Optimal coefficients for elliptic PDEs

TL;DR

This paper investigates the optimization of coefficients in elliptic PDEs to improve specific criteria, focusing on elastic compliance and extending to a broader optimal control framework.

Contribution

It introduces methods for determining optimal coefficients in elliptic PDEs, including elastic compliance and general optimal control problems, within a specified admissible class.

Findings

Derived optimal coefficients for elastic compliance problems.

Extended the approach to general optimal control formulations.

Provided theoretical insights into coefficient optimization in elliptic PDEs.

Abstract

We consider an optimization problem related to elliptic PDEs of the form $-{\rm div}(a(x)\nabla u)=f$ with Dirichlet boundary condition on a given domain $\Omega$. The coefficient $a(x)$ has to be determined, in a suitable given class of admissible choices, in order to optimize a given criterion. We first deal with the case when the cost is the so-called elastic compliance, and then we discuss the more general case when the problem is written as an optimal control problem.