Existence and uniqueness of solutions of Schr\"odinger type stationary equations with very singular potentials without prescribing boundary conditions and some applications
Existence and uniqueness of solutions of Schr\"odinger type stationary equations with very singular potentials without prescribing boundary conditions and some applications
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极奇异位薛定谔型无边界条件平稳方程解的存在唯一性及应用
DOI：
10.7153/dea-2018-10-04
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
J. Rakotoson
中科院分区：
文献类型：
--
作者：
J. Díaz;D. Gómez;J. Rakotoson
基于Schr\“odinger方程解在开有界集$\Omega$上的局部化性质，我们考虑了$\Omega$上的Schr\“odinger方程，其中位势为非常奇异的位势$V（x）\geCd（x，\partial \Omega）^{-r}$，r\ge 2$，对流为$\vec U$.本文证明了当右端数据f（x）$在L^1（\Omega，d（\cdot，\partial \Omega））$中时，即使没有预先给定边界条件，方程的一个很弱解的存在性和唯一性.我们证明，事实上，解决方案必须满足（以适当的方式）狄利克雷条件$u = 0$上$\partial \欧米茄$。这些结果改进了作者与Roger Temam合作的前一篇论文的一些结果。此外，我们还证明了$L ^1（\Omega，d（\cdot，\partial \Omega）^ \alpha）$中伴随算子的$m$-增生性，相应的抛物问题以及$\mathbb R^n$中的复发展Schr\“odinger方程的研究的一些新结果.
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 of the complex evolution Schr\"odinger equation in $\mathbb R^n$.