arXiv subjects
Publications and source records attributed to C. Trombetti.
By means of a suitable weighted rearrangement, we obtain various apriori bounds for the solutions to a Robin problem. Among other things, we derive a family of Faber-Krahn type inequalities.
Artificial Neuronal Networks are models widely used for many scientific tasks. One of the well-known field of application is the approximation of high-dimensional problems via Deep Learning. In the present paper we investigate the Deep Learning techniques applied to Shape Functionals, and we start from the so--called Torsional Rigidity. Our aim is to feed the Neuronal Network with digital approximations of the planar domains where the Torsion problem (a partial differential equation problem) is defined, and look for a prediction of the value of Torsion. Dealing with images, our choice fell on Convolutional Neural Network (CNN), and we train such a network using reference solutions obtained via Finite Element Method. Then, we tested the network against some well-known properties involving the Torsion as well as an old standing conjecture. In all cases, good approximation properties and accuracies occurred.
We show, using symmetrization techniques, that it is possible to prove a comparison principle (we are mainly focused on $L^1$ comparison) between solutions to an elliptic partial differential equation on a smooth bounded set $Ω$ with a rather general boundary condition, and solutions to a suitable related problem defined on a ball having the same volume as $Ω$. This includes for instance mixed problems where Dirichlet boundary conditions are prescribed on part of the boundary, while Robin boundary conditions are prescribed on its complement.
Comparison results of Talenti type for Elliptic Problems with Dirichlet boundary conditions have been widely investigated in the last decades. In this paper, we deal with Robin boundary conditions. Surprisingly, contrary to the Dirichlet case, Robin boundary conditions make the comparison sensitive to the dimension, and while the planar case seems to be completely settled, in higher dimensions some open problems are yet unsolved.
Let $Ω$ be an open convex set in ${\mathbb R}^m$ with finite width, and let $v_Ω$ be the torsion function for $Ω$, i.e. the solution of $-Δv=1, v\in H_0^1(Ω)$. An upper bound is obtained for the product of $\Vert v_Ω\Vert_{L^{\infty}(Ω)}λ(Ω)$, where $λ(Ω)$ is the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(Ω)$. The upper bound is sharp in the limit of a thinning sequence of convex sets. For planar rhombi and isosceles triangles with area $1$, it is shown that $\Vert v_Ω\Vert_{L^{1}(Ω)}λ(Ω)\ge \frac{π^2}{24}$, and that this bound is sharp.
In this paper we prove that the ball maximizes the first eigenvalue of the Robin Laplacian operator with negative boundary parameter, among all convex sets of \mathbb{R}^n with prescribed perimeter. The key of the proof is a dearrangement procedure of the first eigenfunction of the ball on the level sets of the distance function to the boundary of the convex set, which controls the boundary and the volume energies of the Rayleigh quotient.
In this survey we consider the classical overdetermined problem which was studied by Serrin in 1971. The original proof relies on Alexandrov's moving plane method, maximum principles, and a refinement of Hopf's boundary point Lemma. Since then other approaches to the same problem have been devised. Among them we consider the one due to Weinberger which strikes for the elementary arguments used and became very popular. Then we discuss also a duality approach involving harmonic functions, a shape derivative approach and a purely integral approach, all of them not relying on maximum principle. For each one we consider pros and cons as well as some generalizations.
In this short note we consider an unconventional overdetermined problem for the torsion function: let $n\geq 2$ and $Ω$ be a bounded open set in $\mathbb{R}^n$ whose torsion function $u$ (i.e. the solution to $Δu=-1$ in $Ω$, vanishing on $\partialΩ$) satisfies the following property: $\sqrt{M-u(x)}$ is convex, where $M=\max\{u(x)\,:\,x\in\overlineΩ\}$. Then $Ω$ is an ellipsoid.
We prove a quantitative Faber-Krahn inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions. The asymmetry term involves the square power of the Fraenkel asymmetry, multiplied by a constant depending on the Robin parameter, the dimension of the space and the measure of the set.
Let $Ω$ be an open set in Euclidean space with finite Lebesgue measure $|Ω|$. We obtain some properties of the set function $F:Ω\mapsto \R^+$ defined by $$ F(Ω)=\frac{T(Ω)λ_1(Ω)}{|Ω|} ,$$ where $T(Ω)$ and $λ_1(Ω)$ are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical Pólya bound $F(Ω)\le 1,$ and show that $$F(Ω)\le 1- ν_m T(Ω)|Ω|^{-1-\frac2m},$$ where $ν_m$ depends only on $m$. For any $m=2,3,\dots$ and $ε\in (0,1)$ we construct an open set $Ω_ε\subset \R^m$ such that $F(Ω_ε)\ge 1-ε$.
In this paper we prove a sharp inequality of Szegő-Weinberger type for the first nontrivial eigenvalue of the $p-$Laplacian with Neumann boundary conditions. This applies to convex sets with given diameter. Some variants, extensions and limit cases are investigated as well.
In this paper, generalizing to the non smooth case already existing results, we prove that, for any convex planar set $Ω$, the first non-trivial Neumann eigenvalue $μ_1(Ω)$ of the Hermite operator is greater than or equal to 1. Furthermore, and this is our main result, under some additional assumptions on $Ω$, we show that $μ_1(Ω)=1$ if and only if $Ω$ is any strip. The study of the equality case requires, among other things, an asymptotic analysis of the eigenvalues of the Hermite operator in thin domains.
The first nontrivial eigenfunction of the Neumann eigenvalue problem for the $p$-Laplacian, suitable normalized, converges as $p$ goes to $\infty$ to a viscosity solution of an eigenvalue problem for the $\infty$-Laplacian. We show among other things that the limit of the eigenvalue, at least for convex sets, is in fact the first nonzero eigenvalue of the limiting problem. We then derive a number of consequences, which are nonlinear analogues of well-known inequalities for the linear (2-)Laplacian.
In this paper we prove a sharp lower bound for the first nontrivial Neumann eigenvalue $μ_1(Ω)$ for the $p$-Laplace operator in a Lipschitz, bounded domain $Ω$ in $\R^n$. Our estimate does not require any convexity assumption on $Ω$ and it involves the best isoperimetric constant relative to $Ω$.
This paper deals with the Neumann eigenvalue problem for the Hermite operator defined in a convex, possibly unbounded, planar domain $Ω$, having one axis of symmetry passing through the origin. We prove a sharp lower bound for the first eigenvalue $μ_1^{odd}(Ω)$ with an associated eigenfunction odd with respect to the axis of symmetry. Such an estimate involves the first eigenvalue of the corresponding one-dimensional problem. As an immediate consequence, in the class of domains for which $μ_1(Ω)=μ_1^{odd}(Ω)$, we get an explicit lower bound for the difference between $μ(Ω)$ and the first Neumann eigenvalue of any strip.
We prove a Payne-Weinberger type inequality for the $p$-Laplacian Neumann eigenvalues ($p\ge 2$). The inequality provides the sharp upper bound on convex domains, in terms of the diameter alone, of the best constants in Poincaré inequality. The key point is the implementation of a refinement of the classical Pólya-Szegö inequality for the symmetric decreasing rearrangement which yields an optimal weighted Wirtinger inequality.
We prove a long standing conjecture concerning the fencing problem in the plane: among planar convex sets of given area, prove that the disc, and only the disc maximizes the length of the shortest area-bisecting curve. Although it may look intuitive, the result is by no means trivial since we also prove that among planar convex sets of given area the set which maximizes the length of the shortest bisecting chords is the so-called Auerbach triangle.