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:
(I) If u is a solution of problem (1), the energy functional \int_B1 \B0 ∥∇u∥2 dx attains its maximum if and only if B0 and B1 are concentric.
(II) The first eigenvalue λ1 of problem (2) attains its maximum if and only if the balls are concentric.
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.
In [2], we developed a shape calculus on general Riemannian manifolds, and used it to prove the analogues of the results of Hersch, Kesavan and Ramm- Shivakumar on space-forms (complete simply connected Riemannian manifolds of constant sectional curvature). The reflection method works here just as in the Euclidean case, because reflection in a hyperplane is an isometry in any space form, and so it commutes with the Laplacian.
Rank-one Symmetric spaces are a natural generalization of the space-forms as the isometry group acts transitively on the unit tangent bundle. In [8], we showed that if S is a non-compact rank one symmetric space, then the analogues of the results of [23, 27, 2] hold. It is a fact that if every hyperplane reflection in a simply connected Riemannian manifold S is an isometry, then S must be a space-form. So, at first it appears that the reflection technique will not work for more general Riemannian manifolds. But because of the presence (in our situation) of cylindrical symmetry about a geodesic, the arguments used in the reflection method are still valid if we replace reflection maps by geodesic inversions.
In the expository article [4], by the use of ‘approximating polygons’, we solved the isoperimetric problem on the simply connected surface Mκ2 having constant sectional curvature κ (κ = 0, ±1), and prove that ‘circle’ is the unique solution to the isoperimetric problem. We gave an elementary geometric proof which applies to all the three simply connected space forms.
Recently, Emamizadeh and Zivari-Rezapour [18] have tried to generalise the result of [23, 27, 21] to the case of a non-linear differential operator namely the p-Laplacian. But they could succeed only in proving a domain monotonicity result for a weighted eigenvalue problem in which the weights need to satisfy some artificial conditions. In [5], we too generalise the results to the p-Laplacian (1 < p < ∞) without any artificial restrictions, and in the process we simplify greatly the proof even in the case of the Laplacian. The existence and uniqueness of non-negative solution of a particular Boundary Value Problem involving the p-Laplacian with non-vanishing boundary conditions is also derived. As a con- sequence, a Weak Comparison Principle for the p-Laplacian (with non-vanishing boundary condition) necessary for our purpose is proved. The uniqueness of the maximising domain in the nonlinear case was proved later (using our results) in [9].
In [7], we dealt with an obstacle placement problem which can be formulated as the following eigenvalue optimization problem: We consider a disk B ⊂ R2. We wanted to place an obstacle P, having a specific symmetry, within B so as to maximize or minimize the fundamental Dirichlet eigenvalue λ1 on B \ P , in the case when the centers of mass for P and B are non-concentric. We studied the extrema the function λ1(B \ ρ(P)), where ρ runs over the set of all rigid motions of the plane fixing the center of mass for P such that ρ(P) ⊂ B. We considered the case where the obstacle P is invariant under the action of a dihedral group Dn; n ≥ 2; n even. We could also characterize the optimal location of the obstacle w.r.t. its translations within B. For n-odd case, we could find the critical points of the above mentioned function. We do provide numerical evidence supporting our results. We also prove the generalizations of our results from the Euclidean plane to other manifolds, and from the differential equations involving the Laplace operator to the ones involving the Schr¨odinger-type operators.
In [39], in collaboration with Dr Mrityunjoy Ghosh, we proved that, even for the n odd case, the critical points found in [7] and [38] are the only critical points and have characterized the maximising and the minimising configurations completely. Based on relevant numerical observations, these conjectures were formulated in [7] and [38] by us. We also generalised the results of [35] and [7] for the p-Laplacian.
In [40], in collaboration with Sagar Basak and Dr Sheela Verma, we considered mixed Steklov-Dirichlet eigenvalue problem on smooth bounded domains in Riemannian manifolds. Under certain symmetry assumptions on multiconnected domains in R𝑛 with a spherical hole, we obtained isoperimetric inequalities for 𝑘-th Steklov- Dirichlet eigenvalues for 2 ≤ 𝑘 ≤ 𝑛 + 1. We extend Theorem 3.1 of [41] from Euclidean domains to domains in space forms, that is, we obtain sharp lower and upper bounds of the first Steklov-Dirichlet eigenvalue on bounded star-shaped domains in the unit 𝑛-sphere and in the hyperbolic space.
Collaborating with Dr Apala Majumdar (from University of Bath, United Kingdom) on shape optimisation problems for nematic liquid crystals in confined geometries. Nematic liquid crystals are classical examples of mesophases with physical properties intermediate between solids and liquids. We can study pattern formation in confined nematic media in terms of problems in the calculus of variations or boundary-value problems for elliptic partial differential equations. A common question of practical interest is - what is the optimal location of nanoinclusions inside nematic media? For example, we consider an annulus with specified boundary conditions on the inner and outer circles. The corresponding nematic profile, modelled as an energy minimizer, depends on the location of the inner circle relative to the outer circle and one could attack this problem with different approaches, to find the optimal location of the inner circle relative to the outer circle. In a similar vein, we are recasting these practical problems as shape optimisation problems, based on computing the shape derivatives of complex energy functionals in the theory of nematic liquid crystals. Some of this work is in progress. Although, we do have results for a particular inhomogeneous Dirichlet Boundary Value Problem (relevant from the Liquid Crystal point of view) for two different family of domains. One of them can be found in [38].
Research Plan:
We, along with Sagar Basak, Ravi Prakash and Sheela Verma, have proved an optimisation problem for a mixed Steklov-Dirichlet eigenvalue of the Laplacian on a class of doubly connected domains.
We, along with Dr. Rajesh Mahadevan (from Universidad de Concepcion, Chile) and Dr Apala Majumdar (from University of Bath, United Kingdom) now aim to study the behaviour of the energy functional in the case of a more complex inhomogeneous Dirichlet Boundary Value Problem useful from the liquid crystal point of view.
Collaborating with Dr Apala Majumdar (from University of Bath, United Kingdom) on shape optimisation problems for nematic liquid crystals in confined geometries. Nematic liquid crystals are classical examples of mesophases with physical properties intermediate between solids and liquids. We can study pattern formation in confined nematic media in terms of problems in the calculus of variations or boundary-value problems for elliptic partial differential equations. A common question of practical interest is - what is the optimal location of nanoinclusions inside nematic media? For example, we consider an annulus with specified boundary conditions on the inner and outer circles. The corresponding nematic profile, modelled as an energy minimizer, depends on the location of the inner circle relative to the outer circle and one could attack this problem with different approaches, to find the optimal location of the inner circle relative to the outer circle. In a similar vein, we are recasting these practical problems as shape optimisation problems, based on computing the shape derivatives of complex energy functionals in the theory of nematic liquid crystals. This work is in progress. We do have results for a particular inhomogeneous Dirichlet Boundary Value Problem (relevant from the Liquid Crystal point of view) for two different family of domains. We, along with Dr. Rajesh Mahadevan (from Universidad de Concepcion, Chile). We now aim to study the behaviour of the energy functional in the case of a more complex inhomogeneous Dirichlet Boundary Value Problem useful from the liquid crystal point of view.
Following the tradition of developing various theorems in analysis to graph theory, we hope to establish an analogue of the results by Hersch, Kesavan and Ramm-Shivakumar which were discussed above. This is an ongoing project in collaboration with Chandrasheel Bhagwat and Pralhad Shinde, both from IISER Pune. We aim at proving an analogous theorem for the Laplacian (suitably modified) on a specific family of graphs called the Spider Networks. Our conjecture is as follows:
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.
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.
• Try and give another proof of the fact: 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.
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