American Institute of Mathematical Sciences

Advanced
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.  doi: 10.2307/1971480.  Google Scholar

[2]

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

[3]

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

[9]

L. Wang, On the regularity theory of fully nonlinear parabolic equations. II,, Comm. Pure Appl. Math., 45 (1992), 141.   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.  doi: 10.2307/1971480.  Google Scholar

[2]

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

[3]

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

[9]

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

[1]

Wenxiong Chen, Congming Li. A priori estimate for the Nirenberg problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 225-233. doi: 10.3934/dcdss.2008.1.225
[2]

Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control & Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014
[3]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424
[4]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367
[5]

Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307
[6]

John Sylvester. An estimate for the free Helmholtz equation that scales. Inverse Problems & Imaging, 2009, 3 (2) : 333-351. doi: 10.3934/ipi.2009.3.333
[7]

Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014
[8]

J. F. Padial. Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion. Conference Publications, 2011, 2011 (Special) : 1176-1185. doi: 10.3934/proc.2011.2011.1176
[9]

Dan Mangoubi. A gradient estimate for harmonic functions sharing the same zeros. Electronic Research Announcements, 2014, 21: 62-71. doi: 10.3934/era.2014.21.62
[10]

Li-Ming Yeh. Pointwise estimate for elliptic equations in periodic perforated domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1961-1986. doi: 10.3934/cpaa.2015.14.1961
[11]

Jong-Shenq Guo, Satoshi Sasayama, Chi-Jen Wang. Blowup rate estimate for a system of semilinear parabolic equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 711-718. doi: 10.3934/cpaa.2009.8.711
[12]

Peng Gao. Global Carleman estimate for the Kawahara equation and its applications. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1853-1874. doi: 10.3934/cpaa.2018088
[13]

Neal Bez, Chris Jeavons. A sharp Sobolev-Strichartz estimate for the wave equation. Electronic Research Announcements, 2015, 22: 46-54. doi: 10.3934/era.2015.22.46
[14]

Gary Lieberman. A new regularity estimate for solutions of singular parabolic equations. Conference Publications, 2005, 2005 (Special) : 605-610. doi: 10.3934/proc.2005.2005.605
[15]

Liangjun Weng. The interior gradient estimate for some nonlinear curvature equations. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1601-1612. doi: 10.3934/cpaa.2019076
[16]

Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic & Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883
[17]

Stefano Galatolo, Isaia Nisoli, Benoît Saussol. An elementary way to rigorously estimate convergence to equilibrium and escape rates. Journal of Computational Dynamics, 2015, 2 (1) : 51-64. doi: 10.3934/jcd.2015.2.51
[18]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387
[19]

Sun-Sig Byun, Yunsoo Jang. Calderón-Zygmund estimate for homogenization of parabolic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6689-6714. doi: 10.3934/dcds.2016091
[20]

Mikko Kaasalainen. Multimodal inverse problems: Maximum compatibility estimate and shape reconstruction. Inverse Problems & Imaging, 2011, 5 (1) : 37-57. doi: 10.3934/ipi.2011.5.37

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences