Rajesh Mahadevan

This paper is devoted to the computation of certain directional semi-derivatives of eigenvalue functionals of self-adjoint elliptic operators involving a variety of boundary conditions. A uniform treatment of these problems is possible by considering them as a problem of calculating the semi-derivative of a minimum with respect to a parameter. The...

We prove an analog of the Faber–Krahn inequality for the Riesz potential operator. The proof is based on Riesz’s inequality under Steiner symmetrization and the continuity of the first eigenvalue of the Riesz potential operator with respect to the convergence, in the complementary Hausdorff distance, of a family of uniformly bounded non-empty conve...

In this work, we study the existence and nonexistence of positive radial solutions for the quasilinear equation $\mathrm{div}(A(|\nabla u|)\nabla u)+\lambda k(|x|)f(u)=0$ in the exterior of a ball with vanishing boundary conditions using an approach based on a fixed point theorem for operators on Banach Space.

While considering boundary value problems with oscillating coefficients or in oscillating domains, it is important to associate an asymptotic model which accounts for the average behaviour. This model permits to obtain the average behaviour without costly numerical computations implied by the fine scale of oscillations in the original model. The as...

It has been shown by Kesavan ( Proc. R. Soc. Edinb. A 133 (2003), 617–624) that the first eigenvalue for the Dirichlet Laplacian in a punctured ball, with the puncture having the shape of a ball, is maximum if and only if the balls are concentric. Recently, Emamizadeh and Zivari-Rezapour ( Proc. Am. Math. Soc. 136 (2007), 1325–1331) have tried to g...

We consider the problem of distributing two conducting materials with a prescribed volume ratio in a given domain so as to minimize the first eigenvalue of an elliptic operator with Dirichlet conditions. The gap between the two conductivities is assumed to be small (low contrast regime). For any geometrical configuration of the mixture, we provide...

A famous conjecture made by Lord Rayleigh is the following: “The first eigenvalue of the Laplacian on an open domain of given measure with Dirichlet boundary conditions is minimum when the domain is a ball and only when it is a ball”. This conjecture was proved simultaneously and independently by Faber [G. Faber, Beweiss dass unter allen homogenen...

In this article we consider the problem of the optimal distribution of two conducting materials with given volume inside a fixed domain, in order to minimize the first eigenvalue (the ground state) of a Dirichlet operator. It is known, when the domain is a ball, that the solution is radial, and it was conjectured that the optimal distribution of th...

Given a density function f on a compact subset of Rd we look at the problem of finding the best approximation of f by discrete measures ν=∑ciδxi in the sense of the p-Wasserstein distance, subject to size constraints of the form ∑h(ci)⩽α where h is a given weight function. This is an important problem with applications in economic planning of locat...

The study of control problems governed by partial differential equations where the cost of the control is offset by a small parameter ε had been initiated by J. L. Lions [”Remarks on ’cheap control’ ”, Topics numer. Analysis, Proc. Royal Irish Acad. Conf., Dublin 1972, 203-209 (1973; Zbl 0318.49006)], [”Some Methods in the Mathematical Analysis of...

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 minimiz...

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....

In this note we will present an extension of the Krein-Rutman theorem [M.G. Kreǐn, 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 c...

Let be a bounded Lipschitz regular open subset of Rd and let µ,� be two probablity measures on . It is well known that if µ = f dx is absolutely continuous, then there exists, for every p > 1, a unique transport map Tp pushing forward µ onand which realizes the Monge-Kantorovich distance Wp(µ,�). In this paper, we establish an L1 bound for the disp...

In this article we study the asymptotic behaviour of the eigenvalues of a family of nonlinear monotone elliptic operators of the form A(epsilon) = - div(a(epsilon) (x, del u)), which are sub-differentials of even, positively homogeneous convex functionals, under the assumption that the operators G-converge to art operator A(hom) = div(a(hom) (x, de...

In the homogenization of second order elliptic equations with periodic coefficients, it is well known that the rate of convergence of the zero order corrector un - uhom in the L2 norm is 1/n, the same as the scale of periodicity (see Jikov et al [6]). It is possible to have the same rate of convergence in the case of almost periodic coefficients un...

We study the homogenization of integral functionals depending on the Hessian matrix over periodic low-dimensional structures in \mathbbRn\mathbb{R}^n . To that aim, we follow the same approach as in [6], where the case of first order energies was analyzed. Precisely, we identify the thin periodic structure under consideration with a positive meas...

In this article we study the homogenization, of a particular example, of degenerate elliptic equations of second order in the setup of viscosity solutions. These results are an attempt to extend the corresponding results of Evans to degenerate situations.

In this article, we study the homogenization of the family of parabolic equations over periodically perforated domains # # -diva # # # =f(x,t) # u 0 (x) in # # .

Given a density function f on a compact subset of R d we look at the problem of finding the best approximation of f by discrete measures ν = c i δ x i in the sense of the p-Wasserstein distance, subject to size constraints of the form h(c i) α where h is a given weight function. This is an important problem with applications in economic planning of...

We consider the problem of optimal location of production centres to serve a non-uniform distribution of customers. The location is required to be optimal with respect to the cost of transportation which is modeled by a weighted average of the distance function to the nearest production centre. In this Note we study the asymptotic behaviour of the...

In this article, we study the homogenization of the family of parabolic equations over periodically perforated domains\(\begin{gathered} \partial _t b(\tfrac{x}{{d_\varepsilon }},u_\varepsilon ) - div a(u_\varepsilon , \nabla u_\varepsilon ) = f(x,t) in \Omega _\varepsilon x (0, T), \hfill \\ u_\varepsilon ) = 0 on \partial \Omega _\varepsilon x (0...

A new formulation for the limit matrix occurring in the cost functional of an optimal control problem on homogenization is obtained. It is used to obtain an upper bound for this matrix (in the sense of positive definite matrices).

In this article, we study the homogenization of the family of parabolic equations over periodically perforated domains ¶t b(\tfracxe,ue ) - diva(\tfracxe,ue ,Ñue ) = f(x,t) in We (0,T), a(\tfracxe,ue ,Ñue ) ve = 0 on ¶Se (0,T), ue = 0 on ¶W(0,T), ue (x,0) = u0 (x) in We \begin{gathered} \partial _t b(\tfrac{x}{\varepsilon },u_\varepsilon ) - diva(...

In this article, we study the homogenization of the nonlinear degenerate parabolic equation $$ partial_t b({x /varepsilon},u_varepsilon) - mathop{ m div} a({x /varepsilon},{t /varepsilon}, u_varepsilon,abla u_varepsilon)=f(x,t), $$ with mixed boundary conditions(Neumann and Dirichlet) and obtain the limit equation as $varepsilon o 0$. We also prove...

We prove a corrector result for the homogenization of flow in a partially fissured medium. The homogenization problem was studied by Clark and Showalter [3] using the two-scale convergence technique.

The aim of this paper is to provide an alternate treatment of the homogenization of an optimal control problem in the framework of two-scale (multi-scale) convergence in the periodic case. The main advantage of this method is that we are able to show the convergence of cost functionals directly without going through the adjoint equation. We use a c...