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V. I. Smirnov Seminar on Mathematical Physics
March 15, 2021 16:30, St. Petersburg, zoom online-conference
 


On the regularity of $p(x)$-harmonic functions

M. D. Surnachev

Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow

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Abstract: The talk is based on a joint work with Y.A.Altukhov (Vladimir State University). I will present a series of results on the regularity of solutions to divergent elliptic equations of the $p(x)$-Laplacian type. Most of well-known facts for solutions to equations of such type (as well as in the theory of the corresponding functional spaces) requires at least “logarithmic Hölder-continuity” of the nonlinearity characteristics $p(x)$. We deal with the cases when the condition is not valid. Firstly, we are interested in the case when $p(x)$ is discontinuous, but has a distinct geometric structure. Secondly, we study equations with $p(x)$ having modulus of continuity that is weaker than the logarithmic one. Special attention is devoted to the case when $p(x)$ is continuous only at a given point.
 
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