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We show how to capture the gradient concentration of the solutions of Dirichlet-type problems subjected to large sources of order ${1\over \sqrt \varepsilon}$ concentrated on an ε-neighborhood of a hypersurface of the domain. To this end we define the gradient Young-concentration measures generated by sequences of finite energy and establish a very simple characterization of these measures.

We examine how the use of typical techniques from non-convex vector variational problems can help in understanding optimal design problems in conductivity. After describing the main ideas of the underlying analysis and providing some standard material in an attempt to make the exposition self-contained, we show how those ideas apply to a typical optimal desing problem with two different conducting materials. Then we examine the equivalent relaxed formulation to end up with a new problem whose numerical simulation leads to approximated optimal configurations. We include several such simulations in 2d and 3d.

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A characterization of gradient Young-concentration measures generated by solutions of Dirichlet-type problems with large sources

Vector variational problems and applications to optimal design

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A characterization of gradient Young-concentration measures generated by solutions of Dirichlet-type problems with large sources

Vector variational problems and applications to optimal design