Local boundedness of Quasi-minimizers of fully anisotropic scalar variational problems

On the local everywhere bounndedness of the minima of a class of integral functionals of the Calculus of the Variations

“…The Caccioppoli inequalities (1.10) and (1.11) are typical of scalar variational problems with growths of the type q-p; in particular we will refer to the ideas presented in [14,29,[32][33][34] to obtain from the inequalities (1.10) and (1.11) the boundedness theorem Theorem 1.…”

“…with appropriate hypotheses on density G (x, s, ξ); in particular, for the following article, the results given in [37] are of particular relevance. The proof of Theorem 1 given in this article uses, mixing them, various techniques, the results given in [9,10,26,30] are rephrased to obtain Caccioppoli estimates for the minima of the functional (1.5) then we proceed with techniques similar to those presented in [4,5,6] to obtain uniform estimates on Sobolev norms of minima of functionals of the type (1.5); finally, by properly rephrasing the results given in [14,29,[32][33][34] the proof of Theorem1 is obtained. The author hopes, in some forthcoming works, both to develop the case 1<q<2, and to obtain the continuity of the minima of the functional (1.1).…”

Modified Caccioppoli inequalities for minimizer of anisotropic functionals

“…Let η ∈ C ∞ c (B R (x 0 )) such that 0 ≤ η ≤ 1 on B R (x 0 ), η = 1 on B τ (x 0 ), |∇η| ≤ C R−τ on B (x 0 ) where 0 < R 2 < τ < R < R 0 = 1 4 min {1, dist (x 0 , ∂Ω)}, since u ∈ L ∞ loc (Ω) , refer to [1,7], then the function ϕ = η 2 (u − k) γ + , with γ ≥ 1, is a test function and we get…”

“…In particular we will prove that the minimizers of the functional (1) verify some energetic inequalities that we will call modified Cacciolloli inequalities. The study of this anisotropic of functionals has been introduced in [1,9], recently interesting results have been obtained on the boundedness of the minimizers of these functionals, refer to [1,2,3,4,5,7]. At the moment, due to the knowledge of the author, no results are known about the continuity of the minima of the functional (1).…”

“…were introduced and studieddove, the functions g i are N-function, refer to [7]. Furthermore, in [7] a boundedness result was proved for functionals with the following growths…”