Stability of weighted Laplace-Beltrami operators under \(L^ p\)-perturbation of the Riemannian metric. (English) Zbl 0868.35025

Let \((M,G)\) be a Riemannian manifold, where \(G\) is a suitable measurable metric. The author considers operators of the form \(H:f\mapsto- \sigma^{-2}\nabla_G\cdot (\sigma^2\nabla_G f)+f\), where \(\sigma\) is a measurable weight. When \(M\) has no boundary, he shows that \(H^{-1}\) maps \(L^p\) continuously into \(W^{1,p}\) if \(p\) is close to 2; as \(G\), \(\sigma\) vary in \(L^p\), quantitative estimates for the stability of \(H^{-1}\) in trace norms are given. This result is extended to the case when \(M\) has a boundary and mixed Dirichlet-Neumann boundary conditions are imposed. Actually, this paper extends earlier results by the same author published in a previous paper [J. Differ. Equations 124, No. 2, 302-323 (1996; Zbl 0848.35082)], where \(M\) was a bounded domain with Lipschitz boundary in \(\mathbb{R}^n\) and the Dirichlet boundary condition was imposed.


35J15 Second-order elliptic equations
58J32 Boundary value problems on manifolds
35J25 Boundary value problems for second-order elliptic equations


Zbl 0848.35082
Full Text: DOI


[1] Barbatis, G., Spectral stability under L^p-perturbation of the second-order coefficients, J. Diff. Equ., 124, 302-323 (1996) · Zbl 0848.35082 · doi:10.1006/jdeq.1996.0011
[2] Birman, M. S.; Solomjak, M. Z., Spectral asymptotics of non-smooth elliptic operators. I, Trans. Moscow Math. Soc., 27, 1-52 (1972) · Zbl 0296.35065
[3] E. B. Davies,Heat Kernels and Spectral Theory, Cambridge University Press, 1989. · Zbl 0699.35006
[4] Deift, P., Applications of a commutation formula, Duke Math. J., 45, 267-309 (1978) · Zbl 0392.47013 · doi:10.1215/S0012-7094-78-04516-7
[5] Dixmier, J., Les algèbres d’opérateurs dans l’espace Hilbertien (1957), Paris: Gauthier-Villars, Paris · Zbl 0088.32304
[6] Gröger, K., A W^1,p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann., 283, 679-687 (1989) · Zbl 0646.35024 · doi:10.1007/BF01442860
[7] Meyers, N. G., An L^p-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Sci. Norm. Sup. Pisa, 17, 189-206 (1963) · Zbl 0127.31904
[8] Simader, C. G., On Dirichlet’s boundary value problem (1972), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0242.35027
[9] B. Simon,Trace ideals and their applications, London Math. Soc. Lecture Note Series, Vol. 35, Cambridge University Press, 1979. · Zbl 0423.47001
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