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

Schauder type estimates of linearized Mullins-Sekerka problem

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.
Citation: Feiyao Ma, Lihe Wang. Schauder type estimates of linearized Mullins-Sekerka problem. Communications on Pure and 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.

[2]

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

[3]

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

[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.

[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.

[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.

[7]

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

[8]

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

[9]

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

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.

[2]

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

[3]

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

[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.

[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.

[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.

[7]

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

[8]

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

[9]

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

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