On L 1 -Lower Semicontinuity in BV

Abstract

and of the corresponding extension (1.3) to BV (Ω), with the aim of understanding which are the minimal assumptions on f that guarantee lower semicontinuity. Here, the starting point is the result, proved in [20], stating that if the integrand f(x, u, ξ) is a nonnegative, continuous function from Ω× IR× IR , convex in ξ, and such that the (classical) derivatives ∇xf,∇ξf,∇x∇ξf exist and are continuous functions, then the integral functional (1.1) is lower semicontinuous in W 1,1(Ω), with respect to the L1 convergence in the open set Ω. In 1983 De Giorgi, Buttazzo and Dal Maso (see [9]) showed that Serrin’s assumptions can be substantially weakened when dealing with autonomous (i.e. f ≡ f(u, ξ)) functionals of the type (1.1). In this case they were able to prove the L1-lower semicontinuity without even assuming f to be continuous in u for all ξ ∈ IR . Their result is proved by approximating the integrand f with a sequence of affine functions and then using a suitable version of the chain rule in the Sobolev space W 1,1(Ω) in order to get the lower semicontinuity of the approximating functionals. On the other hand, things are more complicate if we allow f to depend explicitely on x, since convexity in ξ and continuity with respect to all variables (x, u, ξ) are not enough to ensure the lower semicontinuity of functional (1.1), as shown by a well known example of Aronszajin (see [19]). An even more surprising counterexample to lower semicontinuity has been recently given by Gori, Maggi and Marcellini in [18] (see also [17]). In their example, which is one dimensional, the integrand f is equal to |a(x, u)ξ − 1| and the function a is Holder continuous in x, uniformly with respect to u. These examples clearly show that in order to get lower semicontinuity we must retain, in the spirit of Serrin’s theorem recalled above, some differentiability of f with respect to x. This is