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 & Control Theory, 2015, 4 (4) : 391-429. doi: 10.3934/eect.2015.4.391
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show all references

References:
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[7]

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[8]

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[9]

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[10]

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[11]

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[12]

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[13]

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[14]

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[15]

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[16]

S. P. Degtyarev, Classical solvability of multidimensional two-phase Stefan problem for degenerate parabolic equations and Schauder's estimates for a degenerate parabolic problem with dynamic boundary conditions,, Nonlinear Differential Equations and Applications NoDEA, 22 (2015), 185. doi: 10.1007/s00030-014-0280-3. Google Scholar

[17]

S. P. Degtyarev, The existence of a smooth interface in the evolutionary elliptic Muskat-Verigin problem with nonlinear source,, (Russian) Ukrainian Mathematical Bulletin, 7 (2010), 301. Google Scholar

[18]

S. P. Degtyarev, The existence of a smooth interface in the evolutionary elliptic Muskat-Verigin problem with nonlinear source,, , (). Google Scholar

[19]

R. Denk, J. Prüss and R. Zacher, Maximal $L_p$ - regularity of parabolic problems with boundary dynamics of relaxation type,, J. Funct. Anal., 255 (2008), 3149. doi: 10.1016/j.jfa.2008.07.012. Google Scholar

[20]

R. Denk and R. Volevich, A new class of parabolic problems connected with Newton's polygon,, Discrete Cont. Dyn. Syst., (2007), 294. Google Scholar

[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. doi: 10.1007/s00028-008-0392-5. Google Scholar

[22]

R. Denk and M. Kaip, General Parabolic Mixed Order Systems in $L_p$ and Applications,, Operator Theory: Advances and Applications, (2013). doi: 10.1007/978-3-319-02000-6. Google Scholar

[23]

P. Dintelmann, Classes of Fourier multipliers and Besov-Nikolskij spaces,, Math. Nachr., 173 (1995), 115. doi: 10.1002/mana.19951730108. Google Scholar

[24]

H. Dong, Gradient estimates for parabolic and elliptic systems from linear laminates,, Arch. Ration. Mech. Anal., 205 (2012), 119. doi: 10.1007/s00205-012-0501-z. Google Scholar

[25]

H. Dong and S. Kim, Partial Scauder estimates for second-order elliptic and parabolic equations,, Calc. Var. Partial Differential Equations, 40 (2011), 481. doi: 10.1007/s00526-010-0348-9. Google Scholar

[26]

J. Escher, Quasilinear parabolic systems with dynamical boundary conditions,, Comm. Partial Differential Equations, 18 (1993), 1309. doi: 10.1080/03605309308820976. Google Scholar

[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. doi: 10.4171/ZAA/1431. Google Scholar

[28]

J. Escher and G. Simonett, Classical solutions of multidimensional Hele-Shaw models,, SIAM J. Math. Anal., 28 (1997), 1028. doi: 10.1137/S0036141095291919. Google Scholar

[29]

J. Escher and G. Simonett, Classical solutions for Hele-Shaw models with surface tension,, Adv. Differential Equations, 2 (1997), 619. Google Scholar

[30]

P. Fife, Schauder estimates under incomplete Hölder continuity assumptions,, Pacific J. Math., 13 (1963), 511. doi: 10.2140/pjm.1963.13.511. Google Scholar

[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. doi: 10.4171/IFB/37. Google Scholar

[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. doi: 10.1512/iumj.2001.50.2118. Google Scholar

[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. Google Scholar

[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. doi: 10.3934/dcds.2008.22.1041. Google Scholar

[35]

S. Gindikin and L. R. Volevich, The Method of Newton's Polyhedron in the Theory of Partial Differential Equations,, Mathematics and its Applications, (1992). doi: 10.1007/978-94-011-1802-6. Google Scholar

[36]

M. Girardi and L. Weis, Operator-valued Fourier multiplier theorems on Besov spaces,, Math. Nachr., 251 (2003), 34. doi: 10.1002/mana.200310029. Google Scholar

[37]

G. R. Goldstein and A. Miranville, A Cahn-Hilliard-Gurtin model with dynamic boundary conditions,, Discrete Contin. Dyn. Syst. Ser. S., 6 (2013), 387. doi: 10.3934/dcdss.2013.6.387. Google Scholar

[38]

K. K. Golovkin, On equivalent normalizations of fractional spaces,, in Automatic Programming, (1962), 364. Google Scholar

[39]

K. K. Golovkin and V. A. Solonnikov, Bounds for integral operators in translation-invariant norms,, in Boundary Value Problems of Mathematical Physics, (1964), 47. Google Scholar

[40]

K. K. Golovkin and V. A. Solonnikov, Estimates of integral operators in translation-invariant norms. II,, in Boundary Value Problems of Mathematical Physics. Part 4, (1966), 5. Google Scholar

[41]

K. K. Golovkin and V. A. Solonnikov, On some estimates of convolutions,, in Boundary-Value Problems of Mathematical Physics and Related Problems of Function Theory. Part 2, (1968), 6. Google Scholar

[42]

B. Grec and E. V. Radkevich, Newton's polygon method and the local solvability of free boundary problems,, Journal of Mathematical Sciences, 143 (2007), 3253. doi: 10.1007/s10958-007-0208-0. Google Scholar

[43]

D. Guidetti, The parabolic mixed Cauchy-Dirichlet problem in spaces of functions which are Hölder continuous with respect to space variables,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 7 (1996), 161. Google Scholar

[44]

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