Uniform boundedness of conditional gauge and Schrödinger equations
Uniform boundedness of conditional gauge and Schrödinger equations
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条件规范和薛定谔方程的一致有界性
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发表时间：
1984
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通讯作者：
Zhongxin Zhao
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作者：
Zhongxin Zhao
AbstractWe prove that for a bounded domainD ⊂Rn withC2 boundary and
$$q in K_n^{loc} (n geqq 3) if E^x exp intlimits_0^{ au _D } {q(x_t )dt} mathop ddag limits_--- infty $$
inD, then
$$mathop {sup }limits_{mathop {x in D}limits_{z in partial D} } E_z^x exp intlimits_0^{ au _D } {q(x_t )dt}< + infty $$
({xt: Brownian motion})The important corollary of this result is that if the Schrödinger equation Δ/2u+qu=0 has a strictly positive solution onD, then for anyD0 ⊂ ⊂D, there exists a constantC=C(n,q,D,D0) such that for anyf εL1(∂D, σ), (σ: area measure on ∂D) we have
$$mathop {sup |}limits_{x in D_0 } u_f (x)| mathop< limits_ = Cintlimits_{partial D} {|f(y)|sigma (dy)} $$
whereuf is the solution of the Schrödinger equation corresponding to the boundary valuef.To prove the main result we set up the following estimate inequalities on the Poisson kernelK(x,z) corresponding to the Laplace operator:
$$C_1 frac{{d(x,partial D)}}{{|x - z|^n }}mathop< limits_ = K(x,z)mathop< limits_ = C_2 frac{{d(x,partial D)}}{{|x - z|^n }},x in D,z in partial D$$
whereC1 andC2 are constants depending onn andD.
AbstractWe prove that for a bounded domainD ⊂Rn withC2 boundary and
$$q in K_n^{loc} (n geqq 3) if E^x exp intlimits_0^{ au _D } {q(x_t )dt} mathop ddag limits_--- infty $$
inD, then
$$mathop {sup }limits_{mathop {x in D}limits_{z in partial D} } E_z^x exp intlimits_0^{ au _D } {q(x_t )dt}< + infty $$
({xt: Brownian motion})The important corollary of this result is that if the Schrödinger equation Δ/2u+qu=0 has a strictly positive solution onD, then for anyD0 ⊂ ⊂D, there exists a constantC=C(n,q,D,D0) such that for anyf εL1(∂D, σ), (σ: area measure on ∂D) we have
$$mathop {sup |}limits_{x in D_0 } u_f (x)| mathop< limits_ = Cintlimits_{partial D} {|f(y)|sigma (dy)} $$
whereuf is the solution of the Schrödinger equation corresponding to the boundary valuef.To prove the main result we set up the following estimate inequalities on the Poisson kernelK(x,z) corresponding to the Laplace operator:
$$C_1 frac{{d(x,partial D)}}{{|x - z|^n }}mathop< limits_ = K(x,z)mathop< limits_ = C_2 frac{{d(x,partial D)}}{{|x - z|^n }},x in D,z in partial D$$
whereC1 andC2 are constants depending onn andD.