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Lower bounds of eigenvalues for a class of bi-subelliptic operators ☆

Research paper by Hua Chen, Yifu Zhou

Indexed on: 04 Apr '17Published on: 22 Feb '17Published in: Journal of Differential Equations



Abstract

Let Ω be a bounded open domain in RnRn with smooth boundary and X=(X1,X2,⋯,Xm)X=(X1,X2,⋯,Xm) be a system of real smooth vector fields defined on Ω with the boundary ∂Ω which is non-characteristic for X. If X   satisfies the Hörmander's condition, then the vector fields are finitely degenerate and the sum of square operators <img height="20" border="0" style="vertical-align:bottom" width="102" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022039617300797-si122.gif">△X=∑i=1mXi2 is a subelliptic operator. Let λkλk be the k  -th eigenvalue for the bi-subelliptic operator <img height="20" border="0" style="vertical-align:bottom" width="23" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022039617300797-si5.gif">△X2 on Ω. In this paper, we introduce the generalized Métivier's condition and study the lower bounds of Dirichlet eigenvalues for the operator <img height="20" border="0" style="vertical-align:bottom" width="23" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022039617300797-si5.gif">△X2 on some finitely degenerate systems of vector fields X   which satisfy the Hörmander's condition or the generalized Métivier's condition. By using the subelliptic estimates, we shall give a explicit lower bound estimates of λkλk which is polynomial increasing in k with the order relating to the Hörmander index or the generalized Métivier index.