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Sunra Mosconi

Publications and source records attributed to Sunra Mosconi.

At least 19 recordsLinked to original sources
arXiv2026

On boundary regularity for the fractional p-Laplacian with unbounded reactions

We consider an elliptic equation driven by the $s$-fractional $p$-Laplacian, set in a smooth bounded domain $\Omega\subset\mathbb{R}^N$ with homogeneous nonlocal Dirichlet conditions and a reaction $f$ lying in $L^q(\Omega)$ for some $q\ge 1$. We prove that the unique solution $u$ is $\alpha$-H\"older continuous up to the boundary, for any $\alpha$ below $p'(s-N/pq)$ if $N/ps N/s$. Also, we prove that if $q>N/s$ then $u/{\rm d}_\Omega^s$ admits a H\"older continuous extension to the closure of $\Omega$, where ${\rm d}_\Omega$ denotes the distance from the boundary. Our results are almost optimal and extend previous regularity theorems known in the linear case.

math.AP↗
arXiv2024

Power law convergence and concavity for the Logarithmic Schr\"odinger equation

We study concavity properties of positive solutions to the Logarithmic Schr\"odinger equation $-\Delta u=u\, \log u^2$ in a general convex domain with Dirichlet conditions. To this aim, we analyse the auxiliary Lane-Emden problems $-\Delta u = \sigma\, (u^q-u)$ and build, for any $\sigma>0$ and $q>1$, solutions $u_q$ such that $u_q^{(1-q)/2}$ is convex. By choosing $\sigma_q=2/(q-1)$ and letting $q \to 1^+$ we eventually construct a solution $u$ of the Logarithmic Schr\"odinger equation such that $\log u$ is concave. This seems to be one of the few attempts at studying concavity properties for superlinear, sign changing sources. To get the result, we both make inspections on the constant rank theorem and develop Liouville theorems on convex epigraphs, which might be useful in other frameworks.

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arXiv2024

Fine boundary regularity for the singular fractional p-Laplacian

We study the boundary weighted regularity of weak solutions $u$ to a $s$-fractional $p$-Laplacian equation in a bounded smooth domain $\Omega$ with bounded reaction and nonlocal Dirichlet type boundary condition, in the singular case $p\in(1,2)$ and with $s\in(0,1)$. We prove that $u/{\rm d}_\Omega^s$ has a $\alpha$-H\"older continuous extension to the closure of $\Omega$, ${\rm d}_\Omega(x)$ meaning the distance of $x$ from the complement of $\Omega$. This result corresponds to that of ref. [28] for the degenerate case $p\ge 2$.

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arXiv2023

Concave solutions to Finsler $p$-Laplace type equations

We prove concavity properties for solutions to anisotropic quasi-linear equations, extending previous results known in the Euclidean case. We focus the attention on nonsmooth anisotropies and in particular we also allow the functions describing the anisotropies to be not even.

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arXiv2023

Lipschitz regularity for solutions of a general class of elliptic equations

We prove local Lipschitz regularity for local minimiser of \[ W^{1,1}(\Omega)\ni v\mapsto \int_\Omega F(Dv)\, dx \] where $\Omega\subseteq {\mathbb R}^N$, $N\ge 2$ and $F:{\mathbb R}^N\to {\mathbb R}$ is a quasiuniformly convex integrand in the sense of Kovalev and Maldonado, i.e. a convex $C^1$-function such that the ratio between the maximum and minimum eigenvalues of $D^2F$ is essentially bounded. This class of integrands inculdes the standard singular/degenerate functions $F(z)=|z|^p$ for any $p>1$ and arises naturally as the closure, with respect to a natural convergence, of the strongly elliptic integrands of the Calculus of Variations.

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arXiv2022

A non-smooth Brezis-Oswald uniqueness result

We classify the non-negative critical points in $W^{1,p}_0(\Omega)$ of \[ J(v)=\int_\Omega H(Dv)-F(x, v)\, dx \] where $H$ is convex and positively $p$-homogeneous, while $t\mapsto \partial_tF(x, t)/t^{p-1}$ is non-increasing. Since $H$ may not be differentiable and $F$ has a one-sided growth condition, $J$ is only l.s.c. on $W^{1,p}_0(\Omega)$. We employ a weak notion of critical point for non-smooth functionals, derive sufficient regularity of the latter without an Euler-Lagrange equation available and focus on the uniqueness part of the results in \cite{BO}, through a non-smooth Picone inequality.

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arXiv2022

Concavity properties for solutions to $p$-Laplace equations with concave nonlinearities

We obtain new concavity results, up to a suitable transformation, for a class of quasi-linear equations in a convex domain involving the $p$-Laplace operator and a general nonlinearity satisfying concavity type assumptions. This provides an extension of results previously known in the literature only for the torsion and the eigenfunction equations. In the semilinear case $p = 2$ the results are already new since they include new admissible nonlinearities.

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arXiv2021

A general notion of uniform ellipticity and the regularity of the stress field for elliptic equations in divergence form

For solutions of ${\rm div}\,(DF(Du))=f$ we show that the quasiconformality of $z\mapsto DF(z)$ is the key property leading to the Sobolev regularity of the stress field $DF(Du)$, in relation with the summability of $f$. This class of nonlinearities encodes in a general way the notion of uniform ellipticity and encompasses all known instances where the stress field is known to be Sobolev regular. We provide examples showing the optimality of this assumption and present three applications: the study of the strong locality of the operator ${\rm div}\,(DF(Du))$, a nonlinear Cordes condition for equations in divergence form, and some partial results on the $C^{p'}$-conjecture.

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arXiv2021

Parabolic Harnack estimates for anisotropic slow diffusion

We prove a Harnack inequality for positive solutions of a parabolic equation with slow anisotropic spatial diffusion. After identifying its natural scalings, we reduce the problem to a Fokker-Planck equation and construct a self-similar Barenblatt solution. We exploit translation invariance to obtain positivity near the origin via a self-iteration method and deduce a sharp anisotropic expansion of positivity. This eventually yields a scale invariant Harnack inequality in an anisotropic geometry dictated by the speed of the diffusion coefficients. As a corollary, we infer Hölder continuity, an elliptic Harnack inequality and a Liouville theorem.

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arXiv2021

On the logistic equation for the fractional p-Laplacian

We consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, with a logistic type reaction depending on a positive parameter. In the subdiffusive and equidiffusive cases, we prove existence and uniqueness of the positive solution when the parameter lies in convenient intervals. In the superdiffusive case, we establish a bifurcation result. A new strong comparison result, of independent interest, plays a crucial role in the proof of such bifurcation result.

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arXiv2019

Sobolev versus Hölder minimizers for the degenerate fractional $p$-Laplacian

We consider a nonlinear pseudo-differential equation driven by the fractional $p$-Laplacian $(-Δ)^s_p$ with $s\in(0,1)$ and $p\ge 2$ (degenerate case), under Dirichlet type conditions in a smooth domain $Ω$. We prove that local minimizers of the associated energy functional in the fractional Sobolev space $W^{s,p}_0(Ω)$ and in the weighted Hölder space $C^0_s(\overlineΩ)$, respectively, do coincide.

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arXiv2018

Fine boundary regularity for the degenerate fractional $p$-Laplacian

We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $p\ge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $Ω$. By means of barriers, a nonlocal superposition principle, and the comparison principle, we prove that any weak solution $u$ of such equation exhibits a weighted Hölder regularity up to the boundary, that is, $u/d^s\in C^α(\overlineΩ)$ for some $α\in(0,1)$, $d$ being the distance from the boundary.

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arXiv2018

Nonlocal problems with critical Hardy nonlinearity

By means of variational methods we establish existence and multiplicity of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian and a combined Sobolev and Hardy nonlinearity at subcritical and critical growth.

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arXiv2017

Asymptotic for optimizers of the fractional Hardy-Sobolev inequality

We consider the optimizers $u$ in the Hardy-Sobolev inequality for the space $\dot{W}^{s,p}({\mathbb R}^N)$ with order of differentiability $s\in ]0,1[$. After proving existence through concentration-compactness, we derive the pointwise asymptotic $u(x)\simeq |x|^{-\frac{N-ps}{p-1}}$ for large $|x|$ and the summability estimate $u\in \dot{W}^{s,γ}({\mathbb R}^N)$ for all $γ>\frac{N(p-1)}{N-s}$. These estimates are optimal in the limit $s\to 1^-$, in which case optimizers are explicitly known.

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arXiv2017

Optimal elliptic regularity: a comparison between local and nonlocal equations

Given $L\geq 1$, we discuss the problem of determining the highest $α=α(L)$ such that any solution to a homogeneous elliptic equation in divergence form with ellipticity ratio bounded by $L$ is in $C^α_{\rm loc}$. This problem can be formulated both in the classical and non-local framework. In the classical case it is known that $α(L)\gtrsim {\rm exp}(-CL^β)$, for some $C, β\geq 1$ depending on the dimension $N\geq 3$. We show that in the non-local case, $α(L)\gtrsim L^{-1-δ}$ for all $δ>0$.

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arXiv2017

On a $(p,q)$-Laplacian problem with parametric concave term and asymmetric perturbation

A Dirichlet problem driven by the $(p,q)$-Laplace operator and an asymmetric concave reaction with positive parameter is investigated. Four nontrivial smooth solutions (two positive, one negative, and the remaining nodal) are obtained once the parameter turns out to be sufficiently small. Under a oddness condition near the origin for the perturbation, a whole sequence of sign-changing solutions, which converges to zero, is produced.

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