arXiv subjects
Publications and source records attributed to Carlo Nitsch.
We prove that the Weinstock inequality for the first nonzero Steklov eigenvalue holds in $\mathbb{R}^n$, for $n\ge 3$, in the class of convex sets with prescribed surface area. The key result is a sharp isoperimetric inequality involving simultanously the surface area, the volume and the boundary momentum of convex sets. As a by product, we also obtain some isoperimetric inequalities for the first Wentzell eigenvalue
We consider two optimization problems in thermal insulation: in both cases the goal is to find a thin layer around the boundary of the thermal body which gives the best insulation. The total mass of the insulating material is prescribed.. The first problem deals with the case in which a given heat source is present, while in the second one there are no heat sources and the goal is to have the slowest decay of the temperature. In both cases an optimal distribution of the insulator around the thermal body exists; when the body has a circular symmetry, in the first case a constant heat source gives a constant thickness as the optimal solution, while surprisingly this is not the case in the second problem, where the circular symmetry of the optimal insulating layer depends on the total quantity of insulator at our disposal. A symmetry breaking occurs when this total quantity is below a certain threshold. Some numerical computations are also provided, together with a list of open questions.
It was recently proved that the elastic energy $E(γ)=\tfrac{1}{2}\int_γκ^2 ds$ of a closed curve $γ$ with curvature $κ$ has a minimizer among all plane, simple, regular and closed curves of given enclosed area $A(γ)$, and that the minimum is attained only for circles. Here we show under which hypothesis the result can be extended to other functionals involving the curvature. As an example we show that the optimal shape remains a circle for the $p$-elastic energy $\int_γ|κ|^p ds$, whenever $p>1$.
Given a smooth simply connected planar domain, the area is bounded away from zero in terms of the maximal curvature alone. We show that in higher dimensions this is not true, and for a given maximal mean curvature we provide smooth embeddings of the ball with arbitrary small volume.
Extremal functions are exhibited in Poincaré trace inequalities for functions of bounded variation in the unit ball ${\mathbb B}^n$ of the $n$-dimensional Euclidean space ${\mathbb R}^n$. Trial functions are subject to either a vanishing mean value condition, or a vanishing median condition in the whole of ${\mathbb B}^n$, instead of just on $\partial {\mathbb B}^n$, as customary. The extremals in question take a different form, depending on the constraint imposed. In particular, under the latter constraint, unusually shaped extremal functions appear. A key step in our approach is a characterization of the sharp constant in the relevant trace inequalities in any admissible domain $Ω\subset {\mathbb R}^n$, in terms of an isoperimetric inequality for subsets of $Ω$.
We consider the problem of optimally insulating a given domain $Ω$ of ${\mathbb{R}}^d$; this amounts to solve a nonlinear variational problem, where the optimal thickness of the insulator is obtained as the boundary trace of the solution. We deal with two different criteria of optimization: the first one consists in the minimization of the total energy of the system, while the second one involves the first eigenvalue of the related differential operator. Surprisingly, the second optimization problem presents a symmetry breaking in the sense that for a ball the optimal thickness is nonsymmetric when the total amount of insulator is small enough. In the last section we discuss the shape optimization problem which is obtained letting $Ω$ to vary too.
We show that the elastic energy $E(γ)$ of a closed curve $γ$ has a minimizer among all plane simple regular closed curves of given enclosed area $A(γ)$, and that the minimum is attained for a circle. The proof is of a geometric nature and deforms parts of $γ$ in a finite number of steps to construct some related convex sets with smaller energy.
The Faber-Krahn inequality in $\mathbb{R}^2$ states that among all open bounded sets of given area the disk minimizes the first Dirichlet Laplacian eigenvalue. There are numerical evidences that for all $N\ge 3$ the first Dirichlet Laplacian eigenvalue of the regular $N$-gon is greater than the one of the regular $(N+1)$-gon of same area. This natural property is also suggested by the fact that the shape of regular polygons becomes more and more "rounded" as $N$ increases and, among sets of given area, disk minimize the eigenvalue. Aiming to settle such a conjecture, in this work we investigate possible ways to estimate the difference between eigenvalues of regular $N$-gons and $(N+1)$-gons.
For a given bounded Lipschitz set $Ω$, we consider a Steklov--type eigenvalue problem for the Laplacian operator whose solutions provide extremal functions for the compact embedding $H^1(Ω)\hookrightarrow L^2(\partial Ω)$. We prove that a conjectured reverse Faber--Krahn inequality holds true at least in the class of Lipschitz sets which are "close" to a ball in a Hausdorff metric sense. The result implies that among sets of prescribed measure, balls are local minimizers of the embedding constant.
We study the optimal constant in a Sobolev inequality for BV functions with zero mean value and vanishing outside a bounded open set. We are interested in finding the best possible embedding constant in terms of the measure of the domain alone. We set up an optimal shape problem and we completely characterize the behavior of optimal domains.
The study of the optimal constant in an Hessian-type Sobolev inequality leads to a fully nonlinear boundary value problem, overdetermined with non standard boundary conditions. We show that all the solutions have ellipsoidal symmetry. In the proof we use the maximum principle applied to a suitable auxiliary function in conjunction with an entropy estimate from affine curvature flow.
Balls are shown to have the smallest optimal constant, among all admissible Euclidean domains, in Poincaré type boundary trace inequalities for functions of bounded variation with vanishing median or mean value.
We prove a sharp upper bound on convex domains, in terms of the diameter alone, of the best constant in a class of weighted Poincaré inequalities. The key point is the study of an optimal weighted Wirtinger inequality.
The Faber-Krahn deficit $δλ$ of an open bounded set $Ω$ is the normalized gap between the values that the first Dirichlet Laplacian eigenvalue achieves on $Ω$ and on the ball having same measure as $Ω$. For any given family of open bounded sets of $\R^N$ ($N\ge 2$) smoothly converging to a ball, it is well known that both $δλ$ and the isoperimetric deficit $δP$ are vanishing quantities. It is known as well that, at least for convex sets, the ratio $\frac{δP}{δλ}$ is bounded by below by some positive constant (see \cite{BNT,PW}), and in this note, using the technique of the shape derivative, we provide the explicit optimal lower bound of such a ratio as $δP$ goes to zero.
Denoting by $\mathcal{E}\subseteq \R^2$ the set of the pairs $(λ_1(Ω),λ_2(Ω))$ for all the open sets $Ω\subseteq\R^N$ with unit measure, and by $Θ\subseteq\R^N$ the union of two disjoint balls of half measure, we give an elementary proof of the fact that $\partial\E$ has horizontal tangent at its lowest point $(λ_1(Θ),λ_2(Θ))$.