Georges Habib (Lebanese University Beirut)
Given a compact Riemannian manifold \((M^n ,g)\) with smooth boundary \(\partial M\), we give an estimate for the quotient \(\frac{\int_{\partial M} f dv_g}{\int_M f dv_g}\) in terms of the Bessel functions. Here \(f\) is a smooth positive function on \(M\) that satisfies some inequality involving the scalar Laplacian. As an application, estimates of type Faber-Krahn are given for the Laplacian with Dirichlet and Robin boundary conditions. Also a new estimate is established for the eigenvalues of the Dirac operator in terms of the zeros of the Bessel functions. This is a joint work with Fida El Chami and Nicolas Ginoux.
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