Partial regularity of $p(x)$-harmonic maps

By M. Alessandra Ragusa, A. Tachikawa, Hiroshi Takabayashi. 2011

51 citations recorded by the source corpus.

Let $(g^{\alpha\beta}(x))$ and $(h_{ij}(u))$ be uniformly elliptic symmetric matrices, and assume that $h_{ij}(u)$ and $p(x) \, (\, \geq 2)$ are sufficiently smooth. We prove partial regularity of minimizers for the functional [ {\mathcal F}(u) = \int_\Omega (g^{\alpha \beta}(x) h_{ij}(u) D_\alpha u^iD_\beta u^j)^{p(x)/2} dx, \] under the non-standard growth conditions of $p(x)$-type. If $g^{\alpha\beta}(x)$ are in the class $VMO$, we have partial H\"older regularity. Moreover, if $g^{\alpha\beta}$ are H\"older continuous, we can show partial $C^{1,\alpha}$-regularity.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/0c5477d0905e63f26ffd806a83ff310be516b4fa

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In Applied Mathematics

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Citations in the corpus

Regularity for general functionals with double phase (2017, result): “3), while further results in this direction can be for instance found in [2, 3, 4, 8, 11, 9, 12, 13, 14, 20, 47, 48, 49, 52]; see also the work of Ragusa & Tachikawa for partial regularity [50, 51, 52].”
Mixed Morrey spaces and their applications to partial differential equations (2017, unknown): “in [3], recently in [24], [20], [12], [13], [14] and others.”