اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدWeighted ${L^p}$-Liouville Theorems for Hypoelliptic Partial Differential Operators on Lie Groups
126
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We prove weighted $L^p$-Liouville theorems for a class of second order hypoelliptic partial differential operators $mathcal{L}$ on Lie groups $mathbb{G}$ whose underlying manifold is $n$-dimensional space. We show that a natural weight is the right-invariant measure $check{H}$ of $mathbb{G}$. We also prove Liouville-type theorems for $C^2$ subsolutions in $L^p(mathbb{G},check{H})$. We provide examples of operators to which our results apply, jointly with an application to the uniqueness for the Cauchy problem for the evolution operator $mathcal{L}-partial_t$.
قيم البحث
اقرأ أيضاً
We prove some $L^p$-Liouville theorems for hypoelliptic second order Partial Differential Operators left translation invariant with respect to a Lie group composition law in $mathbb{R}^n$. Results for both solutions and subsolutions are given.
We consider nonnegative solutions $u:Omegalongrightarrow mathbb{R}$ of second order hypoelliptic equations begin{equation*} mathscr{L} u(x) =sum_{i,j=1}^n partial_{x_i} left(a_{ij}(x)partial_{x_j} u(x) right) + sum_{i=1}^n b_i(x) partial_{x_i} u(x) =
0, end{equation*} where $Omega$ is a bounded open subset of $mathbb{R}^{n}$ and $x$ denotes the point of $Omega$. For any fixed $x_0 in Omega$, we prove a Harnack inequality of this type $$sup_K u le C_K u(x_0)qquad forall u mbox{ s.t. } mathscr{L} u=0, ugeq 0,$$ where $K$ is any compact subset of the interior of the $mathscr{L}$-propagation set of $x_0$ and the constant $C_K$ does not depend on $u$.
We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushimas ergodic theorem for the harmonic functions in the domain of the $ L^{p} $ generator. Secondly we prove analogues of Yaus and Karps Liouville theorems
for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain $ L^{p} $ growth criteria must be constant. As consequence we give an integral criterion for recurrence.
This note contains a representation formula for positive solutions of linear degenerate second-order equations of the form $$ partial_t u (x,t) = sum_{j=1}^m X_j^2 u(x,t) + X_0 u(x,t) qquad (x,t) in mathbb{R}^N times, ]- infty ,T[,$$ proved by a func
tional analytic approach based on Choquet theory. As a consequence, we obtain Liouville-type theorems and uniqueness results for the positive Cauchy problem.
We prove, with a purely analytic technique, a one-side Liouville theorem for a class of Ornstein--Uhlenbeck operators ${mathcal L_0}$ in $mathbb{R}^N$, as a consequence of a Liouville theorem at $t=- infty$ for the corresponding Kolmogorov operator
s ${mathcal L_0} - partial_t$ in $mathbb{R}^{N+1}$. In turn, this last result is proved as a corollary of a global Harnack inequality for non-negative solutions to $({mathcal L_0} - partial_t) u = 0$ which seems to have an independent interest in its own right. We stress that our Liouville theorem for ${mathcal L_0}$ cannot be obtained by a probabilistic approach based on recurrence if $N>2$. We provide a self-contained proof of a Liouville theorem involving recurrent Ornstein--Uhlenbeck stochastic processes in the Appendix.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
