Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userYoung measures, Cartesian maps, and polyconvexity
106
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider the variational problem consisting of minimizing a polyconvex integrand for maps between manifolds. We offer a simple and direct proof of the existence of a minimizing map. The proof is based on Young measures.
rate research
Read More
Let E be a compact set in the plane, g be a K-quasiconformal map, and let 0<t<2. Then H^t (E) = 0 implies H^{t} (g E) = 0, for t=[2Kt]/[2+(K-1)t]. This is a refinement of a set of inequalities on the distortion of Hausdorff dimensions by quasiconformal maps proved by K. Astala in his celebrated paper on area distortion for quasiconformal maps and answers in the positive a Conjecture of K. Astala in op. cit.
It is an open question whether the linear extension complexity of the Cartesian product of two polytopes P, Q is the sum of the extension complexities of P and Q. We give an affirmative answer to this question for the case that one of the two polytopes is a pyramid.
We prove that there exists an open and dense subset $mathcal{U}$ in the space of $C^{2}$ expanding self-maps of the circle $mathbb{T}$ such that the Lyapunov minimizing measures of any $Tin{mathcal U}$ are uniquely supported on a periodic orbit.This answers a conjecture of Jenkinson-Morris in the $C^2$ topology.
We propose a new notion called emph{infinity-harmonic maps}between Riemannain manifolds. These are natural generalizations of the well known notion of infinity harmonic functions and are also the limiting case of $p$% -harmonic maps as $pto infty $. Infinity harmoncity appears in many familiar contexts. For example, metric projection onto the orbit of an isometric group action from a tubular neighborhood is infinity harmonic. Unfortunately, infinity-harmonicity is not preserved under composition. Those infinity harmonic maps that always preserve infinity harmonicity under pull back are called infinity harmonic morphisms. We show that infinity harmonic morphisms are precisely horizontally homothetic mas. Many example of infinity-harmonic maps are given, including some very important and well-known classes of maps between Riemannian manifolds.
In the first part of this paper, we develop the theory of anisotropic curvature measures for convex bodies in the Euclidean space. It is proved that any convex body whose boundary anisotropic curvature measure equals a linear combination of other lower order anisotropic curvature measures with nonnegative coefficients is a scaled Wulff shape. This generalizes the classical results by Schneider [Comment. Math. Helv. textbf{54} (1979), 42--60] and by Kohlmann [Arch. Math. (Basel) textbf{70} (1998), 250--256] to the anisotropic setting. The main ingredients in the proof are the generalized anisotropic Minkowski formulas and an inequality of Heintze--Karcher type for convex bodies. In the second part, we consider the volume preserving flow of smooth closed convex hypersurfaces in the Euclidean space with speed given by a positive power $alpha $ of the $k$th anisotropic mean curvature plus a global term chosen to preserve the enclosed volume of the evolving hypersurfaces. We prove that if the initial hypersurface is strictly convex, then the solution of the flow exists for all time and converges to the Wulff shape in the Hausdorff sense. The characterization theorem for Wulff shapes via the anisotropic curvature measures will be used crucially in the proof of the convergence result. Moreover, in the cases $k=1$, $n$ or $alphageq k$, we can further improve the Hausdorff convergence to the smooth and exponential convergence.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
