## Analysis & PDE

• Anal. PDE
• Volume 3, Number 4 (2010), 359-407.

### Mean curvature motion of graphs with constant contact angle at a free boundary

Alexandre Freire

#### Abstract

We consider the motion by mean curvature of an $n$-dimensional graph over a time-dependent domain in $ℝn$ intersecting $ℝn$ at a constant angle. In the general case, we prove local existence for the corresponding quasilinear parabolic equation with a free boundary and derive a continuation criterion based on the second fundamental form. If the initial graph is concave, we show this is preserved and that the solution exists only for finite time. This corresponds to a symmetric version of mean curvature motion of a network of hypersurfaces with triple junctions with constant contact angle at the junctions.

#### Article information

Source
Anal. PDE, Volume 3, Number 4 (2010), 359-407.

Dates
Revised: 8 October 2009
Accepted: 17 October 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513731094

Digital Object Identifier
doi:10.2140/apde.2010.3.359

Mathematical Reviews number (MathSciNet)
MR2718258

Zentralblatt MATH identifier
1228.35126

#### Citation

Freire, Alexandre. Mean curvature motion of graphs with constant contact angle at a free boundary. Anal. PDE 3 (2010), no. 4, 359--407. doi:10.2140/apde.2010.3.359. https://projecteuclid.org/euclid.apde/1513731094

#### References

• O. Baconneau and A. Lunardi, “Smooth solutions to a class of free boundary parabolic problems”, Trans. Amer. Math. Soc. 356:3 (2004), 987–1005.
• K. Ecker, Regularity theory for mean curvature flow, Progress in Nonlinear Differential Equations and their Applications 57, Birkhäuser, Boston, MA, 2004.
• K. Ecker and G. Huisken, “Interior estimates for hypersurfaces moving by mean curvature”, Invent. Math. 105:3 (1991), 547–569.
• S. D. Eidelman and N. V. Zhitarashu, Parabolic boundary value problems, Operator Theory: Advances and Applications 101, Birkhäuser Verlag, Basel, 1998.
• B. Guan, “Mean curvature motion of nonparametric hypersurfaces with contact angle condition”, pp. 47–56 in Elliptic and parabolic methods in geometry (Minneapolis, 1994), edited by B. Chow et al., A K Peters, Wellesley, MA, 1996.
• G. Huisken, “Flow by mean curvature of convex surfaces into spheres”, J. Differential Geom. 20:1 (1984), 237–266.
• G. Huisken, “Nonparametric mean curvature evolution with boundary conditions”, J. Differential Equations 77:2 (1989), 369–378.
• G. M. Lieberman, Second order parabolic differential equations, World Scientific, River Edge, NJ, 1996.
• C. Mantegazza, M. Novaga, and V. M. Tortorelli, “Motion by curvature of planar networks”, Ann. Sc. Norm. Super. Pisa Cl. Sci. $(5)$ 3:2 (2004), 235–324.
• R. Mazzeo and M. Sáez, “Self-similar expanding solutions of the planar network flow”, preprint, 2007.
• M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Springer, New York, 1984.
• O. Schnürer, A. Azouani, M. Georgi, J. Hell, N. Jangle, A. Koeller, T. Marxen, S. Ritthaler, M. Sáez, and F. S. B. Smith, “Evolution of convex lens-shaped networks under curve shortening flow”, preprint, Lens Seminar, FU Berlin, 2007.
• V. A. Solonnikov, “Lectures on evolution free boundary problems: classical solutions”, pp. 123–175 in Mathematical aspects of evolving interfaces (Funchal, 2000), edited by P. Colli and J. F. Rodrigues, Lecture Notes in Math. 1812, Springer, Berlin, 2003.
• A. Stahl, “Convergence of solutions to the mean curvature flow with a Neumann boundary condition”, Calc. Var. Partial Differential Equations 4:5 (1996), 421–441.
• M. Struwe, “The existence of surfaces of constant mean curvature with free boundaries”, Acta Math. 160:1-2 (1988), 19–64.