Optimal Design in Two-Dimensional Conductivity for a General Cost Depending on the Field

ISSN： 0003-9527

Source： Archive for Rational Mechanics and Analysis, Vol.182, Iss.3, 2006-11, pp. : 367-385

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Abstract

We explore an optimal design problem in two-dimensional conductivity for a rather general cost depending on the underlying field. Through a typical variational reformulation that has been explored recently, we provide a simplified relaxed version which is amenable to numerical simulation, and prove that it is a true relaxation under a main structural hypothesis. Several important cases are covered including a linear cost in the gradient and a convex, isotropic functional (in particular, the p</i>th power of the field for any </equationsource> ). For the isotropic, non-quadratic case, our computations do not require an explicit form of the constrained quasiconvexification of the equivalent vector variational problem. That structural assumption ties together the underlying state equation and the integral cost.