Properties of the $C^1$-smooth functions whose gradient range has topological dimension 1

Properties of the $C^1$-smooth functions whose gradient range has topological dimension 1

Mon, 25/01/2010
17:00 		Mikhail Korobkov (Sobolev Institute of Mathematics) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Properties of the $ C^1 $-smooth functions whose gradient range has topological dimension 1
In the talk we discuss some results of [1]. We apply our previous methods [2] to geometry and to the mappings with bounded distortion. Theorem 1.  Let $ v:\Omega\to\mathbb{R} $ be a $ C^1 $-smooth function on a domain (open connected set) $ \Omega\subset\mathbb{R}^2 $. Suppose
$$ (1)\qquad \operatorname{Int} \nabla v(\Omega)=\emptyset. $$
Then $ \operatorname{meas}\nabla v(\Omega)=0 $. Here $ \operatorname{Int}E $ is the interior of $ {E} $, $ \operatorname{meas} E $ is the Lebesgue measure of $ {E} $. Theorem 1 is a straight consequence of the following two results. Theorem 2 [2].  Let $ v:\Omega\to\mathbb{R} $ be a $ C^1 $-smooth function on a domain $ \Omega\subset\mathbb{R}^2 $. Suppose (1) is fulfilled. Then the graph of $ v $ is a normal developing surface.  Recall that a $ C^1 $-smooth manifold $ S\subset\mathbb{R}^3 $ is called  a normal developing surface [3] if for any $ x_0\in S $ there exists a straight segment $ I\subset S $ (the point $ x_0 $ is an interior point of $ I $) such that the tangent plane to $ S $ is stationary along $ I $. Theorem 3.  The spherical image of any $ C^1 $-smooth normal developing surface $ S\subset\mathbb{R}^3 $ has the area (the Lebesgue measure) zero. Recall that the spherical image of a surface $ S $ is the set $ \{\mathbf{n}(x)\mid x\in S\} $, where $ \mathbf{n}(x) $ is the unit normal vector to $ S $ at the point~$ x $. From Theorems 1–3 and the classical results of A.V. Pogorelov (see [4, Chapter 9]) we obtain the following corollaries Corollary 4. Let the spherical image of a $ C^1 $-smooth surface $ S\subset\mathbb{R}^3 $ have no interior points. Then this surface is a surface of zero extrinsic curvature in the sense of Pogorelov. Corollary 5. Any $ C^1 $-smooth normal developing surface $ S\subset\mathbb{R}^3 $ is a surface of zero extrinsic curvature in the sense of Pogorelov. Theorem 6. Let $ K\subset\mathbb{R}^{2\times 2} $ be a compact set and the topological dimension of $ K $ equals 1. Suppose there exists $ \lambda> 0 $ such that $ \forall A,B\in K, \, \, |A-B|^2\le\lambda\det(A-B). $ Then for any Lipschitz mapping $ f:\Omega\to\mathbb R^2 $ on a domain $ \Omega\subset\mathbb R^2 $ such that $ \nabla f(x)\in K $ a.e. the identity $ \nabla f\equiv\operatorname{const} $ holds. Many partial cases of Theorem 6 (for instance, when $ K=SO(2) $ or $ K $ is a segment) are well-known (see, for example, [5]). The author was supported by the Russian Foundation for Basic Research (project no. 08-01-00531-a).   [1] {Korobkov M.\,V.,} {“Properties of the $ C^1 $-smooth functions whose gradient range has topological dimension~1,” Dokl. Math., to appear.} [2] {Korobkov M.\,V.} {“Properties of the $ C^1 $-smooth functions with nowhere dense gradient range,” Siberian Math. J., 48, No.~6, 1019–1028 (2007).} [3] { Shefel$ {}' $ S.\,Z.,} {“$ C^1 $-Smooth isometric imbeddings,” Siberian Math. J., 15, No.~6, 972–987 (1974).} [4] {Pogorelov A.\,V.,} {Extrinsic geometry of convex surfaces, Translations of Mathematical Monographs. Vol. 35. Providence, R.I.: American Mathematical Society (AMS). VI (1973).} [5] {Müller ~S.,} {Variational Models for Microstructure and Phase Transitions. Max-Planck-Institute for Mathematics in the Sciences. Leipzig (1998) (Lecture Notes, No.~2. http://www.mis.mpg.de/jump/publications.html).}