# Eigenvalue estimates involving Bessel functions

#### 15.04.2021, 16:15  –  Online-Seminar Forschungsseminar Differentialgeometrie

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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