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Uniform boundedness of conditional gauge and Schrödinger equations. (English) Zbl 0545.35087

We prove that for a bounded domain \(D\subset R^ n\) with \(C^ 2\) boundary and \(q\in K_ n^{loc}(n\geq 3)\) if \(E^ x\quad \exp \int^{\tau_ D}_{0}q(x_ t)dt\not\equiv \infty\) in D, then \(\sup_{x\in D,\quad z\in \partial D}\quad E^ x_ z\quad \exp \int^{\tau_ D}_{0}q(x_ t)dt<+\infty (\{x_ t\}:\) Brownian motion). The important corollary of this result is that if the Schrödinger equation \(\Delta u/2+qu=0\) has a strictly positive solution on D, then for any \(D_ 0\subset \subset D\), there exists a constant \(C=C(n,q,D,D_ 0)\) such that for any \(f\in L^ 1(\partial D,\sigma)\), (\(\sigma\) : area measure on \(\partial D)\) we have \[ \sup_{x\in D_ 0} | u_ f(x)| \leq C\int_{\partial D}| f(y)| \sigma(dy), \] where \(u_ f\) is the solution of the Schrödinger equation corresponding to the boundary value f. To prove the main result we set up the following estimate inequalities on the Poisson kernel K(x,z) corresponding to the Laplace operator: \[ C_ 1d(x,\partial D)/| x- z|^ n\leq K(x,z)\leq C_ 2d(x,\partial D)/| x-z|^ n,\quad x\in D,\quad u\in \partial D, \] where \(C_ 1\) and \(C_ 2\) are constants depending on n and D.

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35J10 Schrödinger operator, Schrödinger equation
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