SIAM Journal on Mathematical Analysis, Volume 53, Issue 1, Page 1088-1121, January 2021.
We consider elliptic equations of Schrödinger type with a right-hand side fixed and with the linear part of order zero given by a potential $V$. The main goal is to study the optimization problem for an integral cost depending on the solution $u_V$, when $V$ varies in a suitable class of admissible potentials. These problems can be seen as the natural extension of shape optimization problems to the framework of potentials. The main result is an existence theorem for optimal potentials, and the main difficulty is to work in the whole Euclidean space $\mathbb{R}^d$, which implies a lack of compactness in several crucial points. In the last section we present some numerical simulations.

中文翻译：

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无界域上最优势的存在

SIAM数学分析杂志，第53卷，第1期，第1088-1121页，2021年1月。
我们考虑Schrödinger类型的椭圆方程，其右侧固定且零阶线性部分由潜在的$ V $给出。当$ V $在合适的可允许电位类别中变化时，主要目标是研究取决于解决方案$ u_V $的积分成本的优化问题。这些问题可以看作是形状优化问题在势能框架中的自然延伸。主要结果是最优势的存在性定理，主要困难是在整个欧氏空间$ \ mathbb {R} ^ d $中工作，这意味着在几个关键点上缺乏紧凑性。在最后一节中，我们介绍了一些数值模拟。