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May  2012, 11(3): 1037-1050. doi: 10.3934/cpaa.2012.11.1037

Schauder type estimates of linearized Mullins-Sekerka problem

Feiyao Ma 1, and  Lihe Wang 2,

1. 

Department of Mathematics, Ningbo University, Ningbo, Zhejiang, 315211, China
2. 

Department of Mathematics, University of Iowa, Iowa City, IA 52242

Received  October 2010 Revised  May 2011 Published  December 2011

In this paper we obtain a Caccioppoli type estimate for the model of the linearized Mullins-Sekerka equations by a new technique, then we use this estimate to derive it's Schauder type estimates by polynomial approximation method.
Keywords: Schauder estimate, Caccioppoli estimate., Mullins-Sekerka problem.
Mathematics Subject Classification: Primary: 35B65, 35R35; Secondary: 35H9.
Citation: Feiyao Ma, Lihe Wang. Schauder type estimates of linearized Mullins-Sekerka problem. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1037-1050. doi: 10.3934/cpaa.2012.11.1037
References:
[1]

L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. Math., 130 (1989), 189-213. doi: 10.2307/1971480.  Google Scholar

[2]

L. A. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl., 4 (1998), 383-402. doi: 10.1007/BF02498216.  Google Scholar

[3]

L. A. Caffarelli, "The Obstacle Problem. Lezioni Fermiane," [Fermi Lectures], Accademia Nazionale dei Lincei, Rome, 1998.  Google Scholar

[4]

X. Chen, J. Hong and F. Yi, Existence,uniqueness,and regularity of classical solutions of the mullins-sekerka problem, Comm. In. PDE, 21 (1996), 1705-1727. doi: 10.1016/j.jde.2004.10.028.  Google Scholar

[5]

X. Chen and F. Retich, Local existence and uniqueness of solutions of the stefan problem with surface tension and kinetic undercooling, J. Math. Anal. Appl., 164 (1992), 350-362. doi: 10.1016/0022-247X(92)90119-X.  Google Scholar

[6]

E. Milakis and L. E. Silvestre, Regularity for fully nonlinear elliptic equations with neumann boundary data, Com. in Partial Differential Equations, 31 (2006), 1227-1252. doi: 10.1080/03605300600634999.  Google Scholar

[7]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," vol. 30 of PMS. Princeton University Press, 1971.  Google Scholar

[8]

L. Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math., 45 (1992), 27-76.  Google Scholar

[9]

L. Wang, On the regularity theory of fully nonlinear parabolic equations. II, Comm. Pure Appl. Math., 45 (1992), 141-178.  Google Scholar

show all references

References:
[1]

L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. Math., 130 (1989), 189-213. doi: 10.2307/1971480.  Google Scholar

[2]

L. A. Caffarelli, The obstacle problem revisited, J. Fourier Anal. Appl., 4 (1998), 383-402. doi: 10.1007/BF02498216.  Google Scholar

[3]

L. A. Caffarelli, "The Obstacle Problem. Lezioni Fermiane," [Fermi Lectures], Accademia Nazionale dei Lincei, Rome, 1998.  Google Scholar

[4]

X. Chen, J. Hong and F. Yi, Existence,uniqueness,and regularity of classical solutions of the mullins-sekerka problem, Comm. In. PDE, 21 (1996), 1705-1727. doi: 10.1016/j.jde.2004.10.028.  Google Scholar

[5]

X. Chen and F. Retich, Local existence and uniqueness of solutions of the stefan problem with surface tension and kinetic undercooling, J. Math. Anal. Appl., 164 (1992), 350-362. doi: 10.1016/0022-247X(92)90119-X.  Google Scholar

[6]

E. Milakis and L. E. Silvestre, Regularity for fully nonlinear elliptic equations with neumann boundary data, Com. in Partial Differential Equations, 31 (2006), 1227-1252. doi: 10.1080/03605300600634999.  Google Scholar

[7]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," vol. 30 of PMS. Princeton University Press, 1971.  Google Scholar

[8]

L. Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math., 45 (1992), 27-76.  Google Scholar

[9]

L. Wang, On the regularity theory of fully nonlinear parabolic equations. II, Comm. Pure Appl. Math., 45 (1992), 141-178.  Google Scholar

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