Variational problems on classes of convex domains

TL;DR

This paper establishes the existence of minimizers for various functionals over convex domains within a bounded set, with applications to eigenvalue problems, elliptic equations, and the Newton minimal resistance problem.

Contribution

It proves the existence of minimizers for functionals on convex domains with fixed volume, extending to higher-order elliptic operators and diverse applications.

Findings

Existence of minimizers for eigenvalues of differential operators.

Minimization of integral functionals depending on elliptic solutions.

Application to the Newton problem of minimal resistance.

Abstract

We prove the existence of minimizers for functionals defined over the class of convex domains contained inside a bounded set D of R^N and with prescribed volume. Some applications are given, in particular we prove that the eigenvalues of differential operators of second and fourth order with non-constant coefficients as well as integral functionals depending on the solution of an elliptic equation can be minimized over this class of domains. Another application of this result is related to the famous Newton problem of minimal resistance. In general, all the results we shall develop hold for elliptic operators of any even order larger that 0.