A general lower bound for the relaxation of an optimal design problem with a general quadratic cost functional and a general linear state equation
2012 - U. Fidalgo, P. Pedregal
Convex Anal. 19, 281-294 (2012)
authors IMACI
Abstract. We examine an optimal design problem in conductivity where anisotropy and/or non-ellipticity is a main ingredient both in the state law and the cost functional, which is quadratic in the gradient. We are able to provide a 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. We also explore when this lower bound may turn out to be exact, and formulate a conjecture for the underlying relaxed problem.