SIAM Journal on Control and Optimization, Vol.38, No.2, 384-399, 2000
Minimization of functionals of the gradient by Baire's theorem
We give sufficient conditions for the existence of solutions of the minimum problem P-u0 : Minimize integral(Omega) g(Du(x))dx; u is an element of u(0) + W-0(1,p) (Omega, R), based on the structure of the epigraph of the lower convex envelope of g, which is assumed be lower semicontinuous and to grow at infinity faster than the power p with p larger than the dimension of the space. No convexity conditions are required on g, and no assumptions are made on the boundary datum u(0) is an element of W-0(1,p) (Omega, R).