A mixing effect induced by sources concentrated in a soft junction and the gradient concentration phenomenon

We show that the variational limit of a ε-soft and thin junction problem (𝒫_ε) with sources concentrated in the junction gives rise to a surface energy mixing the internal energy and sources. The surface energy functional possesses an integral representation with respect to the gradient Young-concentration measures generated by sequences (ū_ε)_ε>0 of minimizers of (𝒫_ε).