A general lower bound for the relaxation of an optimal design problem with a general quadratic cost functional and a general linear state equation

摘要

Recently, several particular problems in optimal design have been analyzed by using tools from non-convex, variational problems. As many of those have similarities, but also different features, we pretend to look at a full family of problems that includes most of those particular situations. Specifically, we examine an optimal design problem where anisotropy and/or non-ellipticity is permitted both in the state law, and the cost functional, which is quadratic in the gradient. In this generality, we are able to provide a general lower bound for the relaxed integrand (effective behavior) which is valid in all of these situations. Our philosophy, which has been introduced and implemented in simpler situations, leads to an elementary semi-definite mathematical programming problem for matrices depending on various parameters, that are precisely the variables for the relaxed problem. We also explore when this lower bound may turn out to be exact, and formulate a conjecture for the underlying relaxed problem.