Chorwadwala Anisa Mohmad Husen Department of Mathematics, University of Mumbai, India.

Synopsis of the Ph.D. Thesis ‘‘Study of the Laplacian in a Class of Doubly Connected Domains on the Riemann Sphere S 2 .” Abstract. Fix an open ball B1 of radius r1 in S n (Hn ). Let B0 be any open ball of radius r0 such that B0 ⊂ B1 . For S n we consider r1 < π. Let u be a solution of the problem ∆u = 1 in Ω := B1 \ B0 vanishing on the boundary. It is shown that the energy functional is minimal if and only if the balls are concentric. It is also shown that first Dirichlet eigenvalue of the Laplacian on Ω is maximal if and only if the balls are concentric. ————————————————————————————————————– Notations and Basic Facts : Let (M, g) be a Riemannian manifold, Ω ⊂ M open set such that Ω is smooth compact submanifold of M. For a smooth function f : M −→ R, let ∇f , ∇2 f denote the gradient and the Hessian of f respectively. The (positive) Laplacian ∆ is defined by ∆f = −div(∇f ).

Sobolev Spaces H k(Ω), H0k(Ω) (k ∈ N) : Consider Sk (Ω) =



f : M −→ R, C

∞

Z  j 2 function k∇ f k (x) dVol < ∞ ∀ j = 0, 1, . . . , k .

Define for f ∈ Sk (Ω), kf kk,Ω :=

Ω

P

R k j=0 Ω

 12 k∇j f k2 (x) dVol . The Sobolev Space

H k (Ω) is the completion of Sk (Ω) with respect to k · kk,Ω . The closure of Co∞ (Ω) in H k (Ω) is denoted by H0k (Ω).

Fact - There exists unique weak solution y = y(Ω) ∈ H01 (Ω) of the Dirichlet Boundary Value Problem : ) ∆u = 1 on Ω, (I) u = 0 on ∂Ω. Regularity Theorem - Let L be a linear elliptic operator of order 2 and let f ∈ H k (Ω). If u ∈ H01 (Ω) is such that L u = f then u ∈ H k+2(Ω). Further, if f , coefficients of L ∈ C ∞ (Ω) then u ∈ C ∞ (Ω).

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Let V be a C ∞ -vector field on M having compact support. Let {Φt }t∈R be the 1-parameter family of flow for V . ∀ t ∈ R, let Ωt := Φt(Ω), yt := y(Ωt) ( the unique solution of (I) for Ωt ) and

y t := yt ◦ Φt : Ω → R.

————————————————————————————————————– Following [5] of the Euclidean case, we prove the results of shape calculus on (M, g) which are needed for proving the main theorems listed on page 4 : Let Ω′ ⊂ Ω be a relatively compact open set with Ω′ ⊂ Ω. Lemma 0 Let t 7−→ f (t) be a differentiable curve in L2 (Ω) such that f (0) ∈ H 2 (Ω). Then for t sufficiently close to 0, t 7−→ f (t) ◦ Φt is differentiable in L2 (Ω′ ) at t = 0 and     d d = + g (∇f (0), V ) . (f (t) ◦ Φt ) f (t) dt dt |t=0 |t=0

Shape calculus for Dirichlet Boundary Value Problem :
Proposition I.1 The map t 7−→ y t is a C 1 -curve in H 2 (Ω) ∩ H01 (Ω), || · ||H 2 (Ω) from a neighbourhood of 0 in R.
Definition : y(Ω, ˙ V ) := dtd y t |t=0 ∈ H 2 (Ω) ∩ H01 (Ω) is called the material derivative of y = y(Ω) in the direction of V . Definition : y ′(Ω, V ) := y(Ω, ˙ V ) − g(∇y, V ) ∈ H 2 (Ω) is called the shape derivative of y = y(Ω) in the direction of V . Let Ω′ ⊂ Ω be a relatively compact open set with Ω′ ⊂ Ω. Proposition I.2 The map t 7−→ yt |Ω′ is a C 1 -curve in H 1 (Ω′ ) from a neighbourhood of 0 in R and dtd |t=0 (yt |Ω′ ) = y ′ |Ω′ . Proposition I.3 The shape derivative y ′ = y ′(Ω, V ) ∈ H 2(Ω) satisfies the Dirichlet boundary value problem :  ∆v = 0 in L2 (Ω),  ∂y v|∂Ω = − g(V, n).  ∂n (Here, n is the outward unit normal field on ∂Ω). ————————————————————————————————————– 2

Let y1 := y1 (Ω) be the unique solution of the Dirichlet Eigenvalue Problem : ) ∆u = λu on Ω, (II) u=0 on ∂Ω. corresponding to the first eigenvalue λ1 := λ1 (Ω), characterized by Z y1 > 0 on Ω & y12 dV = 1. Ω

Shape calculus for the Dirichlet Eigenvalue Problem : Let (M, g), V , Φt , Ω, Ωt be as before. Let λ1 (t) := λ1 (Ωt) and y1 (t) := y1 (Ωt) be the first eigenvalue and the unique solution of (II) corresponding to this eigenvalue respectively, for Ωt . We denote y1 (Ω) by y1 and λ1 (Ω) by λ1 . Let y1t := y1 (t) ◦ Φt|Ω

(t ∈ R).

Proposition II.1 The map t 7−→ ( λ1 (t) , y1t ) is a C 1 -curve in R×H 2 (Ω)∩H01 (Ω) from a neighbourhood of 0 in R.
Definition : y˙1 (Ω, V ) := dtd y1t |t=0 ∈ H 2 (Ω) ∩ H01 (Ω) is called the material derivative of y1 = y1 (Ω) in the direction of V . Definition : y1′ (Ω, V ) := y˙1 (Ω, V ) − g(∇y1 , V ) ∈ H 2 (Ω) is called the shape derivative of y1 = y1 (Ω) in the direction of V . Let Ω′ ⊂ Ω be a relatively compact open set with Ω′ ⊂ Ω. Proposition II.2 The map t 7−→
y1 (t)|Ω′ is a C 1 -curve in H 1 (Ω′ ) from a neighbourhood of 0 in R and dtd [y1 (t)|Ω′ ] |t=0 = (y˙ 1 − g(∇y1, V )) |Ω′ ∈ H 2 (Ω′ ). Further, 1 g(V, n). y1′ |∂Ω = − ∂y ∂n Proposition II.3

The shape derivative y1′ ∈ H 2 (Ω) satisfies ∆y1′ = λ1 y1′ + λ′1 y1

in L2 (Ω).

¯ y1′ ∈ C ∞ (Ω). 2 Z  ∂y1 ′ g(V, n) dS. Proposition II.5 λ1 = − ∂n ∂Ω ————————————————————————————————————— Proposition II.4

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Fix p ∈ S n (Hn ). Fix an open ball B1 := B(p, r1 ) of radius r1 in S n (Hn ). Let B0 be any open ball of radius r0 such that B0 ⊂ B1 . For S n we consider r1 < π. Let Ω = B1 \ B0 . Let F denote the family of such domains Ω. R We consider the energy functional E(Ω) := Ω ||∇y(Ω)||2 dV on F , associated to the problems (I). Let λ1 (Ω) be the first Dirichlet eigenvalue of Ω ( see problem (II) on page 3). Following [3] of the Euclidean case, using the results of Shape calculus mentioned above and maximum principles for elliptic PDEs (cf. [4]) we prove our following main results : Put Ω0 = B(p, r1 ) \ B(p, r0 ) for any fixed p ∈ S n (Hn ). Theorem 1 The energy functional E(Ω) on F assumes minimum at Ω if and only if Ω = Ω0 , i.e., when the balls are concentric. Theorem 2 The first Dirichlet eigenvalue λ1 (Ω) on F assumes maximum at Ω if and only if Ω = Ω0 , i.e., when the balls are concentric. —————————————————————————————————————

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[2]

Folland G, Introduction to Partial Differential Equations (New Delhi : Prentice-Hall of India Private Limited) (2001).

[3]

Kesavan S, On Two Functionals Connected to the Laplacian in a Class of Doubly Connected Domains, Proceedings of the Royal Society of Edinburgh 133A (2003) 617-624.

[4]

Protter M and Weinberger H, Maximum Principles in Differential Equations (New Jersey : Prentice-Hall Inc.)(1967).

[5]

Sokolowski J and Zolesio J-P, Introduction to Shape Optimization - Shape Sensitivity Analysis, (Springer-Verlag)(1992).

[6]

Todhunter I, Spherical Trigonometry (London : Macmillan and Co. Ltd.)(1949).

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