# American Institute of Mathematical Sciences

December  2015, 4(4): 391-429. doi: 10.3934/eect.2015.4.391

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

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

Received  March 2015 Revised  October 2015 Published  November 2015

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.
Citation: Sergey P. Degtyarev. On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions. Evolution Equations and Control Theory, 2015, 4 (4) : 391-429. doi: 10.3934/eect.2015.4.391
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##### References:
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Syst., Dynamical Systems and Differential Equations, Proceedings of the 6th AIMS International Conference, suppl., (2007), 294-303. [21] R. Denk and L. R. Volevich, Parabolic boundary value problems connected with Newton's polygon and some problems of crystallization, J. Evol. Equ., 8 (2008), 523-556. doi: 10.1007/s00028-008-0392-5. [22] R. Denk and M. Kaip, General Parabolic Mixed Order Systems in $L_p$ and Applications, Operator Theory: Advances and Applications, 239, Birkhäuser/Springer, 2013. doi: 10.1007/978-3-319-02000-6. [23] P. Dintelmann, Classes of Fourier multipliers and Besov-Nikolskij spaces, Math. Nachr., 173 (1995), 115-130. doi: 10.1002/mana.19951730108. [24] H. Dong, Gradient estimates for parabolic and elliptic systems from linear laminates, Arch. Ration. Mech. Anal., 205 (2012), 119-149. doi: 10.1007/s00205-012-0501-z. [25] H. Dong and S. Kim, Partial Scauder estimates for second-order elliptic and parabolic equations, Calc. Var. Partial Differential Equations, 40 (2011), 481-500. doi: 10.1007/s00526-010-0348-9. [26] J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations, 18 (1993), 1309-1364. doi: 10.1080/03605309308820976. [27] J. Escher and B.-V. Matioc, On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results, Z. Anal. Anwend., 30 (2011), 193-218. doi: 10.4171/ZAA/1431. [28] J. Escher and G. Simonett, Classical solutions of multidimensional Hele-Shaw models, SIAM J. Math. Anal., 28 (1997), 1028-1047. doi: 10.1137/S0036141095291919. [29] J. Escher and G. Simonett, Classical solutions for Hele-Shaw models with surface tension, Adv. Differential Equations, 2 (1997), 619-642. [30] P. Fife, Schauder estimates under incomplete Hölder continuity assumptions, Pacific J. Math., 13 (1963), 511-550. doi: 10.2140/pjm.1963.13.511. [31] A. Friedman, B. Hu and J. J. L. Velazquez, A Stefan problem for a protocell model with symmetry-breaking bifurcations of analitic solutions, Interfaces Free Bound., 3 (2001), 143-199. doi: 10.4171/IFB/37. [32] A. Friedman and J. J. L. Velazquez, A free boundary problem associated with crystallization of polymers in a temperature field, Indiana Univ. Math. J., 50 (2001), 1609-1649. doi: 10.1512/iumj.2001.50.2118. [33] E. Frolova, Solvability in Sobolev spaces of a problem for a second order parabolic equation with time derivative in the boundary condition, Portugal Math., 56 (1999), 419-441. [34] C. G. Gal and H. Wu, Asymptotic behavior of Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation, Discrete Contin. Dyn. Syst., 22 (2008), 1041-1063. doi: 10.3934/dcds.2008.22.1041. [35] S. Gindikin and L. R. 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