Research Statement

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Questions of the following type arise quite naturally from what we see around us. Why are soap bubbles that float in air approximately spherical? Why does a herd of reindeer form a round shape when attacked by wolves? Of all the geometric objects having a certain property, which ones have the greatest area or volume; and of all objects having a certain property, which ones have the least perimeter or surface area? These problems have been stimulating mathematical thought for a long time now. Mathematicians have been trying to answer the above questions and this has led to a branch of mathematical analysis known as shape optimization problems.

A typical shape optimization problem is, as the name suggests, to find a shape which is optimal in the sense that it minimizes a certain cost functional while satisfying given constraints. Mathematically speaking, it is to find a domain Ω that minimizes a functional J(Ω), possibly subject to a constraint of the form G(Ω) = 0. In other words, it involves minimizing a functional J(Ω) over a family F of admissible domains Ω.

One may refer to [2], for a mini review of the kind of shape optimization problems that we (along with our collaborators) have worked on. This mini review includes a brief history of the isoperimetric problems and gives a brief survey of our results up to April 2017. In [2], we also discuss the key ideas used in proving these results in the Euclidean case. We also mention how we generalized the results which were known in the Euclidean case to other geometric spaces. We also describe how we extended these results from the linear setting to a non-linear one. We describe briefly the difficulties faced in proving these generalized versions and how we overcame these difficulties.

We give an updated brief research description here including our results post April 2017 and describe a few on-going and future research problems under consideration.

We are interested in shape-optimisation problems including the isoperimetric problems (see [3]). The details of our research work is as described below.

Let S be a Riemannian manifold with metric g and Laplace-Beltrami operator ∆. Let B1 be an open (geodesic) ball in S. Let B0 be an open ball whose closure is contained in B1. Let Ω = B1 \ \bar{B 0}. Consider the following problems:

−∆u = 1 in Ω, u=0 on ∂Ω,               (1)

and

−∆u = λu in Ω, u=0 on ∂Ω.             (2)

In the case that S is Euclidean space of dimension n, Hersch [21] (for n = 2), Ramm-Shivakumar [27] (for n = 2), and Kesavan [23] (for n ≥ 2) proved the following:

The proofs in [23, 27] rely on shape differentiation [28] and the moving plane method [11, 19]. In the application of the moving plane method in [23, 27], the commutativity of the Laplacian and the reflection in a hyperplane is used.

Research Plan:

Let v be a node in a spider network G. Let Gv be be the subnetwork obtained by deleting the node v and all the edges stemming from v. Let λ1(v) denote the smallest positive eigenvalue of the Laplacian on Gv. Then, λ1(v) is maximum when v is the origin i.e., the vertex at the center of the original spider network G.

We have also formulated some different versions of this conjecture where we change our network slightly and look at the spectrum of the reduced Laplacian instead of the usual one. We do have the computational evidence for the aforementioned conjecture for spider networks of various sizes (approximately for sizes up to 1000). We plan to use techniques analogous to the hyperplane reflection principle, Rayleigh quotients and Hadamard perturbation theory from the geometric analysis to solve this problem.

• With Jyotsna Prajapat (University of Mumbai) and Anoop T V (IIT Madras), we plan to generalise the results of [35] and [7] for the p-Laplacian.

Potential Future Research Problems:

• We have studied the following problem. Fix V_0 > 0. Let Ω be a domain in E^n of the formU \ \overline{B} where B is a fixed ball while U is any domain (with C^2 boundary) such that \overline{B}⊂ U.

Consider the following boundary value problems :

−∆u = 1 in Ω, u = 0 on ∂B,  ∂u/∂n = 0 on ∂U,                                        (3)

and

−∆u = λu in Ω, u = 0 on ∂B,  ∂u/∂n = 0 on ∂U.                                        (4)

Here n denotes outward unit normal to Ω on ∂Ω. We proved the following:

(I’) If u is a solution of problem (3), the energy functional \int_Ω ||∇u||^2 dx attains its maximum if and only if U is a ball concentric with B.

(II’) The first eigenvalue λ_1 of problem (4) attains its maximum if and only if U is a ball concentric with B.

We have also studied the behaviour of the above two functionals associated to Problems (3) and (4) with Ω of the form B \ \overline{U} where U is a domain with C^2 boundary such that \overline{U} ⊂ B and such that Vol(Ω) = V_0. As one of the boundary component is asymmetric with Neumann boundary condition on it, the usual reflection technique involved in moving plane method does not work for this family. We have come up with a naive argument which uses the maximum principles for elliptic operators. We have also started studying these problems for the p-Laplacian. The difficulty involved in studying these problems for the p-Laplacian is that the radiality of the solutions to Problems (3) and (4) for the p-Laplace operator in place of the Laplacian is not known for a symmetric annular region.

• To prove the analogues of the results of [23, 27, 2, 8] for compact rank one symmetric spaces. We plan to use the unified approach to all rank one symmetric spaces of compact type via Clifford algebra given by Koranyi and Ricci [24].

• We are studying the behaviour of the above mentioned functionals associated to the Dirichlet Boundary Value Problems (1) and (2) over each of the families {U\\overline{B}} and {B\\overline{U}} as described above. We expect the extremum to be attained when U is a ball concentric with B. The difficulty here is to come up with a test function for an arbitrary domain in the family under consideration with same L^2-norm as the solution for the symmetric domain but with lesser energy. We try to overcome this difficulty by considering concentric curves in the domain Ω (viz. concentric circles intersected with the domain Ω) and constructing a test function v such that the mean L^2 integral of the corresponding solution on each these curve C equals the mean L^2 integral of the test function v on a full sphere whose radius equals the radius of C. We need to check that this test function will have a lesser L^2-norm for the gradient as compared to the most symmetric candidate.

• With Professor Rajesh Mahadevan, there was a plan to consider a problem which is listed as problem 42 in Yau’s list (cf. [34]). The problem is as follows: In [32], K. Uhlenbeck proved that, for a generic metric on a compact manifold, every eigenvalue is simple with no multiplicity. It can be asked that is it possible to give an effective condition on the metric to assure that the metric is generic in this sense? Metrics with a special property are unlikely to generic. For example, do Einstein metrics have degenerate eigenvalue? (An approach is to find nontrivial operators which commute with the Laplacian. Characterize these manifolds.) can one describe domains where all eigenvalues are simple. Does there exist a domain with only finite number of multiple eigenvalues? If there is a compact group acting on the manifold, the spectrum clearly can not be simple. But we can replace the question by understanding whether the representation of the group is irreducible or not. The Weyl estimate tells us the principal term of the asymptotic behaviour of the counting functions for the eigenvalues. Does the error reflect some part of the topology of the manifold? For example, can the torus admit a metric whose spectrum behaves like the spectrum of the sphere after discarding the Weyl term? Does the growth of the multiplicity of λ_1 depend on the topology of the manifold? What type of topological information can we extract from the spectrum on functions alone?

• In [36] the authors have found a lower estimate of (λ2 − λ1) d^2 for a convex domain where λ_1 and λ_2 are the first two eigenvalues and d is the diameter of the domain. What is the best constant for such a lower estimate and is there an extremal domain? Can one find a lower estimate of λ_2 − λ_1 for non-convex domains or for manifolds with boundary? We also wanted to study the gap problem mentioned in section 3 of [37].

• To see if results analogous to [23, 27, 2, 8] hold for the functional whose Euler Lagrange equation is the following semi-linear elliptic boundary value problem:

−∆u = λu + λ u^α in Ω, u = 0 on ∂Ω, for suitable α.

We need to develop the shape calculus for this Boundary Value Problem.

• To study the extremum of the above-mentioned functionals for having two punctures, i.e., when is of the form B \ (B1 ∪ B2), where B1, B2 are geodesic balls such that B1, B2 are disjoint with their union completely contained in the geodesic ball B. Here, we plan to use the calculus of variation for functions in two variables.

• Prove that The regular N-gon has the least first eigenvalue among all the N-gons of given area for N ≥ 5.

• Prove that Among all parallelograms with given distances between their opposite sides, the rectangle maximizes λ_1.

Bibliography