کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
8896660 | 1630593 | 2018 | 33 صفحه PDF | دانلود رایگان |
عنوان انگلیسی مقاله ISI
Stable solutions of symmetric systems involving hypoelliptic operators
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موضوعات مرتبط
مهندسی و علوم پایه
ریاضیات
اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
چکیده انگلیسی
Let X and Y be two noncommuting vector fields in an open set Ω in a manifold M equipped with a sub-Riemannian structure. We examine stable solutions of the following symmetric systemÎXYui=Hi(u1,â¯,um)in Ω for 1â¤iâ¤m, when the operator ÎXY is the Hörmander's operator given by ÎXY(â
):=X(Xâ
)+Y(Yâ
) and HiâC1(Rm). We prove the following identity for any wâC2(Ω)|âXYXw|2+|âXYYw|2â|X|âXYw||2â|Y|âXYw||2={|âXYw|2[A2+B2]in {|âXYw|>0}â©Î©,0a.e.in {|âXYw|=0}â©Î©, where A is the intrinsic curvature of the level sets of w and B is connected with the intrinsic normal and the intrinsic tangent direction to the level sets and also with the Lie bracket [X,Y]. We then apply this to establish a geometric Poincaré inequality for stable solutions of the above system for general vector fields X and Y. This inequality enables us to analyze the level sets of stable solutions. In addition, we provide certain reduction of dimensions results which can be regarded as counterparts of the classical De Giorgi type results. This is remarkable since the classical one-dimensional symmetry results do not hold for general vector fields. Our approaches can be applied, but not limited, to the Grushin vector fields X=(1,0) and Y=(0,x) in R2 and the Heisenberg vector fields X=(1,0,ây2) and Y=(0,1,x2) in R3 and their multidimensional extensions. These specific vector fields generate nonelliptic operators which are hypoelliptic.
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Functional Analysis - Volume 274, Issue 12, 15 June 2018, Pages 3470-3502
Journal: Journal of Functional Analysis - Volume 274, Issue 12, 15 June 2018, Pages 3470-3502
نویسندگان
Mostafa Fazly,