On existence of optimal potentials on unbounded domains

TL;DR

This paper investigates the existence of optimal potentials for elliptic Schrödinger equations on unbounded domains, extending shape optimization to potential optimization, and provides numerical simulations.

Contribution

It proves an existence theorem for optimal potentials in unbounded domains, addressing compactness issues and extending shape optimization concepts.

Findings

Existence of optimal potentials established.

Overcomes compactness challenges in unbounded domains.

Includes numerical simulations demonstrating theoretical results.

Abstract

We consider elliptic equations of Schr\"odinger 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 uV , 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 Rd, which implies a lack of compactness in several crucial points. In the last section we present some numerical simulations.