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Sharp Reilly-type inequalities for a class of elliptic operators on submanifolds. (English) Zbl 1415.58008

Summary: Let \(M\) be an \(n(> 2)\)-dimensional closed orientable submanifold in an \((n + p)\)-dimensional space form \(\mathbb{R}^{n + p}(c)\). We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on \(M\) defined by \(L_T f = - \operatorname{div}(T {\nabla} f)\), where \(T\) is a general symmetric, positive definite and divergence-free \((1, 1)\)-tensor on \(M\). The upper bound is given in terms of an integration involving tr \(T\) and \(| H_T |^2\), where tr \(T\) is the trace of the tensor \(T\) and \(H_T = \sum_{i = 1}^n A(T e_i, e_i)\) is a normal vector field associated with \(T\) and the second fundamental form \(A\) of \(M\). Furthermore, we give the sufficient and necessary conditions when the upper bound is attained. Our main theorem can be viewed as an extension of the famous “Reilly inequality”. The operator \(L_T\) can be regarded as a natural generalization of the well-known operator \(L_r\) which is the linearized operator of the first variation of the \((r + 1)\)-th mean curvature for hypersurfaces in a space form. As applications of our main theorem, we generalize the results of J.-F. Grosjean [ibid. 13, No. 3, 267–276 (2000; Zbl 0977.53052)] and H. Li and X. Wang [Proc. Am. Math. Soc. 140, No. 1, 291–307 (2012; Zbl 1252.53071)] in codimension one to arbitrary codimension.

MSC:

58C40 Spectral theory; eigenvalue problems on manifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
35P15 Estimates of eigenvalues in context of PDEs
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