propose a (q,t)-analogue of the Catalan number $C_n$. These (q,t)-Catalan numbers count on the one hand ad-nilpotent ideals with respect to dimension and class of nilpotence, and on the other hand admit interpretations in terms of natural statistics on Dyck paths.

math.RA↗