Regularity for minimizers of double phase functionals with mild transition and regular coefficients

We prove sharp regularity results for minimizers of the functional P(w, Omega) := integral(Omega) b(x, w) [vertical bar Dw vertical bar(p) + a(x)vertical bar Dw vertical bar(p) log (e + vertical bar Dw vertical bar)] dx, with w is an element of W-1,W-1(Omega), p > 1, a is an element of L-infinity(Omega), a(.) >= 0, and 0 < nu <= b(., .) <= L. P is a double phase functional with mild transition between vertical bar Du vertical bar(p) and vertical bar Du vertical bar(p) log (e + vertical bar Du vertical bar). First, under suitable conditions on the moduli of continuity of a(.) and b(., .), we prove that local minimizers are of class C-0,C-alpha for every alpha is an element of (0, 1), then that they are of class C-1,C-alpha for some alpha > 0, provided the functions a(.) and b(., .) are Holder continuous. (C) 2020 Elsevier Inc. All rights reserved.