A local Faber-Krahn inequality and applications to Schrodinger equations

作者：Lierl Janna; Steinerberger Stefan^*

来源：Communications in Partial Differential Equations, 2018, 43(1): 66-81.

DOI：10.1080/03605302.2017.1423330

• Faber-Krahn inequality
• Feynman-Kac formula
• Schrodinger operator
• Primary: 35J10
• 46E35 Secondary: 47D08

We prove a local Faber-Krahn inequality for solutions u to the Dirichlet problem for Delta + V on an arbitrary domain Omega in R-n. Suppose a solution u assumes a global maximum at some point x(0) is an element of Omega and u(x(0)) > 0. Let T(x(0)) be the smallest time at which a Brownian motion, started at x(0), has exited the domain Omega with probability >= 1/2. For nice (e.g., convex) domains, T(x(0)) asymptotic to d(x(0,) partial derivative Omega)(2) but we make no assumption on the geometry of the domain. Our main result is that there exists a ball B of radius asymptotic to T(x(0))(1/2) such that
parallel to V parallel to L-2(n,1) ((Omega boolean AND B)) >= c(n) > 0,
provided that n >= 3. In the case n = 2, the above estimate fails and we obtain a substitute result. The Laplacian may be replaced by a uniformly elliptic operator in divergence form. This result both unifies and strenghtens a series of earlier results.

• 出版日期2018