In this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we consider T (Ω)/(M (Ω)|Ω|) and M (Ω)λ 1 (Ω), where Ω is a bounded open set of R d with finite Lebesgue measure |Ω|, M (Ω) denotes the maximum of the torsion function, T (Ω) the torsion, and λ 1 (Ω) the first Dirichlet eigenvalue. Particular attention is devoted to the subclass of convex sets.
We study a 2d-variational problem, in which the cost functional is an integral depending on the gradient through a convex but not strictly convex integrand, and the admissible functions have zero gradient on the complement of a given domain D. We are interested in establishing whether solutions exist whose gradient "avoids" the region of non-strict convexity. Actually, the answer to this question is related to establishing whether homogenization phenomena occur in optimal thin torsion rods. We provide some existence results for different geometries of D, and we study the nonstandard free boundary problem with a gradient obstacle, which is obtained through the optimality conditions.
For $\Omega$ varying among open bounded sets in ${\mathbb R} ^n$, we consider
shape functionals $J (\Omega)$ defined as the infimum over a Sobolev space of
an integral energy of the kind $\int _\Omega[ f (\nabla u) + g (u) ]$, under
Dirichlet or Neumann conditions on $\partial \Omega$. Under fairly weak
assumptions on the integrands $f$ and $g$, we prove that, when a given domain
$\Omega$ is deformed into a one-parameter family of domains $\Omega
_\varepsilon$ through an initial velocity field $V\in W ^ {1, \infty} ({\mathbb
R} ^n, {\mathbb R} ^n)$, the corresponding shape derivative of $J$ at $\Omega$
in the direction of $V$ exists. Under some further regularity assumptions, we
show that the shape derivative can be represented as a boundary integral
depending linearly on the normal component of $V$ on $\partial \Omega$. Our
approach to obtain the shape derivative is new, and it is based on the joint
use of Convex Analysis and Gamma-convergence techniques. It allows to deduce,
as a companion result, optimality conditions in the form of conservation laws.Comment: Mathematical Programming, September 201
In this paper we look for the domains minimizing the h-th eigenvalue of the Dirichlet-Laplacian λ h with a constraint on the diameter. Existence of an optimal domain is easily obtained, and is attained at a constant width body. In the case of a simple eigenvalue, we provide non standard (i.e., non local) optimality conditions. Then we address the question whether or not the disk is an optimal domain in the plane, and we give the precise list of the 17 eigenvalues for which the disk is a local minimum. We conclude by some numerical simulations showing the 20 first optimal domains in the plane.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.