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:
[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]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467

[2]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[3]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[4]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[5]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[6]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[7]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[8]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108

[9]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

[10]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

[11]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075

[12]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

[13]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[14]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

[15]

Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099

[16]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102

[17]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[18]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351

[19]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[20]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]