Papers by Rajesh Mahadevan
In this article we deal with the problem of distributing two conducting materials in a given doma... more In this article we deal with the problem of distributing two conducting materials in a given domain, with their proportions being xed, so as to minimize the rst eigenvalue of a Dirichlet operator. When the design region is a ball, it is known that there is an optimal distribution of materials which does not involve the mixing of the materials.
Let Ω be a bounded Lipschitz regular open subset of R d and let µ, ν be two probablity measures o... more Let Ω be a bounded Lipschitz regular open subset of R d and let µ, ν be two probablity measures on Ω. It is well known that if µ = f dx is absolutely continuous, then there exists, for every p > 1, a unique transport map T p pushing forward µ on ν and which realizes the Monge-Kantorovich distance W p (µ, ν). In this paper, we establish an L ∞ bound for the displacement map T p x − x which depends only on p, on the shape of Ω and on the essential infimum of the density f .
We study the homogenization of integral functionals depending on the Hessian matrix over periodic... more We study the homogenization of integral functionals depending on the Hessian matrix over periodic low-dimensional structures in R n . To that aim, we follow the same approach as in , where the case of first order energies was analyzed. Precisely, we identify the thin periodic structure under consideration with a positive measure µ, and we associate with µ an integral functional initially defined just for smooth functions. We prove that, under a suitable connectedness assumption on µ, the homogenized energy is an integral functional of the same kind, now with respect to the Lebesgue measure, whose effective density is obtained by solving an infimum problem on the periodicity cell. Such a problem presents basic differences from the first order case, as it involves both the microscopic displacement and the microscopic bending (Cosserat field). This feature is a consequence of the relaxation result for second order energies on thin structures proved in . In the case when the initial energy density is quadratic and isotropic, we apply the main homogenization theorem to obtain some bounds on the eigenvalues of the homogenized tensor and to compute explicitly the effective density for several examples of geometries in the plane.
Homogenization of some low-cost control problems
Nonlinear Analysis-theory Methods & Applications, Jan 1, 2007
In this note we will present an extension of the Krein–Rutman theorem [M.G. Kreĭn, M.A. Rutman, L... more In this note we will present an extension of the Krein–Rutman theorem [M.G. Kreĭn, M.A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl. (26) (1950). [9]] for an abstract non-linear, compact, positively 1-homogeneous operator on a Banach space having the properties of being increasing with respect to a convex cone KK and such that there is a non-zero u∈Ku∈K for which MTu≽u for some positive constant MM. This will provide a uniform framework for recovering the Krein–Rutman-like theorems proved for many non-linear differential operators of elliptic type, like the pp-Laplacian, cf. Anane [A. Anane, Simplicité et isolation de la première valeur propre du pp-laplacien avec poids (Simplicity and isolation of the first eigenvalue of the pp-Laplacian with weight), C. R. Acad. Sci. Paris 305 (16) (1987) 725–728 (in French)], the Hardy–Sobolev operator, cf. Sreenadh [K. Sreenadh, On the eigenvalue problem for the Hardy–Sobolev operator with indefinite weights, Electron. J. Differential Equations (33) (2002) 1–12], Pucci’s operator, cf. Felmer and Quaas [P. Felmer, A. Quaas, Positive radial solutions to a ‘semilinear’ equation involving the Pucci’s operator, J. Differential Equations 199 (2) (2004) 376–393]. Our proof follows the same lines as in the linear case, cf. Rabinowitz [P. Rabinowitz, Théorie du Degré Topologique et Applications à des Problèmes aux Limites Non Linéaires, Lecture Notes Lab. Analyse Numérique, Université Paris VI, 1975], and is based on a bifurcation theorem.
In this article we deal with the problem of distributing two conducting materials in a given doma... more In this article we deal with the problem of distributing two conducting materials in a given domain, with their proportions being fixed, so as to minimize the first eigenvalue of a Dirichlet operator. When the design region is a ball, it is known that there is an optimal distribution of materials which does not involve the mixing of the materials. However, the optimal configuration even in this simple case is not known. As a step in the resolution of this problem, in this paper, we develop the shape derivative analysis for this two-phase eigenvalue problem in a general domain. We also obtain a formula for the shape derivative in the form of a boundary integral and obtain a simple expression for it in the case of a ball. We then present some numerical calculations to support our conjecture that the optimal distribution in a ball should consist in putting the material with higher conductivity in a concentric ball at the centre.
Applied Mathematics and Optimization, Jan 1, 2009
The pioneering works of Murat and Tartar (Topics in the mathematical modeling of composite materi... more The pioneering works of Murat and Tartar (Topics in the mathematical modeling of composite materials. PNLDE 31. Birkhäuser, Basel, 1997) go a long way in showing, in general, that problems of optimal design may not admit solutions if microstructural designs are excluded from consideration. Therefore, assuming, tactilely, that the problem of minimizing the first eigenvalue of a two-phase conducting material with the conducting phases to be distributed in a fixed proportion in a given domain has no true solution in general domains, Cox and Lipton only study conditions for an optimal microstructural design (Cox and Lipton in Arch. Ration. Mech. Anal. 136:101–117, 1996). Although, the problem in one dimension has a solution (cf. Kreĭn in AMS Transl. Ser. 2(1):163–187, 1955) and, in higher dimensions, the problem set in a ball can be deduced to have a radially symmetric solution (cf. Alvino et al. in Nonlinear Anal. TMA 13(2):185–220, 1989), these existence results have been regarded so far as being exceptional owing to complete symmetry. It is still not clear why the same problem in domains with partial symmetry should fail to have a solution which does not develop microstructure and respecting the symmetry of the domain. We hope to revive interest in this question by giving a new proof of the result in a ball using a simpler symmetrization result from Alvino and Trombetti (J. Math. Anal. Appl. 94:328–337, 1983).
Discrete and Continuous Dynamical Systems, Jan 1, 2005
Additives for Polymers, Jan 1, 2011
Given a density function f on a compact subset of RdRd we look at the problem of finding the best... more Given a density function f on a compact subset of RdRd we look at the problem of finding the best approximation of f by discrete measures ν=∑ciδxiν=∑ciδxi in the sense of the p-Wasserstein distance, subject to size constraints of the form ∑h(ci)⩽α∑h(ci)⩽α where h is a given weight function. This is an important problem with applications in economic planning of locations and in information theory. The efficiency of the approximation can be measured by studying the rate at which the minimal distance tends to zero as α tends to infinity. In this paper, we introduce a rescaled distance which depends on a small parameter and establish a representation formula for its limit as a function of the local statistics for the distribution of the ciciʼs. This allows to treat the asymptotic problem as α→∞α→∞ for a quite large class of weight functions h.Etant donnée une densité à support compact f sur RdRd, on cherche la meilleure approximation de f par des mesures discrètes ν=∑ciδxiν=∑ciδxi au sens de la distance de Wasserstein dʼordre p, sous une contrainte du type ∑h(ci)⩽α∑h(ci)⩽α, où h est une fonction de poids donnée. Cʼest un problème important que lʼon rencontre notamment dans le domaine de lʼéconomie urbaine et en théorie de lʼinformation. Lʼefficacité de lʼapproximation peut être mesurée en étudiant à quelle vitesse la distance minimale tend vers zéro quand α tend vers lʼinfini. Dans cet article on introduit une distance normalisée dépendant dʼun petit paramètre et on établit une formule de représentation pour sa limite en fonction de la statistique locale de distribution des coefficients cici. On peut ainsi traiter le problème asymptotique α→∞α→∞ pour une très large classe de fonctions h.
Uploads
Papers by Rajesh Mahadevan