Quadratic tilt-excess decay and strong maximum principle for varifolds

Schätzle, Reiner

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 1, pp. 171-231.

In this paper, we prove that integral $n$-varifolds $\mu$ in codimension 1 with $H_{\mu }\in L_{\mathrm{loc}}^p\left(\mu \right)$, $p>n$, $p\ge 2$ have quadratic tilt-excess decay ${\mathrm{tiltex}}_{\mu }(x,\varrho ,T_x\mu )=O_x\left({\varrho }^2\right)$for $\mu$-almost all $x$, and a strong maximum principle which states that these varifolds cannot be touched by smooth manifolds whose mean curvature is given by the weak mean curvature $H_{\mu }$, unless the smooth manifold is locally contained in the support of $\mu$.

Classification : 49Q15, 35J60, 53A10

[All72] W. K. Allard, On the first variation of a varifold, Ann. of Math. 95 (1972), 417-491. | MR | Zbl

[Bra78] K. Brakke, “The motion of a surface by its mean curvature”, Princeton University Press, 1978. | MR | Zbl

[Cab00] X. Cabré, oral communication.

[Caf89] L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math., 130 (1989), 189-213. | MR | Zbl

[CafCab] L. A. Caffarelli - X. Cabré, “Fully Nonlinear Elliptic equations”, American Mathematical Society, 1996. | MR | Zbl

[CafCKS96] L. A. Caffarelli - M. G. Crandall - M. Kocan - A. Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math. 49, (1996), 365-397. | MR | Zbl

[CIL] M. G. Crandall - H. Ishii - P.-L. Lions, User's Guide to Viscosity Solutions of second Order Partial Differential Equations, Bull. Amer. Math. Soc. 27 (1992), 1-67. | MR | Zbl

[DuSt94] F. Duzaar - K. Steffen, Comparison principles for hypersurfaces of prescribed mean curvature, J. Reine Angew. Math. 457 (1994), 71-83. | MR | Zbl

[Es93] L. Escauriaza, $W^{2,n}$ apriori estimates for solutions of fully non-linear equations, Indiana Univ. Math. J. 42 (1993), 413-423. | MR | Zbl

[Ev82] L. C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982), 333-363. | MR | Zbl

[F] H. Federer, “Geometric Measure Theory”, Springer Verlag, Grund. Math. Wiss., Band 153, Berlin - Heidelberg - New York, 1969. | MR | Zbl

[GT] D. Gilbarg - N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Springer Verlag, Berlin - Heidelberg - New York - Tokyo 1983. | MR | Zbl

[Il96] T. Ilmanen, A strong maximum principle for singular minimal hypersurfaces, Calc. Var. Partial Differential Equations, 4 (1996), 443-467. | MR | Zbl

[Kry83] N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations, Math. USSR Izv. 20 (1983), 459-492. | Zbl

[Mo77] M. P. Moschen, Principio di Massimo Forte per le Frontiere di Misura Minima, Ann. Univ. Ferrara, Sez. VII - Sc. Mat. 23 (1977), 165-168. | MR | Zbl

[Resh68] Y. G. Reshetnyak, Generalized derivatives and differentiability almost everywhere, Math. USSR-Sb. 4 (1968), 293-302. | MR | Zbl

[Sch01] R. Schätzle, Hypersurfaces with mean curvature given an ambient Sobolev function, J. Differential Geom. 58 (2001), 371-420. | MR | Zbl

[Sim] L. Simon, “Lectures on Geometric Measure Theory”, Proceedings of the Centre for Mathematical Analysis Australian National University, Volume 3, 1983. | MR | Zbl

[Sim87] L. Simon, A strict maximum principle for area minimizing hypersurfaces, J. Differential Geom. 26 (1987), 327-335. | MR | Zbl

[SW89] B. Solomon - B. White, A Strong Maximum Principle for Varifolds that are Stationary with Respect to Even Parametric Elliptic Functionals, Indiana Univ. Math. J. 38 (1989), 683-691. | MR | Zbl

[T89] N. S. Trudinger, On the twice differentiability of viscosity solutions of nonlinear elliptic equations, Bull. Austral. Math. Soc. 39 (1989), 443-447. | MR | Zbl

[Wa92] L. Wang, On the regularity theory of fully nonlinear parabolic equations I, Comm. Pure Appl. Math. 45 (1992), 27-76. | MR | Zbl

[Wh34] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63-89. | JFM | MR | Zbl

[Wi04] N. Winter, “$W^{2,p}$-Randabschätzungen für Lösungen von voll nicht-linearen elliptischen Gleichungen”, diploma thesis, 2004.

[Zie] W. Ziemer, “Weakly Differentiable Functions”, Springer Verlag, 1989. | MR | Zbl