论文标题
统一域中的Schrödinger操作员的积极解决方案和谐波测量
Positive solutions and harmonic measure for Schrödinger operators in uniform domains
论文作者
论文摘要
我们对方程\ begin {equation*} \ left \ {\ oken {arigned} - \ triangle u＆=ωu\，\，\＆\ mbox {in} \，\，\ partialω，\ end {对齐} \ right。 \ end {equation*}在一个有限的统一域$ω\ subset {\ bf r}^n $中，其中$ω$是$ω$中的本地有限的borel量度，而$ f \ ge 0 $相对于Harmonic Matues $ d H^{x} $ on $ \ partial partial partial g partial $。 我们还提供了足够且匹配的必要条件，以实现$ m^{*}（mω）（mω）（z）（z）= \int_Ωm（x，z）m（x，x）\，dΩ（x）\，dmom（x）$ on $ \ \partialΩ$与$ f \ f \ f \ f \ f \ f \ f \ f \ f \ x_0} $ cd，x_0} $ cd， $ x_0 \ inω，m（x）= \ min（1，g（x，x_0））$，$ g $是绿色的函数。 这些结果为与$ω$上的Schrödingeroperator $ - \ triangle -ω$相关的谐波度量的双边界限提供了双边界限，在$ F = 1 $的情况下，这是量规函数存在的标准。给出了在梯度中具有二次增长的riccati类型椭圆方程的应用。
We give bilateral pointwise estimates for positive solutions of the equation \begin{equation*} \left\{ \begin{aligned} -\triangle u & = ωu \, \,& & \mbox{in} \, \, Ω, \quad u \ge 0, \\ u & = f \, \, & &\mbox{on} \, \, \partial Ω, \end{aligned} \right. \end{equation*} in a bounded uniform domain $Ω\subset {\bf R}^n$, where $ω$ is a locally finite Borel measure in $Ω$, and $f\ge 0$ is integrable with respect to harmonic measure $d H^{x}$ on $\partialΩ$. We also give sufficient and matching necessary conditions for the existence of a positive solution in terms of the exponential integrability of $M^{*} (m ω)(z)=\int_ΩM(x, z) m(x)\, d ω(x)$ on $\partialΩ$ with respect to $f \, d H^{x_0}$, where $M(x, \cdot)$ is Martin's function with pole at $x_0\in Ω, m(x)=\min (1, G(x, x_0))$, and $G$ is Green's function. These results give bilateral bounds for the harmonic measure associated with the Schrödinger operator $-\triangle - ω$ on $Ω$, and in the case $f=1$, a criterion for the existence of the gauge function. Applications to elliptic equations of Riccati type with quadratic growth in the gradient are given.
