Analysis Seminar

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Boundary value problems for Schrodinger equations with strongly singular potentials

Speaker: Moshe Marcus, Technion

Location: Warren Weaver Hall 1302

Date: Thursday, October 4, 2018, 11 a.m.

Synopsis:

We consider Schrodinger operators of the form $L^V=\Delta +V$ in domains $\Omega$ in $R^N$ where $V =  \mu/\delta_F(x)^2$, $F$ a compact subset of the boundary of 

$\Omega$, $\delta_F(x)=\dist (x,F)$. Finally  $\mu$ is a constant strictly smaller than the Hardy constant relative to $V$ in $\Omega$ (denoted by $C_H(V)$).

If $\Omega$ is a bounded Lipschitz domain $c_H(V)>0$ and the condition $\mu<c_H(V)$ implies that $L^V$ is weakly coercive in the sense of Ancona (Ann. Math. 1988). 

Therefore $L^V$ possesses a Green kernel and a positive eigenfunction and the Boundary Harnack Principle is applicable.

In this talk we introduce a notion of normalized boundary trace and - under various restrictions on $F$ -  discuss existence, uniqueness and  a-priori estimates of solutions of boundary value problems 

for $L^V$ where the prescribed boundary data is in the sense of the normalized boundary trace. Applications to some related nonlinear problems will also be discussed.  

Recently derived  sharp estimates of Green and Martin kernels of $L^V$ play a crucial role in this discussion.