Lower semicontinuity problems in Sobolev spaces with respect to a measure

For every finite nonnegative measure ft we introduce the Sobolev spaces W^'(Q,K) and we study the lower semicontinuity of functionals of the form where the integrand / is quasiconvex.

[1]  L. Ambrosio.  On the lower semicontinuity of quasiconvex integrals in SBV W , R k , 1994 .

[2]  Irene Fonseca,et al.  Relaxation of quasiconvex functional in BV(Ω, ℝp) for integrands f(x, u,∇;u) , 1993 .

[3]  L. Ambrosio,et al.  On the relaxation in BV(Ω; Rm) of quasi-convex integrals , 1992 .

[4]  Irene Fonseca,et al.  Quasi-convex integrands and lower semicontinuity in L 1 , 1992 .

[5]  L. Modica.  The gradient theory of phase transitions and the minimal interface criterion , 1987 .

[6]  Nicola Fusco,et al.  Semicontinuity problems in the calculus of variations , 1984 .

[7]  Paolo Marcellini,et al.  Semicontinuity problems in the calculus of variations , 1980 .

[8]  J. Ball.  Convexity conditions and existence theorems in nonlinear elasticity , 1976 .

[9]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[10]  H. Fédérer.  Geometric Measure Theory , 1969 .

[11]  G. Buttazzo,et al.  Relaxation results for some free discontinuity problems. , 1995 .

[12]  Andrea Braides,et al.  The interaction between bulk energy and surface energy in multiple integrals , 1994, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[13]  Ana Cristina Barroso,et al.  Anisotropic singular perturbations—the vectorial case , 1994, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[14]  E. Giorgi,et al.  Un nuovo tipo di funzionale del calcolo delle variazioni , 1988 .

[15]  A. I. Vol'pert,et al.  Analysis in classes of discontinuous functions and equations of mathematical physics , 1985 .

[16]  C. B. Morrey.  Multiple Integrals in the Calculus of Variations , 1966 .