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On the Ashbaugh-Benguria conjecture about lower-order Dirichlet eigenvalues of the Laplacian. (English) Zbl 1487.35261

Suppose \(\Omega \subset \mathbb{R}^n\) is bounded and \(u\) solves \[ \begin{cases} \Delta u = -\lambda u & \text{ in }\Omega,\\ u = 0 &\text{ on }\partial \Omega. \end{cases} \] The spectrum generated via the PDE is real and discrete and \[ 0<\lambda_1<\lambda_2\le \lambda_3 \ldots \rightarrow \infty \] with the convention that the eigenvalues are repeated with multiplicities. If \(\Omega=B_1\), \[ \lambda_1(B_1)=j_{\frac{n}{2}-1, 1}^2 \] \[ \lambda_2(B_1)=\lambda_3(B_1)=\ldots=\lambda_{n+1}(B_1)= j_{\frac{n}{2}, 1}^2 \] where \(j_{p,k}\) represents the \(k^{th}\) positive root of the Bessel function \(J_p(x)\) of the first kind with order \(p\). In the late \(19^{th}\) century Lord Rayleigh conjectured that the first eigenvalue under a fixed mass constraint is minimized by the ball and this is the Faber-Krahn inequality which was independently proved by Faber and Krahn in the 1920s: \[ \lambda_1(\Omega) \ge \Big (\frac{|B_1|}{|\Omega|} \Big)^{\frac{2}{n}} j_{\frac{n}{2}-1, 1}^2 \] with equality if and only if \(\Omega\) is a ball. Ashbaugh and Benguria conjectured in 1993 that \[ \frac{\lambda_1}{\lambda_2-\lambda_1}+ \dots +\frac{\lambda_1}{\lambda_{n+1}-\lambda_1} \ge \frac{n}{\big(\frac{j_{\frac{n}{2}, 1}}{j_{\frac{n}{2}-1, 1}} \big)^2-1}, \] where equality holds if and only if \(\Omega\) is a ball.
In the article, Wang and Xia prove: if \(\Omega\) is bounded with a smooth boundary, \[ \frac{\lambda_1}{\lambda_2-\lambda_1}+ \dots +\frac{\lambda_1}{\lambda_{n}-\lambda_1} \ge \frac{n-1}{\big(\frac{j_{\frac{n}{2}, 1}}{j_{\frac{n}{2}-1, 1}} \big)^2-1}; \] also, the authors identify that equality holds if and only if \(\Omega\) is a ball. The method contains the variational representation of the eigenvalues, the identification of a convenient basis, and a suitable test function constructed via \[ \frac{J_{\frac{n}{2}}(\beta t)}{J_{\frac{n}{2}-1}(\alpha t)}. \] A differential inequality in a paper by M. S. Ashbaugh and R. D. Benguria [Ann. Math. (2) 135, No. 3, 601–628 (1992; Zbl 0757.35052)] enters the proof in a central way. Moreover, Wang and Xia generalize the main result to linear operators which satisfy some assumptions.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
58C40 Spectral theory; eigenvalue problems on manifolds

Citations:

Zbl 0757.35052
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References:

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