# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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. doi: 10.1016/j.na.2012.02.001. Google Scholar [2] M. P. Do Carmo, "Riemannian Geometry,", Mathematics: Theory & Applications, (1992). Google Scholar [3] J. Escher and G. Simonett, Classical solutions for Hele-Shaw models with surface tension,, Adv. Differential Equations, 2 (1997), 619. Google Scholar [4] J. Escher and G. Simonett, A center manifold analysis for the Mullins-Sekerka model,, J. Differential Equations, 143 (1998), 267. doi: 10.1006/jdeq.1997.3373. Google Scholar [5] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition. Classics in Mathematics, (1998). Google Scholar [6] E. I. Hanzawa, Classical solutions of the Stefan problem,, Tôhoku Math. Jour., 33 (1981), 297. doi: 10.2748/tmj/1178229399. Google Scholar [7] M. Kimura, Geometry of hypersurfaces and moving hypersurfaces in $\mathbbR^m$ for the study of moving boundary problems,, Topics in Mathematical Modeling, (2008), 39. Google Scholar [8] M. Köhne, J. Prüss and M. Wilke, On quasilinear parabolic evolution equations in weighted $L_p$-spaces,, J. Evol. Eqns., 10 (2010), 443. doi: 10.1007/s00028-010-0056-0. Google Scholar [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. Google Scholar [10] W. Kühnel, "Differential Geometry. Curves-Surfaces-Manifolds,", Student Mathematical Library, 16 (2002). Google Scholar [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. doi: 10.3934/eect.2012.1.171. Google Scholar [12] J. Prüss, G. Simonett and M. Wilke, On thermodynamically consistent Stefan problems with variable surface energy,, submitted, (). Google Scholar [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. doi: 10.1007/s00205-012-0571-y. Google Scholar

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##### 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. doi: 10.1016/j.na.2012.02.001. Google Scholar [2] M. P. Do Carmo, "Riemannian Geometry,", Mathematics: Theory & Applications, (1992). Google Scholar [3] J. Escher and G. Simonett, Classical solutions for Hele-Shaw models with surface tension,, Adv. Differential Equations, 2 (1997), 619. Google Scholar [4] J. Escher and G. Simonett, A center manifold analysis for the Mullins-Sekerka model,, J. Differential Equations, 143 (1998), 267. doi: 10.1006/jdeq.1997.3373. Google Scholar [5] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Reprint of the 1998 edition. Classics in Mathematics, (1998). Google Scholar [6] E. I. Hanzawa, Classical solutions of the Stefan problem,, Tôhoku Math. Jour., 33 (1981), 297. doi: 10.2748/tmj/1178229399. Google Scholar [7] M. Kimura, Geometry of hypersurfaces and moving hypersurfaces in $\mathbbR^m$ for the study of moving boundary problems,, Topics in Mathematical Modeling, (2008), 39. Google Scholar [8] M. Köhne, J. Prüss and M. Wilke, On quasilinear parabolic evolution equations in weighted $L_p$-spaces,, J. Evol. Eqns., 10 (2010), 443. doi: 10.1007/s00028-010-0056-0. Google Scholar [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. Google Scholar [10] W. Kühnel, "Differential Geometry. Curves-Surfaces-Manifolds,", Student Mathematical Library, 16 (2002). Google Scholar [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. doi: 10.3934/eect.2012.1.171. Google Scholar [12] J. Prüss, G. Simonett and M. Wilke, On thermodynamically consistent Stefan problems with variable surface energy,, submitted, (). Google Scholar [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. doi: 10.1007/s00205-012-0571-y. Google Scholar
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