November  2013, 33(11&12): 5407-5428. doi: 10.3934/dcds.2013.33.5407

On the manifold of closed hypersurfaces in $\mathbb{R}^n$

1. 

Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, D-60120 Halle

2. 

Department of Mathematics, Vanderbilt University, Nashville, TN 37240

Received  August 2011 Revised  December 2011 Published  May 2013

Several results from differential geometry of hypersurfaces in $\mathbb{R}^n$ are derived to form a tool box for the direct mapping method. The latter technique has been widely employed to solve problems with moving interfaces, and to study the asymptotics of the induced semiflows.
Citation: Jan Prüss, Gieri Simonett. On the manifold of closed hypersurfaces in $\mathbb{R}^n$. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5407-5428. doi: 10.3934/dcds.2013.33.5407
References:
[1]

M. Bergner, J. Escher and F. Lippoth, On the blow up scenario for a class of parabolic moving boundary problems, Nonlinear Anal., 75 (2012), 3951-3963. doi: 10.1016/j.na.2012.02.001.

[2]

M. P. Do Carmo, "Riemannian Geometry," Mathematics: Theory & Applications, Birkhäuser, Basel, 1992.

[3]

J. Escher and G. Simonett, Classical solutions for Hele-Shaw models with surface tension, Adv. Differential Equations, 2 (1997), 619-642.

[4]

J. Escher and G. Simonett, A center manifold analysis for the Mullins-Sekerka model, J. Differential Equations, 143 (1998), 267-292. doi: 10.1006/jdeq.1997.3373.

[5]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition. Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[6]

E. I. Hanzawa, Classical solutions of the Stefan problem, Tôhoku Math. Jour., 33 (1981), 297-335. doi: 10.2748/tmj/1178229399.

[7]

M. Kimura, Geometry of hypersurfaces and moving hypersurfaces in $\mathbbR^m$ for the study of moving boundary problems, Topics in Mathematical Modeling, J. Necas Center for Mathematical Modeling, Lecture Notes, 4, Prague (2008), 39-93.

[8]

M. Köhne, J. Prüss and M. Wilke, On quasilinear parabolic evolution equations in weighted $L_p$-spaces, J. Evol. Eqns., 10 (2010), 443-463. doi: 10.1007/s00028-010-0056-0.

[9]

M. Köhne, J. Prüss and M. Wilke, Qualitative behaviour of solutions for the two-phase Navier-Stokes equations with surface tension,, Math. Ann., ().  doi: 10.1007/s00208-012-0860-7.

[10]

W. Kühnel, "Differential Geometry. Curves-Surfaces-Manifolds," Student Mathematical Library, 16, American Mathematical Society, Providence, RI, 2002.

[11]

J. Prüss, Y. Shibata, S. Shimizu and G. Simonett, On well-posedness of incompressible two-phase flows with phase transition: The case of equal densities, Evol. Eqns. & Control Th., 1 (2012), 171-194. doi: 10.3934/eect.2012.1.171.

[12]

J. Prüss, G. Simonett and M. Wilke, On thermodynamically consistent Stefan problems with variable surface energy,, submitted, (). 

[13]

J. Prüss, G. Simonett and R. Zacher, Qualitative behaviour of solutions for thermodynamically consistent Stefan problems with surface tension, Arch. Ration. Mech. Anal., 207 (2013), 611-667. doi: 10.1007/s00205-012-0571-y.

show all references

References:
[1]

M. Bergner, J. Escher and F. Lippoth, On the blow up scenario for a class of parabolic moving boundary problems, Nonlinear Anal., 75 (2012), 3951-3963. doi: 10.1016/j.na.2012.02.001.

[2]

M. P. Do Carmo, "Riemannian Geometry," Mathematics: Theory & Applications, Birkhäuser, Basel, 1992.

[3]

J. Escher and G. Simonett, Classical solutions for Hele-Shaw models with surface tension, Adv. Differential Equations, 2 (1997), 619-642.

[4]

J. Escher and G. Simonett, A center manifold analysis for the Mullins-Sekerka model, J. Differential Equations, 143 (1998), 267-292. doi: 10.1006/jdeq.1997.3373.

[5]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition. Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[6]

E. I. Hanzawa, Classical solutions of the Stefan problem, Tôhoku Math. Jour., 33 (1981), 297-335. doi: 10.2748/tmj/1178229399.

[7]

M. Kimura, Geometry of hypersurfaces and moving hypersurfaces in $\mathbbR^m$ for the study of moving boundary problems, Topics in Mathematical Modeling, J. Necas Center for Mathematical Modeling, Lecture Notes, 4, Prague (2008), 39-93.

[8]

M. Köhne, J. Prüss and M. Wilke, On quasilinear parabolic evolution equations in weighted $L_p$-spaces, J. Evol. Eqns., 10 (2010), 443-463. doi: 10.1007/s00028-010-0056-0.

[9]

M. Köhne, J. Prüss and M. Wilke, Qualitative behaviour of solutions for the two-phase Navier-Stokes equations with surface tension,, Math. Ann., ().  doi: 10.1007/s00208-012-0860-7.

[10]

W. Kühnel, "Differential Geometry. Curves-Surfaces-Manifolds," Student Mathematical Library, 16, American Mathematical Society, Providence, RI, 2002.

[11]

J. Prüss, Y. Shibata, S. Shimizu and G. Simonett, On well-posedness of incompressible two-phase flows with phase transition: The case of equal densities, Evol. Eqns. & Control Th., 1 (2012), 171-194. doi: 10.3934/eect.2012.1.171.

[12]

J. Prüss, G. Simonett and M. Wilke, On thermodynamically consistent Stefan problems with variable surface energy,, submitted, (). 

[13]

J. Prüss, G. Simonett and R. Zacher, Qualitative behaviour of solutions for thermodynamically consistent Stefan problems with surface tension, Arch. Ration. Mech. Anal., 207 (2013), 611-667. doi: 10.1007/s00205-012-0571-y.

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