We investigate extremality properties of shape functionals which are products of Newtonian capacity $\cp(\overline{\Om})$, and powers of the torsional rigidity $T(\Om)$, for an open set $\Om\subset \R^d$ with compact closure $\overline{\Om}$, and prescribed Lebesgue measure. It is shown that if $\Om$ is convex then $\cp(\overline{\Om})T^q(\Om)$ is (i) bounded from above if and only if $q\ge 1$, and (ii) bounded from below and away from $0$ if and only if $q\le \frac{d-2}{2(d-1)}$. Moreover a convex maximiser for the product exists if either $q>1$, or $d=3$ and $q=1$. A convex minimiser exists for $q< \frac{d-2}{2(d-1)}$. If $q\le 0$, then the product is minimised among all bounded sets by a ball of measure $1$.
In this paper we generalize some classical estimates involving the torsional rigidity and the principal frequency of a convex domain to a class of functionals related to some anisotropic non linear operators.
We investigate the existence of a maximiser among open, bounded, convex sets in $\R^d,\,d\ge 3$ for the product of torsional rigidity and Newtonian capacity (or logarithmic capacity if $d=2$), with constraints involving Lebesgue measure or a combination of Lebesgue measure and perimeter.
We prove a sharp upper bound for the first Dirichlet eigenvalue of a class of nonlinear elliptic operators which includes the p-Laplace and the pseudo-p-Laplace operators. Moreover, we prove a stability result by means of a suitable isoperimetric deficity. Finally, we give a sharp lower bound for the anisotropic p-torsional rigidity.
A one-parameter family of variational problems is introduced that interpolates between torsional rigidity and the first Dirichlet eigenvalue of the Laplacian. The associated partial differential equation is derived, which is shown to have positive solutions in many cases. Results are obtained regarding extremal domains and regarding variations of the domain or the parameter.
In this paper we provide a comparison result between the solutions to the torsion problem for the Hermite operator with Robin boundary conditions and the one of a suitable symmetrized problem.
An inequality on torsional rigidity is established. For tangential polygons this inequality is stronger than an inequality of Polya and Szego for convex domains. (A survey of related work, not in the journal submission, is presented in the arXiv version.)
Motivated by the connection between the first eigenvalue of the Dirichlet-Laplacian and the torsional rigidity, the aim of this paper is to find a physically coherent and mathematically interesting new concept for boundary torsional rigidity, closely related to the Steklov eigenvalue. From a variational point of view, such a new object corresponds to the sharp constant for the trace embedding of $W^{1,2}(Ω)$ into $L^1(\partialΩ)$. We obtain various equivalent variational formulations, present some properties of the state function and obtain some sharp geometric estimates, both for planar simpl…
In this paper we study optimal estimates for two functionals involving the anisotropic $p$-torsional rigidity $T_p(Ω)$, $1<p<+\infty$. More precisely, we study $Φ(Ω)=\frac{T_p(Ω)}{|Ω|M(Ω)}$ and $Ψ(Ω)=\frac{T_p(Ω)}{|Ω|[R_{F}(Ω)]^{\frac{p}{p-1}}}$, where $M(Ω)$ is the maximum of the torsion function $u_Ω$ and $R_F(Ω)$ is the anisotropic inradius of $Ω$.
The Lq-Minkowski problem of anisotropic p-torsional rigidity
School of Mathematics and Statistics, Ningxia University（宁夏大学数学与统计学院）；Ningxia Basic Science Research Center of Mathematics, Ningxia University（宁夏大学数学基础科学研究中心）；Department of Mathematics, Lanzhou University of Technology（兰州理工大学数学系）
In this paper we study the (nonlocal) torsional rigidity in the ambient space of random walk spaces. We get the relation of the (nonlocal) torsional rigidity of a set $Ω$ with the spectral $m$-heat content of $Ω$, what gives rise to a complete description of the nonlocal torsional rigidity of $Ω$ by using uniquely probability terms involving the set $Ω$; and recover the first eigenvalue of the nonlocal Laplacian with homogeneous Dirichlet boundary conditions by a limit formula using these probability term. For the random walk in $\R^N$ associated with a non singular kernel, we get a nonlocal v…
In this paper, we introduce the so-called $L_p$ $q$-torsional measure for $p\in\mathbb{R}$ and $q>1$ by establishing the $L_p$ variational formula for the $q$-torsional rigidity of convex bodies without smoothness conditions. Moreover, we achieve the existence of solutions to the $L_p$ Minkowski problem $w.r.t.$ the $q$-torsional rigidity for discrete measure and general measure when $0<p<1$ and $q>1$.
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 $F(Ω_ε)\ge 1-ε$.
Let $Ω_0$ denote the unit ball of $\mathbb{R}^N$ ($N\ge 2$) centered at the origin. We suppose that $Ω_0$ contains a core, given by a smaller concentric ball $D_0$, made of a (possibly) different material. We discover that, depending on the relative hardness of the two materials, this radial configuration can either be a local maximizer for the torsional rigidity functional $E$ or a saddle shape. In this paper we consider perturbations that simultaneously act on the boundaries $\partial D_0$ and $\partialΩ_0$. This gives rise to resonance effects that are not present when $\partial D_0$ or $\p…
We prove the existence and uniqueness up to translations of the solution to a Minkowski type problem for the torsional rigidity in the class of open bounded convex subsets of the $n$-dimensional Euclidean space. For the existence part we apply the variational method introduced by Jerison for analogous problems concerning other variational functionals. Uniqueness follows from the Brunn--Minkowski inequality for the torsional rigidity and corresponding equality conditions.
In this paper the Orlicz-Minkowski problem for torsional rigidity, a generalization of the classical Minkowski problem, is studied. Using the flow method, we obtain a new existence result of solutions to this problem for general measures.
We consider the torsional rigidity and the principal eigenvalue related to the $p$-Laplace operator. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in various classes of domains. The limit cases $p=1$ and $p=\infty$ are also analyzed, which amount to consider the Cheeger constant of a domain and functionals involving the distance function from the boundary.
We develop the theory of torsional rigidity -- a quantity routinely considered for Dirichlet Laplacians on bounded planar domains -- for Laplacians on metric graphs with at least one Dirichlet vertex. Using a variational characterization that goes back to Pólya, we develop surgical principles that, in turn, allow us to prove isoperimetric-type inequalities: we can hence compare the torsional rigidity of general metric graphs with that of intervals of the same total length. In the spirit of the Kohler-Jobin Inequality, we also derive sharp bounds on the ground-state energy of a quantum graph in…
We consider shape functionals of the form $F_q(Ω)=P(Ω)T^q(Ω)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $P(Ω)$ denotes the perimeter of $Ω$ and $T(Ω)$ is the torsional rigidity of $Ω$. The minimization and maximization of $F_q(Ω)$ is considered on various classes of admissible domains $Ω$: in the class $\mathcal{A}_{all}$ of all domains, in the class $\mathcal{A}_{convex}$ of convex domains, and in the class $\mathcal{A}_{thin}$ of thin domains.
We obtain upper bounds on the heat content and on the torsional rigidity of a complete Riemannian manifold M, assuming a generalized Hardy inequality for the Dirichlet Laplacian on M.
We obtain bounds for the expected loss of torsional rigidity of a cylinder $Ω_L=(-L/2,L/2) \times Ω\subset \R^3$ of length $L$ due to a Brownian fracture that starts at a random point in $Ω_L,$ and runs until the first time it exits $Ω_L$. These bounds are expressed in terms of the geometry of the cross-section $Ω\subset \R^2$. It is shown that if $Ω$ is a disc with radius $R$, then in the limit as $L \rightarrow \infty$ the expected loss of torsional rigidity equals $cR^5$ for some $c\in (0,\infty)$. We derive bounds for $c$ in terms of the expected Newtonian capacity of the trace of a Browni…
We consider the problem of minimising or maximising the quantity $λ(Ø)T^q(Ø)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $λ(Ø)$ denotes the first eigenvalue of the Dirichlet Laplacian on $H^1_0(Ø)$, while $T(Ø)$ is the torsional rigidity of $Ø$. The optimisation problem above is considered in the class of {\it all domains} $Ø$, in the class of {\it convex domains} $Ø$, and in the class of {\it thin domains}. The full Blaschke-Santaló diagram for $λ(Ø)$ and $T(Ø)$ is obtained in dimension one, while for higher dimensions we provide some bounds.
Let $T^m$ be the $m$-dimensional unit torus, $m \in N$. The torsional rigidity of an open set $Ω\subset T^m$ is the integral with respect to Lebesgue measure over all starting points $x \in Ω$ of the expected lifetime in $Ω$ of a Brownian motion starting at $x$. In this paper we consider $Ω= T^m \backslash β[0,t]$, the complement of the path $β[0,t]$ of an independent Brownian motion up to time $t$. We compute the leading order asymptotic behaviour of the expectation of the torsional rigidity in the limit as $t \to \infty$. For $m=2$ the main contribution comes from the components in $T^2 \bac…
Kohler-Jobin meets Ehrhard: the sharp lower bound for the Gaussian principal frequency while the Gaussian torsional rigidity is fixed, via rearrangements
Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II（那不勒斯费德里科二世大学）；Department of Mathematics, Purdue University（普渡大学）
Comparison results for the $p$-torsional rigidity on convex domains
American University of Sharjah（沙迦美国大学）；University of Craiova（克卢日-纳波卡大学）；”Gheorghe Mihoc - Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy（罗马尼亚科学院“格奥尔基·米霍克-卡尤斯·雅各布”数学统计与应用数学研究所）；National University of Science and Technology Politehnica of Bucharest（布加勒斯特理工大学国家科学技术大学）
For every positive regular Borel measure, possibly infinite valued, vanishing on all sets of $p$-capacity zero, we characterize the compactness of the embedding $W^{1,p}({\bf R}^N)\cap L^p ({\bf R}^N,μ)\hr L^q({\bf R}^N)$ in terms of the qualitative behavior of some characteristic PDE. This question is related to the well posedness of a class of geometric inequalities involving the torsional rigidity and the spectrum of the Dirichlet Laplacian introduced by Polya and Szegö in 1951. In particular, we prove that finite torsional rigidity of an arbitrary domain (possibly with infinite measure), i…
We prove capacity inequalities involving the total mean curvature of hypersurfaces with boundary in convex cones and the mass of asymptotically flat manifolds with non-compact boundary. We then give the analogous of Pölia-Szegö, Alexandrov-Fenchel and Penrose type inequalities in this setting. Among the techniques used in this paper are the inverse mean curvature flow for hypersurfaces with boundary.
Let $Ω\subset\mathbb{R}^n$, $n\geq 2$, be a bounded, open and convex set and let $f$ be a positive and non-increasing function depending only on the distance from the boundary of $Ω$. We consider the $p-$torsional rigidity associated to $Ω$ for the Poisson problem with Dirichlet boundary conditions, denoted by $T_{f,p}(Ω)$. Firstly, we prove a Pólya type lower bound for $T_{f,p}(Ω)$ in any dimension; then, we consider the planar case and we provide two quantitative estimates in the case $f\equiv 1 $.
We discuss the minimization of a Kohler-Jobin type scale-invariant functional among open, convex, bounded sets, namely $\min T_2(Ω) ^{\frac{1}{N+2}}h_1(Ω)$ among open convex bounded sets $Ω\subset \mathbb R^N$, where $T_2(Ω)$ denotes the torsional rigidity of a set $Ω$ and $h_1(Ω)$ its Cheeger constant. We prove the existence of an optimal set and we conjecture that the ball is the unique minimizer. We provide a sufficient condition for the validity of the conjecture, and an application of the conjecture to prove a quantitative inequality for the Cheeger constant. We also show lack of existenc…
