RUS  ENG ЖУРНАЛЫ   ПЕРСОНАЛИИ   ОРГАНИЗАЦИИ   КОНФЕРЕНЦИИ   СЕМИНАРЫ   ВИДЕОТЕКА   ЛИЧНЫЙ КАБИНЕТ
Видеотека
Архив
Популярное видео

Поиск
RSS
Новые поступления





Для просмотра файлов Вам могут потребоваться






Международная молодежная конференция «Геометрия и управление»
18 апреля 2014 г. 12:00, г. Москва, МИАН
 


Geometric and Analytic Properties of Carnot–Carathéodory Spaces under Minimal Smoothness

Sergey Basalaev

Novosibirsk State University, Novosibirsk, Russia
Видеозаписи:
Flash Video 158.2 Mb
Flash Video 946.9 Mb
MP4 158.2 Mb
Материалы:
Adobe PDF 209.0 Kb
Adobe PDF 95.6 Kb

Количество просмотров:
Эта страница:114
Видеофайлы:51
Материалы:57

Sergey Basalaev


Видео не загружается в Ваш браузер:
  1. Установите Adobe Flash Player    

  2. Проверьте с Вашим администратором, что из Вашей сети разрешены исходящие соединения на порт 8080
  3. Сообщите администратору портала о данной ошибке

Аннотация: We describe geometric and analytical results in the theory of non-holonomic spaces under minimal smoothness, which we define following works [1, 2].
$ $
Definition. Fix a connected Riemannian $C^\infty$-manifold $\mathbb{M}$ of topological dimension $N$. The manifold $\mathbb{M}$ is called the Carnot–Carathéodory space if the tangent bundle $T\mathbb{M}$ has a filtration
$$ H \mathbb{M} = H_1 \mathbb{M} \subsetneq H_2 \mathbb{M} \subsetneq …\subsetneq H_M \mathbb{M} = T \mathbb{M} $$
by subbundles such that every point $x_0 \in \mathbb{M}$ has a neighborhood $U(x_0) \subset \mathbb{M}$ equipped with a collection of $C^1$-smooth vector fields $X_1, …, X_N$ enjoying the following two properties:
$ $
(1) At every point $x \in U(x_0)$ we have a subspace
$$ H_i \mathbb{M} (x) = H_i (x) = \mathrm{span} \{ X_1(x), …, X_{\dim H_i}(x) \} \subset T_x \mathbb{M} $$
of the dimension $\dim H_i$ independent of $x$, $i = 1, …, M$.
$ $
(2) The inclusion $[H_i, H_j] \subset H_{i+j}$ holds for $i+j \leq M$.
$ $
Moreover, the Carnot–Carathéodory space is called the Carnot manifold if the following third condition holds:
$ $
(3) $H_{j+1} = \mathrm{span} \{ H_j, [H_1, H_j], …, [H_k, H_{j+1-k}] \}$, where $k = \lfloor \tfrac{j+1}{2} \rfloor$ for $j = 1, …, M-1$.
$ $
Since it is not a priori known whether Carnot manifolds carry Carnot–Carathéodory metric, the Nagel–Stein–Wainger “Box” metric $d_\infty(x,y)$ is used instead in their study. Using results on fine properties of Carnot–Carathéodory spaces [2] we show that Carnot–Carathéodory metric is well-defined proving an analogue of Carathéodory– Rashevskiĭ–Chow theorem.
$ $
Theorem [3]. 1) For every point $g \in \mathbb{M}$ there is a neighborhood $U$ and $C > 0$ such that every point $x \in U$ can be represented as
$$ x = \exp(a_L X_{i_L}) \circ …\circ \exp(a_1 X_{i_1})(g) $$
with $i_k \in \{ 1, …, \dim H_1 \}$ and $|a_k| \leq C d_\infty(x,g)$ for $k = 1, …, L$. Here $L = L(\mathbb{M})$ does not depend on the points $g$, $x$.
$ $
2) In a connected Carnot manifold any two points can be joined by an absolutely continuous curve consisting of finitely many segments of integral lines of vector fields $X_1, …, X_{\dim H_1}$.
$ $
This result in turn was utilized in [4] to prove local equivalence of “Box” metric $d_\infty$ and Carnot–Carathéodory metric $d_{cc}$ which immediately implies that:
$\bullet$ locally there is $C>0$ such that $B(x, C^{-1} r) \subset \mathrm{Box}(x,r) \subset B(x, Cr)$;
$\bullet$ the Hausdorff dimension of $\mathbb{M}$ is $\nu = \sum\limits_{k=1}^N \deg X_k$;
$\bullet$ the Hausdorff measure $\mathcal{H}^\nu$ is locally doubling.
As an application of these results to a theory of Sobolev spaces we obtain the Poincaré inequality for John domains in Carnot manifolds.
$ $
Theorem [5]. Let $x_0 \in \mathbb{M}$ and $1 \leq p < \infty$. There are $C_p > 0$ and $r_0 > 0$ such that for every John domain $\Omega \subset B(x_0, r_0)$ of class $J(a,b)$ and every $f \in C^\infty(\overline{\Omega})$ we have
$$ \Vert f - f_\Omega \Vert_{L_p(\Omega)} \leq (\tfrac{b}{a})^\nu \mathrm{diam} (\Omega) \Vert (X_1 f, …, X_{\dim H_1} f) \Vert_{L_p(\Omega)} $$
where $f_\Omega = \tfrac{1}{|\Omega|} \int_\Omega f$ and $\nu$ is the Hausdorff dimension of $\mathbb{M}$.

Материалы: slides.pdf (209.0 Kb), abstract.pdf (95.6 Kb)

Язык доклада: английский

Список литературы
  1. Vodopyanov S. K, Geometry of Carnot–Carathéodory spaces and differentiability of mappings. In: The Interaction of Analysis and Geometry, Amer. Math. Soc. Providence, 2007. P. 247–302.
  2. Karmanova M., Vodopyanov S, Geometry of Carnot–Carathéodory spaces, differentiability, coarea and area formulas. In: Analysis and Mathematical Physics. Trends in Mathematics. Birkhäuser, Basel, 2009. P. 233–335.
  3. Basalaev S. G., Vodopyanov S. K, Approximate differentiability of mappings of Carnot–Carathéodory spaces. // Eurasian Math. J. 2013. V. 4, N. 2. P. 10–48.
  4. Karmanova M., Vodopyanov S, On local approximation theorem on equiregular Carnot–Carathéodory spaces. In: Proc. INDAM Meeting on Geometric Control and Sub-Riemannian Geometry. Springer INDAM Ser., 2014. V. 5. P. 241–262.
  5. Basalaev S, The Poincaré inequality for $C^{1,\alpha}$-smooth vector fields. // Siberian Math. J. 2014. V. 55, N. 2. P. 210–224.


ОТПРАВИТЬ: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru
 
Обратная связь:
 Пользовательское соглашение  Регистрация  Логотипы © Математический институт им. В. А. Стеклова РАН, 2017