A shape optimization problem for the first mixed Steklov-Dirichlet eigenvalue

Abstract

We consider a shape optimization problem for the first mixed Steklov-Dirichlet eigenvalues of domains bounded by two balls in two-point homogeneous space. We give a geometric proof which is motivated by Newton's shell theorem.

Shape monotonicity of the first Steklov-Dirichlet eigenvalue on eccentric annuli

“…It is then proved by Ftouhi [20] that the first Steklov eigenvalue on eccentric annuli attains the maximum at the concentric case. For the Steklov-Dirichlet eigenvalue problem, as stated previously, the maximality at the concentric annulus was shown by Santhanam and Verma [39], Seo [41], and Ftouhi [20]. However, to the best of our knowledge, the monotonicity is not known either for the Steklov or for the Steklov-Dirichlet eigenvalue problems on eccentric annuli.…”

“…Proof. We have the uniform boundedness for R n (t) thanks to | tanh s s | ≤ 1 and σ t 1 ≤ σ 0 1 ; the maximality of σ t 1 at t = 0 was verified in [41,20]. In other words, there exist a positive constant C independent of ε and n such that…”

“…We denote the first eigenvalue by σ t 1 . The first eigenvalue σ t 1 attains the maximum at t = 0, the concentric case, as shown in [20,39,41]; the maximal value is (r 2 (ln r 2 − ln r 1 )) −1 in two dimensions and r 1 r 2 (r 2 −r 1 ) in three dimensions. Beyond this, we ask the following:…”

“…Recently, Santhanam and Verma considered the Steklov-Dirichlet eigenvalue on eccentric annuli in R n , n > 2, with the zero Dirichlet condition on the inner boundary and showed that the first eigenvalue attains the maximum when the annulus is concentric [39]. Seo extended this maximality to some two-point homogeneous spaces including R 2 [41] (see also the work by Ftouhi [20]).…”