Spectral gap for the Cauchy process on convex, symmetric domains
نویسندگان
چکیده
Let D ⊂ R be a bounded convex domain which is symmetric relative to both coordinate axes. Assume that [−a, a] × [−b, b], a ≥ b > 0 is the smallest rectangle (with sides parallel to the coordinate axes) containing D. Let {λn}n=1 be the eigenvalues corresponding to the semigroup of the Cauchy process killed upon exiting D. We obtain the following estimate on the spectral gap: λ2 − λ1 ≥ Cb a2 , where C is an absolute constant. The estimate is obtained by proving new weighted Poincaré inequalities and appealing to the connection between the eigenvalue problem for the Cauchy process and a mixed boundary value problem for the Laplacian in one dimension higher known as the mixed Steklov problem established in [5].
منابع مشابه
Eigenvalue gaps for the Cauchy process and a Poincaré inequality
A connection between the semigroup of the Cauchy process killed upon exiting a domain D and a mixed boundary value problem for the Laplacian in one dimension higher known as the mixed Steklov problem, was established in [6]. From this, a variational characterization for the eigenvalues λn, n ≥ 1, of the Cauchy process in D was obtained. In this paper we obtain a variational characterization of ...
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