## On the weak lower semicontinuity of energies with polyconvex integrands.(English)Zbl 0829.49011

Let $$f: \Omega \times \mathbb{R}^N \times \mathbb{R}^{N \times N}\to [0, \infty)$$ be a Borel measurable function such that $$f(x, u, \xi)= a(x,u) g(x,\xi)$$ and $$g(x, \cdot)$$ is polyconvex in the last variable $$\xi$$ for almost every $$x\in \Omega$$. It is shown that if $$f$$ is continuous, if $$a$$ is bounded away from zero and if $$F(u):= \int_\Omega a(x,u) g(x, \nabla u(x)) dx$$, $$u\in W^{1, N} (\Omega, \mathbb{R}^N)$$, then $$F$$ is weakly lower semicontinuous in $$W^{1,p}$$, $$p> N-1$$, in the sense that $$F(u)\leq \liminf_{n\to\infty} F(u_\nu)$$ for $$u_\nu$$, $$u\in W^{1, N} (\Omega, \mathbb{R}^N)$$ such that $$u_\nu \rightharpoonup u$$ in $$W^{1,p}$$. On the contrary, if $$g$$ is only a Carathéodory function then in general $$F$$ is not weakly lower semicontinuous in $$W^{1,p}$$ for $$N>p >N-1$$. Precisely, it is shown that if $$F(u):= \int_K |\text{det} (\nabla u(x)) |dx$$ where $$K$$ is a compact set, then $$F$$ is weakly lower semicontinuous in $$W^{1,p}$$, $$N>p> N-1$$, if and only if $$\text{meas} (\partial K)=0$$.

### MSC:

 49J45 Methods involving semicontinuity and convergence; relaxation