Geodesic
H\"older Continuity and Harnack's Inequality for $p(x)$-Harmonic Functions
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 308 (2020), pp. 7-27

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A sufficient condition for a measurable exponent $p(x)$ is obtained that implies the Hölder continuity and Harnack's inequality for $p(x)$-harmonic functions. 
@article{TM_2020_308_a0,
     author = {Yu. A. Alkhutov and M. D. Surnachev},
     title = {H\"older {Continuity} and {Harnack's} {Inequality} for $p(x)${-Harmonic} {Functions}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {7--27},
     publisher = {mathdoc},
     volume = {308},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2020_308_a0/}
}
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Yu. A. Alkhutov; M. D. Surnachev. H\"older Continuity and Harnack's Inequality for $p(x)$-Harmonic Functions. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 308 (2020), pp. 7-27. http://geodesic.mathdoc.fr/item/TM_2020_308_a0/