Partial regularity of 𝑝(𝑥)-harmonic maps

Abstract

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.

Partial regularity of minimizers of p (x )-growth functionals with 1 < p (x ) < 2

“…Here, Ragusa-Tachikawa-Takabayashi [13] showed the following lemma on the higher integrability of such local minimizers.…”

“…For the p(x)-growth case, Ragusa-Tachikawa-Takabayashi [13] showed that a minimizer u of the functional…”

“…is of class C 1,α assuming that (g αβ (x)) and (h ij (u)) are uniformly elliptic symmetric matrices, and that h ij (u) and p(x)( 2) are sufficiently smooth. Based on [6,13], we prove that in the case that p(x) is continuous and satisfies 1 < p(x) < 2, a minimizer u of E is in the class C 0,α (Ω 0 ) and that H m−γ1 (Ω − Ω 0 ) = 0 holds, where γ 1 := inf Ω p(x) and Ω 0 is an open subset of Ω.…”

On some regularity results of minimizers of energy functionals

“…The study in cooperation by Ragusa and Tachikawa is continued obtaining with Takabayashi, in [25], partial regularity results of minimizers of p(x)-energy functionals.…”

Embedding of vector-valued Morrey spaces and separable differential operators

“…On the other hand, the study of Morrey spaces has received considerable attention in the last thirty years in different research areas (see e.g. [8][9][10]16,17,[19][20][21]23,28,31,36,43]). A further motivation comes from the fact that, to our knowledge, nothing is known concerning Morrey estimates for such operator-valued Fourier multipliers and embedding properties of abstract Sobolev-Morrey spaces.…”