## Sharp regularity for functionals with ($$p$$,$$q$$) growth.(English)Zbl 1072.49024

In this very interesting paper the authors discuss the regularity of solutions to non-autonomous anisotropic variational problems. They impose a condition on the growth exponents which ensures higher integrability results for various classes of energy densities. The main point here is to prove the absence of a Lavrentiev phenomenon. By presenting a counterexample they also show the sharpness of this condition which surprizingly differs from the one to be expected from the known results in the autonomous case. In the final sections they focus on the relaxed functional and give an answer to a question of Marcellini concerning isolated singularities.

### MSC:

 49N60 Regularity of solutions in optimal control 49J25 Optimal control problems with equations with ret. arguments (exist.) (MSC2000) 49J45 Methods involving semicontinuity and convergence; relaxation
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### References:

 [1] Acerbi, E.; Bouchitté, G.; Fonseca, I., Relaxation of convex functionalsthe gap problem, Ann. IHP (anal. non lineare), 20, 359-380, (2003) · Zbl 1025.49012 [2] Acerbi, E.; Fusco, N., Regularity of minimizers of non-quadratic functionalsthe case 1
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