The Stekloff Problem for Rotationally Invariant Metrics on the Ball

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1 Revista Colombiana de Matemáticas Volumen 47(2032, páginas 8-90 The Steklo Problem or Rotationally Invariant Metrics on the Ball El problema de Steklo para métricas rotacionalmente invariantes en la bola Óscar Andrés Montaño Carreño Universidad del Valle, Cali, Colombia Abstract. Let (B r, g be a ball o radius r > 0 in R n (n 2 endowed with a rotationally invariant metric ds (sdw 2, where dw 2 represents the standard metric on S n, the (n dimensional unit sphere. Assume that B r has non negative sectional curvature. In this paper we prove that i h(r > 0 is the mean curvature on B r and ν is the irst eigenvalue o the Steklo problem, then ν h(r. Equality ( ν h(r holds only or the standard metric o R n. Key words and phrases. Steklo eigenvalue, Rotationally invariant metric, Nonnegative sectional curvature. 200 Mathematics Subject Classiication. 35P5, 53C20, 53C42, 53C43. Resumen. Sea (B r, g una bola de radio r > 0 en R n (n 2 dotada con una métrica g rotacionalmente invariante ds (sdw 2, donde dw 2 representa la métrica estándar sobre S n, la esera unitaria (n dimensional. Asumamos que B r tiene curvatura seccional no negativa. En este artículo demostramos que si h(r > 0 es la curvatura media sobre B r y ν es el ( primer valor propio del problema de Steklo, entonces ν h(r. La igualdad ν h(r se tiene sólo si g es la métrica estándar de R n. Palabras y rases clave. Valor propio de Steklo, métrica rotacionalmente invariante, curvatura seccional no negativa.. Introduction Let (M n, g be a compact Riemannian maniold with boundary. The Steklo problem is the ollowing: ind a solution or the equation 8

2 82 ÓSCAR ANDRÉS MONTAÑO CARREÑO ϕ 0 in M, ϕ η νϕ on M, ( where ν is a real number. Problem ( or bounded domains in the plane was introduced by Steklo in 902 (see [7]. His motivation came rom physics. The unction ϕ represents a steady state temperature on M such that the lux on the boundary is proportional to the temperature. Problem ( is also important in conductivity and harmonic analysis as it was initially studied by Calderón in []. This is because the set o eigenvalues or the Steklo problem is the same as the set o eigenvalues o the well-known Dirichlet-Neumann map. This map associates to each unction u deined on the boundary M, the normal derivative o the harmonic unction on M with boundary data u. The Steklo problem is also important in conormal geometry in the problem o conormal deormation o a Riemannian metric on maniolds with boundary. The set o eigenvalues consists o an ininite sequence 0 < ν ν 2 ν 3 such that ν i. The irst non zero eigenvalue ν has the ollowing variational characterization, ν min ϕ { M ϕ 2 dυ M ϕ2 dσ : ϕ C ( M, M ϕ dσ 0 }. (2 For convex domains in the plane, Payne (see [6] showed that ν k o, where k o is the minimum value o the curvature on the boundary o the domain. Escobar (see [2] generalized Payne s Theorem (see [6] to maniolds 2-dimensionals with non negative Gaussian curvature. In this case Escobar showed that ν k 0, where k g k 0 and k g represents the geodesic curvature o the boundary. In higher dimensions, n 3, or non negative Ricci curvature maniolds, Escobar shows that ν > 2 k 0, where k 0 is a lower bound or any eigenvalue o the second undamental orm o the boundary. Escobar (see [3] established the ollowing conjecture. Conjecture.. Let (M n, g be a compact Riemannian with boundary and dimension n 3. Assume that Ric(g 0 and that the second undamental orm π satisies π k 0 I on M, k 0 > 0. Then ν k 0. Equality holds only or Euclidean ball o radius k 0. We propose to prove the conjecture or rotationally invariant metrics on the ball. Volumen 47, Número 2, Año 203

3 THE STEKLOFF PROBLEM ON THE BALL Preliminaries Throughout this paper B r will be the n dimensional ball o radius r > 0 parametrized by X(s, w sy (w, 0 s r, (3 where Y (w is a standard parametrization o the unit sphere (n dimensional, S n, given by Y (w Y (w,..., w n (sin w n sin w n 2 sin w,..., sin w n cos w n 2, cos w n. (B r, g will be the ball endowed with a rotationally invariant metric, i.e., such that in the parametrization (3 has the orm ds (sdw 2, where dw 2 represents the standard metric on S n, with (0 0, (0 and (s > 0 or 0 < s r. I D is the Levi-Civita connection associated to the metric g, and X s (s, w Y (w, X i (s, w w i X(s, w sy i (w, i,..., n, are the coordinate ields corresponding to the parametrization (3, it is easy to veriy the ollowing identities: D Xs X s 0. (4 D Xi X s X i. (5 D Xn X n X s. (6 D Xn 2 X n 2 sin 2 w n X s sin w n cos w n X n. (7 D Xn X n 2 cos w n sin w n X n 2. (8 All calculations depend on the deinition o the metric and the relation between the metric and the connection. To coordinate ields, this relation is given by g ( D Xi X j, X k 2 X jg ( X i, X k + 2 X ig ( X j, X k 2 X kg ( X i, X j. (9 These identities are necessary to calculate the mean curvature and the sectional curvatures. As an example we show the identity (8. and On the one hand g ( D Xn X n 2, X s 0, g ( D Xn X n 2, X i 0, i,..., n 3. Revista Colombiana de Matemáticas

4 84 ÓSCAR ANDRÉS MONTAÑO CARREÑO On the other hand and g ( D Xn X n 2, X n 2 X n 2g ( X n, X n 2 X n 2 2 (s 0, g ( D Xn X n 2, X n 2 2 X n g ( X n 2, X n 2 From the above we conclude that D Xn X n 2 2 X n 2 sin 2 w n 2 sin w n cos w n. 3. Curvatures cos wn sin w n X n 2. Let X s (r, w Y (w be the outward normal vector ield to the boundary o B r, B r. The identity (5 implies that g ( ( D Xi X s, X i g X i, X i ( g ( X i, X i. Hence, the mean curvature h(r is given by h(r n g ( g ( (r X i, X i DXi X s, X i n (r. (0 i Proposition 3.. The sectional curvature K ( X i, X s, i,..., n is given by K ( X i, X s (s (s. ( Proo. From (5, g ( D Xi X s, X i dw 2( Y i, Y i. Dierentiating with respect to X s we get g ( ( ( D Xs D Xi X s, X i g DXi X s, D Xi X s + 2 dw 2 ( Y i, Y i + dw 2( Y i, Y i ( g X i, X i + ( 2 dw 2 ( Y i, Y i + dw 2( Y i, Y i dw 2( Y i, Y i. From (4, then g ( D Xi D Xs X s, X i g ( DXi 0, X i 0, Volumen 47, Número 2, Año 203

5 THE STEKLOFF PROBLEM ON THE BALL 85 K ( { ( ( } X i, X s g DXi D Xs X s, X i g DXs D Xi X s, X i g(x i, X i. Proposition 3.2. The sectional curvature K ( X n, X n 2 is given by K ( ( (s 2 X n, X n 2 2. (2 (s Proo. From identities (6,(7 and (8, we get and g ( D Xn X n, D Xn 2 X n 2 ( 2 sin 2 w n, From equation (9 it ollows that g ( D Xn X n 2, D Xn X n 2 2 cos 2 w n. g ( D Xn 2 X n 2, X n 2 X n g ( X n 2, X n 2 2 sin w n cos w n. Dierentiating with respect to X n we get g ( D Xn D Xn 2 X n 2, X n Thereore g ( D Xn 2 X n 2, D Xn X n + 2 ( sin 2 w n cos 2 w n. g(d Xn D Xn 2 X n 2, X n On the other hand, ( 2 sin 2 w n + 2 (sin 2 w n cos 2 w n. (3 g ( D Xn X n 2, X n 2 X n 2g ( X n, X n 0. Dierentiating this equation with respect to X n 2 we get g(d Xn 2 D Xn X n 2, X n g(d Xn X n 2, D Xn X n 2 2 cos 2 w n. (4 From equations (3 and (4 the proposition ollows. Revista Colombiana de Matemáticas

6 86 ÓSCAR ANDRÉS MONTAÑO CARREÑO 4. The First Nonconstant Eigenunction or the Steklo Problem on B r In the ollowing theorem Escobar characterized the irst eigenunction o a geodesic ball which has a rotationally invariant metric (see [4]. Theorem 4.. Let B r be a ball in R n endowed with a rotationally invariant metric ds (sdw 2, where dw 2 represents the standard metric on S n, with (0 0, (0 and (s > 0 or 0 < s r. The irst non constant eigenunction or the Steklo problem on B r has the orm ϕ(s, w ψ(se(w, (5 where e(w satisies the equation e + (n e 0 on S n and the unction ψ satisies the dierential equation d ( n n (s dds (s ds ψ(s (n ψ(s 2 0 in (0, r, (6 (s with the conditions ψ (r ν ψ(r, ψ(0 0. (7 Proo. We use separation o variables and observe that the space L 2 (B r is equal to the space L 2 (0, r L 2 (S n. Let {e i }, i 0,, 2,..., be a complete orthogonal set o eigenunctions or the Laplacian on S n with associated eigenvalues λ i such that 0 λ 0 < λ λ 2 λ 3. For i, let ψ i be the unction satisying ( d n n (s d (s ds ds ψ i(s (n ψ i(s 2 0 in (0, r, (s ψ i (r β iψ i (r, ψ i (0 0. Let u 0 and u i ψ i (se i (w or i, 2,.... The set {u i } or i 0,, 2,... orms an orthogonal basis or L 2 (B r. Recall that the irst non-zero Steklo eigenvalue has the variational characterization ϕ 2 B ν min r dv B r ϕ 2 dσ. Since or i β i Br ϕdσ0 B r u i 2 n ds dw B r u 2 i n dw ( d ds ψ 2 i n r ds + λ i 0 (ψ i 2 n 3 ds r 0 ψ 2 i (r n (r Volumen 47, Número 2, Año 203

7 THE STEKLOFF PROBLEM ON THE BALL 87 and λ i λ n, we get that β i β. Because the competing unctions in the variational characterization o ν are orthogonal to the constant unctions on B r, we easily ind that ν β. Using the ormula g ϕ 2 ϕ s + (n ϕ 2 s + ϕ, where is the 2 standard Laplacian on S n, the equation (6 ollows. s When n 2, the unction ψ has the orm ψ(s ce du (u or c constant. The irst eigenvalue and the mean curvature are given by ν ψ (r ψ(r (r and h(r (r (r. From this we observe: Remark 4.2. When (s s + s 3 or (s sinh(s (the hyperbolic space with curvature since (r > then ν < h(r. Thereore or n 2, the condition that B r has non-negative sectional curvature is necessary. Remark 4.3. From Proposition 3., the condition o non negative sectional curvature implies that (s 0, and thereore is decreasing. Since (0, then (r. Hence, or n 2 the condition o non negative sectional curvature implies ν h(r. As examples o these metrics we have (s s (standard metric, (s sin(s (constant sectional curvature equal to and (s s s Main Theorem Theorem 5.. Let (B r, g be a ball in R n (n 3 endowed with a rotationally invariant metric. Assume that B r has non-negative sectional curvature and mean curvature on B r, h(r > 0. Then the irst non-zero eigenvalue o the Steklo problem ν satisies ν h(r. Equality holds only or the standard metric o R n. Proo. The coordinate unctions are eigenunctions o the Laplacian on S n. From the equation (5 it ollows that ϕ(s, w ψ(s cos w n is an eigenunction associated to the irst eigenvalue ν. Consider the unction F 2 ϕ 2. Since ϕ is a harmonic unction and Ric ( ϕ, ϕ 0, the Weizenböck ormula (see [5] F Hess(ϕ 2 ( + g ϕ, ( ϕ + Ric( ϕ, ϕ implies that F 0, and hence F is a subharmonic unction. Thereore, the maximum o F is achieved at some point P (r, θ B r. Hop s Maximum Principle implies that F s (r, θ > 0 or F is constant. Since F (s, w {( 2 ( 2 } ϕ ϕ s w n (ψ 2{ ( 2 2 ψ cos 2 w n + sin 2 w n } Revista Colombiana de Matemáticas

8 88 ÓSCAR ANDRÉS MONTAÑO CARREÑO and F is a non-constant unction, then Evaluating F s (r, θ ψ ψ cos 2 θ n + ψ F w n (s, w at the point P we ind that F w n (r, θ The equation (9 implies that or or I ( ψ sin 2 θ n > 0. (8 ( (ψ 2 ( 2 ψ sin θ n cos θ n 0. (9 ( 2 ψ(r ( ψ (r 2 0, (r sin θ n 0 and cos 2 θ n, sin 2 θ n and cos θ n 0. ( 2 ψ(r ( ψ (r 2 0, (r given that ψ(r 0 (ψ(r 0 implies ϕ 0 on B r and thus, ϕ is a constant unction on B r which is a contradiction, it ollow rom (7 that (ν 2 ( ψ 2 ( 2 (r. (20 ψ(r (r The condition h(r > 0 and (0 implies that (r > 0. Since B r has non negative sectional curvature then (2 implies that ( 2. Then ( 2 (r From (20 and (2 it ollows that ν h(r. Equality holds only or (r. I ( 2 (r ( h(r 2. (2 (r sin θ n 0 and cos 2 θ n, then F ( ( r, θ,..., θ n 2, θ n F r, θ,..., θ n 2, π { (ψ ( } 2 ψ 0 Volumen 47, Número 2, Año 203

9 thus (ν 2 THE STEKLOFF PROBLEM ON THE BALL 89 ( ψ 2 ( 2 (r ψ(r (r Equality holds only or (r. I rom (8 we have since Thus ( ψ The inequality is strict. ( 2 (r ( h(r 2. (r sin 2 θ n and cos θ n 0, ψ F s (P > 0 ( ν ψ ψ 2 ( ψ > 0. ( 2 ψ ( ν h(r > 0. In any case we conclude that ν h(r. I equality is attained then (r. Since the sectional curvature is non-negative, then ( implies that (s 0. (0 (r and (s 0 implies. Since (0 0, then (s s. Consequently g is the standard metric on R n. Reerences [] A. P. Calderón, On an Inverse Boundary Value Problem, Seminar in Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, Brazil, Soc. Brasileira de Matemática, 980, pp [2] J. F. Escobar, The Geometry o the First Non-Zero Steklo Eigenvalue, Journal o Functional Analysis 50 (997, [3], An Isoperimetric Inequality and the First Steklov Eigenvalue, Journal o Functional Analysis 65 (999, 0 6. [4], A Comparison Theorem or the First Non-Zero Steklov Eigenvalue, Journal o Functional Analysis 78 (2000, [5], Topics in PDE s and Dierential Geometry, XII Escola de Geometria Dierencial, Goiania/Ed. da UFG, [6] L. E. Payne, Some Isoperimetric Inequalities or Harmonic Functions, SIAM J. Math. Anal. (970, Revista Colombiana de Matemáticas

10 90 ÓSCAR ANDRÉS MONTAÑO CARREÑO [7] M. W. Steklo, Sur les problemes ondamentaux de la physique mathematique, Ann. Sci. École Norm 9 (902, (r. (Recibido en mayo de 202. Aceptado en octubre de 203 Departamento de Matemáticas Universidad del Valle Facultad de Ciencias Carrera 00, calle 3 Cali, Colombia [email protected] Volumen 47, Número 2, Año 203