solidot新版网站常见问题,请点击这里查看。
消息
本文已被查看1737次
Existence and uniqueness of solutions of Schr\"odinger type stationary equations with very singular potentials without prescribing boundary conditions and some applications. (arXiv:1710.06679v1 [math.
来源于:arXiv
Motivated mainly by the localization over an open bounded set $\Omega$ of
$\mathbb R^n$ of solutions of the Schr\"odinger equations, we consider the
Schr\"odinger equation over $\Omega$ with a very singular potential $V(x) \ge C
d (x, \partial \Omega)^{-r}$ with $r\ge 2$ and a convective flow $\vec U$. We
prove the existence and uniqueness of a very weak solution of the equation,
when the right hand side datum $f(x)$ is in $L^1 (\Omega, d(\cdot, \partial
\Omega))$, even if no boundary condition is a priori prescribed. We prove that,
in fact, the solution necessarily satisfies (in a suitable way) the Dirichlet
condition $u = 0$ on $\partial \Omega$. These results improve some of the
results of the previous paper by the authors in collaboration with Roger Temam.
In addition, we prove some new results dealing with the $m$-accretivity in $L^1
(\Omega, d(\cdot, \partial \Omega)^ \alpha)$, where $\alpha \in [0,1]$, of the
associated operator, the corresponding parabolic problem and the study 查看全文>>