The evolution of H-surfaces with a Plateau boundary condition

Duzaar, Frank; Scheven, Christoph

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 1, pp. 109-157

In this paper we consider the heat flow associated to the classical Plateau problem for surfaces of prescribed mean curvature. To be precise, for a given Jordan curve $\Gamma \subset {ℝ}^3$, a given prescribed mean curvature function $H:{ℝ}^3\to ℝ$ and an initial datum $u_o:B\to {ℝ}^3$ satisfying the Plateau boundary condition, i.e. that $u_o{|}_{\partial B}:\partial B\to \Gamma$ is a homeomorphism, we consider the geometric flow

$${\partial }_{t}u-\Delta u=-2(H\circ u)D_{1}u\times D_{2}u\text{in}B\times (0,\infty ),$$

$$u(·,0)=u_o\text{on}B,u(·,t){|}_{\partial B}:\partial B\to \Gamma \text{is}\text{weakly}\text{monotone}\text{for}\text{all}t>0.$$

We show that an isoperimetric condition on H ensures the existence of a global weak solution. Moreover, we establish that these global solutions sub-converge as $t\to \infty$ to a conformal solution of the classical Plateau problem for surfaces of prescribed mean curvature.

[1] E. Acerbi, G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J. 136 no. 2 (2007), 285 -320 | MR | Zbl

[2] F. Duzaar, G. Mingione, Second order parabolic systems, optimal regularity, and singular sets of solutions, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 22 (2005), 705 -751 | MR | EuDML | Zbl | Numdam

[3] V. Bögelein, F. Duzaar, C. Scheven, Weak solutions to the heat flow for surfaces of prescribed mean curvature, Trans. Amer. Math. Soc. 365 (2013), 4633 -4677 | MR | Zbl

[4] V. Bögelein, F. Duzaar, C. Scheven, Global solutions to the heat flow for m-harmonic maps and regularity, Indiana Univ. Math. J. 61 no. 6 (2012), 2157 -2210 | MR | Zbl

[5] L.A. Caffarelli, I. Peral, On $W^{1,p}$ estimates for elliptic equations in divergence form, Commun. Pure Appl. Math. 51 no. 1 (1998), 1 -21 | Zbl

[6] K. Chang, J. Liu, Heat flow for the minimal surface with Plateau boundary condition, Acta Math. Sin. Engl. Ser. 19 no. 1 (2003), 1 -28 | MR | Zbl

[7] K. Chang, J. Liu, Another approach to the heat flow for Plateau problem, J. Differ. Equ. 189 (2003), 46 -70 | MR | Zbl

[8] K. Chang, J. Liu, An evolution of minimal surfaces with Plateau condition, Calc. Var. Partial Differ. Equ. 19 (2004), 117 -163 | MR | Zbl

[9] Y. Chen, S. Levine, The existence of the heat flow of H-systems, Discrete Contin. Dyn. Syst. 8 no. 1 (2002), 219 -236 | MR | Zbl

[10] U. Dierkes, S. Hildebrandt, A. Küster, O. Wohlrab, Minimal Surfaces, vols. I and II, Grundlehren Math. Wiss. vols. 295 and 296 , Springer (1992) | MR

[11] F. Duzaar, J. Grotowski, Existence and regularity for higher dimensional H-systems, Duke Math. J. 101 no. 3 (2000), 459 -485 | MR | Zbl

[12] F. Duzaar, K. Steffen, Existence of hypersurfaces with prescribed mean curvature in Riemannian manifolds, Indiana Univ. Math. J. 45 (1996), 1045 -1093 | MR | Zbl

[13] F. Duzaar, K. Steffen, Parametric surfaces of least H-energy in a Riemannian manifold, Math. Ann. 314 (1999), 197 -244 | MR | Zbl

[14] L. Evans, R. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math. , CRC Press, Boca Raton, FL (1992) | MR | Zbl

[15] H. Federer, Geometric Measure Theory, Springer, Berlin (1969) | MR | Zbl

[16] R. Gulliver, J. Spruck, The Plateau problem for surfaces of prescribed mean curvature in a cylinder, Invent. Math. 13 (1971), 169 -178 | MR | EuDML | Zbl

[17] R. Gulliver, J. Spruck, Existence theorems for parametric surfaces of prescribed mean curvature, Indiana Univ. Math. J. 22 (1972), 445 -472 | MR | Zbl

[18] J. Haga, K. Hoshino, N. Kikuchi, Construction of harmonic map flows through the method of discrete Morse flows, Comput. Vis. Sci. 7 (2004), 53 -59 | MR | Zbl

[19] E. Heinz, Über die Existenz einer Fläche konstanter mittlerer Krümmung mit gegebener Berandung, Math. Ann. 137 (1954), 258 -287 | MR | EuDML | Zbl

[20] E. Heinz, F. Tomi, Zu einem Satz von Hildebrandt über das Randverhalten von Minimalflächen, Math. Z. 111 (1969), 372 -386 | MR | EuDML | Zbl

[21] S. Hildebrandt, Randwertprobleme für Flächen mit vorgeschriebener mittlerer Krümmung und Anwendungen auf die Kapillaritätstheorie I, Math. Z. 112 (1969), 205 -213 | MR | EuDML | Zbl

[22] S. Hildebrandt, Einige Bemerkungen über Flächen vorgeschriebener mittlerer Krümmung, Math. Z. 115 (1970), 169 -178 | MR | EuDML | Zbl

[23] S. Hildebrandt, H. Kaul, Two-dimensional variational problems with obstructions, and Plateau's problem for H-surfaces in a Riemannian manifold, Commun. Pure Appl. Math. 25 (1972), 187 -223 | MR | Zbl

[24] M. Hong, D. Hsu, The heat flow for H-systems on higher dimensional manifolds, Indiana Univ. Math. J. 59 no. 3 (2010), 761 -789 | MR | Zbl

[25] C. Imbusch, M. Struwe, Variational principles for minimal surfaces, Topics in Nonlinear Analysis, Prog. Nonlinear Differ. Equ. Appl. vol. 35 , Birkäuser, Basel (1999), 477 -498 | MR | Zbl

[26] N. Kikuchi, An approach to the construction of Morse flows for variational functionals, Nematics, Orsay, 1990, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. vol. 332 , Kluwer Acad. Publ., Dodrecht (1991), 195 -199 | MR | Zbl

[27] C. Leone, M. Misawa, A. Verde, A global existence result for the heat flow of higher dimensional H-systems, J. Math. Pures Appl. 97 no. 3 (2012), 282 -294 | MR | Zbl

[28] G. Mingione, The Calderón–Zygmund theory for elliptic problems with measure data, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6 no. 2 (2007), 195 -261 | MR | EuDML | Zbl | Numdam

[29] C.B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, Heidelberg, New York (1966) | MR | Zbl

[30] R. Moser, Weak solutions of a biharmonic map heat flow, Adv. Calc. Var. 2 (2009), 73 -92 | MR | Zbl

[31] F. Müller, A. Schikorra, Boundary regularity via Uhlenbeck–Rivière decomposition, Analysis 29 (2010), 199 -220 | MR | Zbl

[32] L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (3) 20 (1966), 733 -737 | MR | EuDML | Zbl | Numdam

[33] O. Rey, Heat flow for the equation of surfaces with prescribed mean curvature, Math. Ann. 293 (1991), 123 -146 | MR | EuDML | Zbl

[34] T. Rivière, Conservation laws for conformally invariant variational problems, Invent. Math. 168 (2007), 1 -22 | MR | Zbl

[35] J. Simon, Compact sets in the space $L^p(0,T;B)$ , Ann. Mat. Pura Appl. (4) 146 (1987), 65 -96 | MR | Zbl

[36] C. Scheven, Partial regularity for stationary harmonic maps at a free boundary, Math. Z. 253 no. 1 (2006), 135 -157 | MR | Zbl

[37] L. Simon, Lectures on Geometric Measure Theory, Proc. Centre Math. Anal., Aust. Natl. Univ. vol. 3 , Australian National Univ. (1983) | MR | Zbl

[38] K. Steffen, Isoperimetric inequalities and the problem of Plateau, Math. Ann. 222 (1976), 97 -144 | MR | EuDML | Zbl

[39] K. Steffen, On the existence of surfaces with prescribed mean curvature and boundary, Math. Z. 146 (1976), 113 -135 | MR | EuDML | Zbl

[40] K. Steffen, H. Wente, The nonexistence of branch points in solutions to certain classes of Plateau-type variational problems, Math. Z. 163 no. 3 (1978), 211 -238 | MR | EuDML | Zbl

[41] M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv. 60 (1985), 558 -581 | MR | EuDML | Zbl

[42] M. Struwe, The existence of surfaces of constant mean curvature with free boundaries, Acta Math. 160 (1988), 19 -64 | MR | Zbl

[43] M. Struwe, Plateau's Problem and the Calculus of Variations, Math. Notes vol. 35 , Princeton University Press, Princeton, NJ (1988) | MR | Zbl

[44] F. Tomi, Ein einfacher Beweis eines Regularitätssatzes für schwache Lösungen gewisser elliptischer Systeme, Math. Z. 112 (1969), 214 -218 | MR | EuDML | Zbl

[45] H. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl. 26 (1969), 318 -344 | MR | Zbl