In this paper we study the relaxation with respect to the L1 norm of integral functionals of the type F(u), where F(u) is the integral on a bounded open set of Rn Omega of an integrand f(x, u,∇u) and u ∈ W1,1(Omega; Sd−1), where Sd−1 denotes the unit sphere in Rd, n and d being any positive integers, and f satisfies linear growth conditions in the gradient variable. In analogy with the unconstrained case, we show that if, in addition, f is quasiconvex in the gradient variable and satisfies some technical continuity hypotheses, then the relaxed functional admits an integral representation on BV (Omega; Sd−1), via a suface energy density K on the jump set S(u), defined by a suitable Dirichlet-type problem (for the complete integral formula look at the abstract in the pdf of the paper)
Relaxation in BV of integral functionals defined on Sobolev functions with values in the unit sphere
ALICANDRO, Roberto;CORBO ESPOSITO, Antonio;
2007-01-01
Abstract
In this paper we study the relaxation with respect to the L1 norm of integral functionals of the type F(u), where F(u) is the integral on a bounded open set of Rn Omega of an integrand f(x, u,∇u) and u ∈ W1,1(Omega; Sd−1), where Sd−1 denotes the unit sphere in Rd, n and d being any positive integers, and f satisfies linear growth conditions in the gradient variable. In analogy with the unconstrained case, we show that if, in addition, f is quasiconvex in the gradient variable and satisfies some technical continuity hypotheses, then the relaxed functional admits an integral representation on BV (Omega; Sd−1), via a suface energy density K on the jump set S(u), defined by a suitable Dirichlet-type problem (for the complete integral formula look at the abstract in the pdf of the paper)I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
