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Luigi Provenzano

Publications and source records attributed to Luigi Provenzano.

47 records · Page 3Linked to original sources
arXiv2017

Eigenvalues of elliptic operators with density

We consider eigenvalue problems for elliptic operators of arbitrary order $2m$ subject to Neumann boundary conditions on bounded domains of the Euclidean $N$-dimensional space. We study the dependence of the eigenvalues upon variations of mass density and in particular we discuss the existence and characterization of upper and lower bounds under both the condition that the total mass is fixed and the condition that the $L^{\frac{N}{2m}}$-norm of the density is fixed. We highlight that the interplay between the order of the operator and the space dimension plays a crucial role in the existence of eigenvalue bounds.

math.SP↗
arXiv2017

On vibrating thin membranes with mass concentrated near the boundary: an asymptotic analysis

We consider the spectral problem \begin{equation*} \left\{\begin{array}{ll} -Δu_{\varepsilon}=λ(\varepsilon)ρ_{\varepsilon}u_{\varepsilon} & {\rm in}\ Ω\\ \frac{\partial u_{\varepsilon}}{\partialν}=0 & {\rm on}\ \partialΩ\end{array}\right. \end{equation*} in a smooth bounded domain $Ω$ of $\mathbb R^2$. The factor $ρ_{\varepsilon}$ which appears in the first equation plays the role of a mass density and it is equal to a constant of order $\varepsilon^{-1}$ in an $\varepsilon$-neighborhood of the boundary and to a constant of order $\varepsilon$ in the rest of $Ω$. We study the asymptotic behavior of the eigenvalues $λ(\varepsilon)$ and the eigenfunctions $u_{\varepsilon}$ as $\varepsilon$ tends to zero. We obtain explicit formulas for the first and second terms of the corresponding asymptotic expansions by exploiting the solutions of certain auxiliary boundary value problems.

math.AP↗
arXiv2016

Weyl-type bounds for Steklov eigenvalues

We present upper and lower bounds for Steklov eigenvalues for domains in $\mathbb{R}^{N+1}$ with $C^2$ boundary compatible with the Weyl asymptotics. In particular, we obtain sharp upper bounds on Riesz-means and the trace of corresponding Steklov heat kernel. The key result is a comparison of Steklov eigenvalues and Laplacian eigenvalues on the boundary of the domain by applying Pohozaev-type identities on an appropriate tubular neigborhood of the boundary and the min-max principle. Asymptotically sharp bounds then follow from bounds for Riesz-means of Laplacian eigenvalues.

math.SP↗
arXiv2016

On the stability of some isoperimetric inequalities for the fundamental tones of free plates

We provide a quantitative version of the isoperimetric inequality for the fundamental tone of a biharmonic Neumann problem. Such an inequality has been recently established by Chasman adapting Weinberger's argument for the corresponding second order problem. Following a scheme introduced by Brasco and Pratelli for the second order case, we prove that a similar quantitative inequality holds also for the biharmonic operator. We also prove the sharpness of both such an inequality and the corresponding one for the biharmonic Steklov problem.

math.SP↗
arXiv2016

A note on the Neumann eigenvalues of the biharmonic operator

We study the dependence of the eigenvalues of the biharmonic operator subject to Neumann boundary conditions on the Poisson's ratio. In particular, we prove that the Neumann eigenvalues are Lipschitz continuous with respect to $σ\in [0,1[$ and that all the Neumann eigenvalues tend to zero as $σ\rightarrow 1^-$. Moreover, we show that the Neumann problem defined by setting $σ=1$ admits a sequence of positive eigenvalues of finite multiplicity which are not limiting points for the Neumann eigenvalues with $σ\in[0,1[$ as $σ\rightarrow 1^-$, and which coincide with the Dirichlet eigenvalues of the biharmonic operator.

math.SP↗
arXiv2016

Neumann to Steklov eigenvalues: asymptotic and monotonicity results

We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of mass concentration at the boundary of a ball. We discuss the asymptotic behavior of the Neumann eigenvalues and find explicit formulas for their derivatives at the limiting problem. We deduce that the Neumann eigenvalues have a monotone behavior in the limit and that Steklov eigenvalues locally minimize the Neumann eigenvalues.

math.SP↗
arXiv2015

On the eigenvalues of a biharmonic Steklov problem

We consider an eigenvalue problem for the biharmonic operator with Steklov-type boundary conditions. We obtain it as a limiting Neumann problem for the biharmonic operator in a process of mass concentration at the boundary. We study the dependence of the spectrum upon the domain. We show analyticity of the symmetric functions of the eigenvalues under isovolumetric perturbations and prove that balls are critical points for such functions under measure constraint. Moreover, we show that the ball is a maximizer for the first positive eigenvalue among those domains with a prescribed fixed measure.

math.SP↗
arXiv2015

A few shape optimization results for a biharmonic Steklov problem

We derive the equation of a free vibrating thin plate whose mass is concentrated at the boundary, namely a Steklov problem for the biharmonic operator. We provide Hadamard-type formulas for the shape derivatives of the corresponding eigenvalues and prove that balls are critical domains under volume constraint. Finally, we prove an isoperimetric inequality for the first positive eigenvalue.

math.OC↗
arXiv2014

Viewing the Steklov eigenvalues of the Laplace operator as critical Neumann eigenvalues

We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of boundary mass concentration. We discuss the asymptotic behavior of the Neumann eigenvalues in a ball and we deduce that the Steklov eigenvalues minimize the Neumann eigenvalues. Moreover, we study the dependence of the eigenvalues of the Steklov problem upon perturbation of the mass density and show that the Steklov eigenvalues violates a maximum principle in spectral optimization problems.

math.SP↗
arXiv2014

A maximum principle in spectral optimization problems for elliptic operators subject to mass density perturbations

We consider eigenvalue problems for general elliptic operators of arbitrary order subject to homogeneous boundary conditions on open subsets of the euclidean N-dimensional space. We prove stability results for the dependence of the eigenvalues upon variation of the mass density and we prove a maximum principle for extremum problems related to mass density perturbations which preserve the total mass.

math.SP↗