The functionals of double phase typeare introduced in the epoch-making paper by Colombo-Mingione [1] for constants p and q, and investigated by them and Baroni. They obtained sharp regularity results for minimizers of such functionals. In this paper we treat the case that the exponents are functions of x and partly generalize their regularity results.

The paper is devoted to the study of the regularity on the boundary ∂Ω of a bounded open set Ω ⊂ R m for minimizers u for p(x)-energy functionals of the following typewhere (g αβ (x)) and (G ij (u)) are symmetric positive definite matrices whose entries are continuous functions and p(x) ≥ 2 is a continuous function. The authors prove that such minimizers u have no singular points on the boundary. © 2014 RésuméDans cet article, les auteurs étudient la régularité sur la frontière ∂Ω d'un ouvert borné Ω ⊂ R m des minimiseurs u des fonctionnelles d'énergie p(x) du type suivant : (u)) sont des matrices symétriques définies positives dont les éléments sont des fonctions continues et p(x) ≥ 2 est une fonction continue. Les auteurs prouvent que ces minimiseurs u n'ont pas de point singulier sur la frontière ∂Ω.

Let (g αβ (x)) and (h ij (u)) be uniformly elliptic symmetric matrices, and assume that h ij (u) and p(x) (≥ 2) are sufficiently smooth. We prove partial regularity of minimizers for the functional F (u) = Ω (g αβ (x)h ij (u)D α u i D β u j) p(x)/2 dx, under the nonstandard growth conditions of p(x)-type. If g αβ (x) are in the class V MO, we have partial Hölder regularity. Moreover, if g αβ are Hölder continuous, we can show partial C 1,α-regularity.

The paper investigates the partial regularity of the minimizers for quadratic functionals whose integrands have V M O coefficients in principal part and nonlinear terms that are Carathéodory functions. We use some majorizations for the functional, rather than the well known Euler equation associated to it.

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