Energies with respect to a measure and applications to low dimensional structures

Abstract.We consider functionals of the form $$F(u)=\int f(x,Du)\,d\mu\qquad\big(u\in\mathbb{D}(\vec R^n)\big)$$ where $\mu$ is a finite Borel measure on $\vec R^n$, and we characterize their relaxation $\overline{F}$ with respect to the weak convergence in a suitable Sobolev space $W^{1,p}_\mu$. Applications to low dimensional structures and junctions are given.