On Fourier multipliers in function spaces with partial Höldercondition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions

Sergey P.  Degtyarev

doi:10.3934/eect.2015.4.391

Article Contents

2015, Volume 4, Issue 4: 391-429. Doi: 10.3934/eect.2015.4.391

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On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions

Sergey P. Degtyarev^1,

1.
Institute for Applied Mathematics and Mechanics NASU, State Institute for Applied Mathematics and Mechanics, R.Luxenburg Str., 74, Donetsk, 83114

Received: March 2015

Revised: October 2015

Published: December 2015

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Abstract

We give relatively simple sufficient conditions on a Fourier multiplier so that it maps functions with the Hölder property with respect to a part of the variables to functions with the Hölder property with respect to all variables. By using these these sufficient conditions we prove solvability in Hölder classes of the initial-boundary value problems for the linearized Cahn-Hilliard equation with dynamic boundary conditions of two types. In addition, Schauders estimates are derived for the solutions corresponding to the problem under study.

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Mathematics Subject Classification: Primary: 42B15, 42B37; Secondary: 35G15, 35G16, 35R35.

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