arXiv subjects

Benedetta Noris

Publications and source records attributed to Benedetta Noris.

At least 19 recordsLinked to original sources

arXiv2025

Miminization of the first eigenvalue of the Dirichlet Laplacian with a small volume obstacle

We consider the well-known shape optimization problem with spectral cost: minimizing the first eigenvalue of the Dirichlet Laplacian among all subdomains $\Omega$ having prescribed volume and contained in a fixed box $D$; equivalently, we look for the best way to remove a compact set (obstacle) $K\subset\overline{D}$ of Lebesgue measure $|K|=\varepsilon$, $0<\varepsilon<|D|$, in order to minimize the first Dirichlet eigenvalue of the set $\Omega = D \setminus K$. In the small volume regime $\varepsilon\to0$, we prove that the optimal obstacles accumulate, in a suitable sense, to points of $\partial D$ where $|\nabla \phi_0|$ is minimal, where $\phi_0$ denotes the first eigenfunction of the Dirichlet Laplacian on $D$. Moreover, we provide a fairly detailed description of the convergence of the optimal eigenvalues, eigenfunctions and free boundaries. Our results are based on sharp estimates of the optimal eigenvalues, in terms of a suitable notion of relative capacity.

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arXiv2025

An Orlicz space approach to exponential elliptic problems in higher dimensions

We consider semilinear elliptic problems of the form \[ -\Delta u + \lambda u = f(x,u), \quad u\in H^1_0(A), \] where $A\subset\mathbb{R}^N$, $N\geq3$, is either a bounded or unbounded annulus, and $\lambda \geq0$. We study a broad class of nonlinearities $f$ with superlinear growth at infinity, including exponential- and power-type ones. Under suitable assumptions, we establish the existence of a positive nonradial solution via techniques in the spirit of Szulkin's nonsmooth critical point theory, applied within a convex cone in Orlicz spaces. Notably, the Trudinger-Moser inequality fails in the whole Sobolev space $H^1_0(A)$.

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arXiv2024

On Aharonov-Bohm operators with multiple colliding poles of any circulation

This paper deals with quantitative spectral stability for Aharonov-Bohm operators with many colliding poles of whichever circulation. An equivalent formulation of the eigenvalue problem is derived as a system of two equations with real coefficients, coupled through prescribed jumps of the unknowns and their normal derivatives across the segments joining the poles with the collision point. Under the assumption that the sum of all circulations is not integer, the dominant term in the asymptotic expansion for eigenvalues is characterized in terms of the minimum of an energy functional associated with the configuration of poles. Estimates of the order of vanishing of the eigenvalue variation are then deduced from a blow-up analysis, yielding sharp asymptotics in some particular examples.

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arXiv2024

Continuous dependence for p-Laplace equations with varying operators

For the following Neumann problem in a ball $$\begin{cases} -Δ_p u+u^{p-1}=u^{q-1}\quad&\text{in }B,\\ u>0,\,u\text{ radial}\quad&\text{in }B,\\ \frac{\partial u}{\partial ν}=0\quad&\text{on }\partial B, \end{cases}$$ with $1<p<q<\infty$, we prove continuous dependence on $p$, for radially nondecreasing solutions. As a byproduct, we obtain an existence result for nonconstant solutions in the case $p\in(1,2)$ and $q$ larger than an explicit threshold.

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arXiv2023

Quantitative spectral stability for Aharonov-Bohm operators with many coalescing poles

The behavior of simple eigenvalues of Aharonov-Bohm operators with many coalescing poles is discussed. In the case of half-integer circulation, a gauge transformation makes the problem equivalent to an eigenvalue problem for the Laplacian in a domain with straight cracks, laying along the moving directions of poles. For this problem, we obtain an asymptotic expansion for eigenvalues, in which the dominant term is related to the minimum of an energy functional associated with the configuration of poles and defined on a space of functions suitably jumping through the cracks. Concerning configurations with an odd number of poles, an accurate blow-up analysis identifies the exact asymptotic behaviour of eigenvalues and the sign of the variation in some cases. An application to the special case of two poles is also discussed.

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arXiv2023

Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains

We deal with the following semilinear equation in exterior domains \[-\Delta u + u = a(x)|u|^{p-2}u,\qquad u\in H^1_0({A_R}), \] where ${A_R} := \{x\in\mathbb{R}^N:\, |x|>{R}\}$, $N\ge 3$, $R>0$. Assuming that the weight $a$ is positive and satisfies some symmetry and monotonicity properties, we exhibit a positive solution having the same features as $a$, for values of $p>2$ in a suitable range that includes exponents greater than the standard Sobolev critical one. In the special case of radial weight $a$, our existence result ensures multiplicity of nonradial solutions. We also provide an existence result for supercritical $p$ in nonradial exterior domains.

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arXiv2023

A supercritical elliptic equation in the annulus

By a combination of variational and topological techniques in the presence of invariant cones, we detect a new type of positive axially symmetric solutions of the Dirichlet problem for the elliptic equation $$ -Δu + u = a(x)|u|^{p-2}u $$ in an annulus $A \subset \mathbb R^N$ ($N\ge3$). Here $p>2$ is allowed to be supercritical and $a(x)$ is an axially symmetric but possibly nonradial function with additional symmetry and monotonicity properties, which are shared by the solution $u$ we construct. In the case where $a$ equals a positive constant, we detect conditions, only depending on the exponent $p$ and on the inner radius of the annulus, that ensure that the solution is nonradial.

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arXiv2022

Asymptotics for a high-energy solution of a supercritical problem

In this paper we deal with the equation \[-Δ_p u+|u|^{p-2}u=|u|^{q-2}u\] for $1 p$, under Neumann boundary conditions in the unit ball of $\mathbb R^N$. We focus on the three positive, radial, and radially non-decreasing solutions, whose existence for $q$ large is proved in [13]. We detect the limit profile as $q\to\infty$ of the higher energy solution and show that, unlike the minimal energy one, it converges to the constant $1$. The proof requires several tools borrowed from the theory of minimization problems and accurate a priori estimates of the solutions, which are of independent interest.

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arXiv2022

Eigenvalues of the Laplacian with moving mixed boundary conditions: the case of disappearing Neumann region

We deal with eigenvalue problems for the Laplacian with varying mixed boundary conditions, consisting in homogeneous Neumann conditions on a vanishing portion of the boundary and Dirichlet conditions on the complement. By the study of an Almgren type frequency function, we derive upper and lower bounds of the eigenvalue variation and sharp estimates in the case of a strictly star-shaped Neumann region.

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arXiv2021

Multiplicity of solutions on a Nehari set in an invariant cone

For $1<p<2$ and $q$ large, we prove the existence of two positive, nonconstant, radial and radially nondreacreasing solutions of the supercritical equation \[-Δ_p u+u^{p-1}=u^{q-1}\] under Neumann boundary conditions, in the unit ball of $\mathbb R^N$. We use a variational approach in an invariant cone. We distinguish the two solutions upon their energy: one is a ground state inside a Nehari-type subset of the cone, the other is obtained via a mountain pass argument inside the Nehari set. As a byproduct of our proofs, we detect the limit profile of the low energy solution as $q\to\infty$ and show that the constant solution 1 is a local minimum on the Nehari set.

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arXiv2020

Eigenvalues of the Laplacian with moving mixed boundary conditions: the case of disappearing Dirichlet region

In this work we consider the homogeneous Neumann eigenvalue problem for the Laplacian on a bounded Lipschitz domain and a singular perturbation of it, which consists in prescribing zero Dirichlet boundary conditions on a small subset of the boundary. We first describe the sharp asymptotic behaviour of a perturbed eigenvalue, in the case in which it is converging to a simple eigenvalue of the limit Neumann problem. The first term in the asymptotic expansion turns out to depend on the Sobolev capacity of the subset where the perturbed eigenfunction is vanishing. Then we focus on the case of Dirichlet boundary conditions imposed on a subset which is scaling to a point; by a blow-up analysis for the capacitary potentials, we detect the vanishing order of the Sobolev capacity of such shrinking Dirichlet boundary portion.

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arXiv2019

Multiplicity of solutions for the Minkowski-curvature equation via shooting method

In this paper we prove the existence and the multiplicity of radial positive oscillatory solutions for a nonlinear problem governed by the mean curvature operator in the Lorentz-Minkowski space. The problem is set in a ball $B_R$ of $\mathbb R^N$ and is subject to Neumann boundary conditions. The main tool used is the shooting method for ODEs.

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arXiv2019

On simple eigenvalues of the fractional Laplacian under removal of small fractional capacity sets

We consider the eigenvalue problem for the restricted fractional Laplacian in a bounded domain with homogeneous Dirichlet boundary conditions. We introduce the notion of fractional capacity for compact subsets, with the property that the eigenvalues are not affected by the removal of zero fractional capacity sets. Given a simple eigenvalue, we remove from the domain a family of compact sets which are concentrating to a set of zero fractional capacity and we detect the asymptotic expansion of the eigenvalue variation; this expansion depends on the eigenfunction associated to the limit eigenvalue. Finally, we study the case in which the family of compact sets is concentrating to a point.

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arXiv2018

Normalized solutions for Nonlinear Schrödinger systems on bounded domains

We analyze $L^2$-normalized solutions of nonlinear Schrödinger systems of Gross-Pitaevskii type, on bounded domains, with homogeneous Dirichlet boundary conditions. We provide sufficient conditions for the existence of orbitally stable standing waves. Such waves correspond to global minimizers of the associated energy in the $L^2$-subcritical and critical cases, and to local ones in the $L^2$-supercritical case. Notably, our study includes also the Sobolev-critical case.

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arXiv2018

Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions

We analyze existence, multiplicity and oscillatory behavior of positive radial solutions to a class of quasilinear equations governed by the Lorentz-Minkowski mean curvature operator. The equation is set in a ball or an annulus of $\mathbb R^N$, is subject to homogeneous Neumann boundary conditions, and involves a nonlinear term on which we do not impose any growth condition at infinity. The main tool that we use is the shooting method for ODEs.

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arXiv2018

A priori bounds and multiplicity of positive solutions for $p$-Laplacian Neumann problems with sub-critical growth

Let $1 0 \mbox{ in } Ω, \quad \partial_νu = 0 \mbox{ on } \partialΩ. \] We suppose that $f(0)=f(1)=0$ and that $f$ is negative between the two zeros and positive after. In case $Ω$ is a ball, we also require that $f$ grows less than the Sobolev-critical power at infinity. We prove a priori bounds of radial solutions, focusing in particular on solutions which start above 1. As an application, we use the shooting technique to get existence, multiplicity and oscillatory behavior (around 1) of non-constant radial solutions.

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arXiv2017

Radial positive solutions for p-Laplacian supercritical Neumann problems

This paper deals with existence and multiplicity of positive solutions for a quasilinear problem with Neumann boundary conditions, set in a ball. The problem admits at least one constant non-zero solution and it involves a nonlinearity that can be supercritical in the sense of Sobolev embeddings. The main tools used are variational techniques and the shooting method for ODE's. These results are contained in [6,3].

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arXiv2017

Estimates for eigenvalues of Aharonov-Bohm operators with varying poles and non-half-interger circulation

We study the behavior of eigenvalues of a magnetic Aharonov-Bohm operator with non-half-integer circulation and Dirichlet boundary conditions in a planar domain. As the pole is moving in the interior of the domain, we estimate the rate of the eigenvalue variation in terms of the vanishing order of the limit eigenfunction at the limit pole. We also provide an accurate blow-up analysis for scaled eigenfunctions and prove a sharp estimate for their rate of convergence.

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