American Institute of Mathematical Sciences

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 & Applied Analysis, 2012, 11 (3) : 1037-1050. doi: 10.3934/cpaa.2012.11.1037
References:
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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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