[Extension de la formule de Reilly avec applications aux estimées de valeurs propres pour les laplaciens avec dérive]
In this Note, we extend the Reilly formula for drifting Laplacian operator and apply it to study eigenvalue estimate for drifting Laplacian operators on compact Riemannian manifolds' boundary. Our results on eigenvalue estimates extend previous results of Reilly and Choi and Wang.
Dans cette Note, nous étendons la formule de Reilly au cas des opérateurs Laplaciens avec dérive, et l'appliquons à l'étude d'estimées de valeurs propres pour de tels opérateurs sur des variétés riemanniennes compactes à bord. Nos estimées généralisent des résultats antérieurs de Reilly ainsi que de Choi et Wang.
Accepté le :
Publié le :
Li Ma 1 ; Sheng-Hua Du 2
@article{CRMATH_2010__348_21-22_1203_0, author = {Li Ma and Sheng-Hua Du}, title = {Extension of {Reilly} formula with applications to eigenvalue estimates for drifting {Laplacians}}, journal = {Comptes Rendus. Math\'ematique}, pages = {1203--1206}, publisher = {Elsevier}, volume = {348}, number = {21-22}, year = {2010}, doi = {10.1016/j.crma.2010.10.003}, language = {en}, }
TY - JOUR AU - Li Ma AU - Sheng-Hua Du TI - Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians JO - Comptes Rendus. Mathématique PY - 2010 SP - 1203 EP - 1206 VL - 348 IS - 21-22 PB - Elsevier DO - 10.1016/j.crma.2010.10.003 LA - en ID - CRMATH_2010__348_21-22_1203_0 ER -
Li Ma; Sheng-Hua Du. Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians. Comptes Rendus. Mathématique, Volume 348 (2010) no. 21-22, pp. 1203-1206. doi : 10.1016/j.crma.2010.10.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.10.003/
[1] Some Nonlinear Problems in Riemannian Geometry, Springer Monogr. Math., Springer-Verlag, Berlin, 1998
[2] Diffusion hypercontractivities, Séminaire de Probabilités XIX, 1983/1984 (Lect. Notes in Math.), Volume vol. 1123, Springer, Berlin (1985), pp. 177-206
[3] Volume comparison theorems without Jacobi fields, Current Trends in Potential Theory, Theta Ser. Adv. Math., vol. 4, Theta, Bucharest, 2005, pp. 115-122
[4] A first eigenvalue estimate for minimal hypersurfaces, J. Diff. Geom., Volume 18 (1983), pp. 559-562
[5] Hamilton's Ricci Flow, Lectures in Contemporary Mathematics, vol. 3, Science Press and American Mathematical Society, 2006
[6] Riemannian Geometry, Universitext, Springer-Verlag, Berlin, 2004
[7] The formation of singularities in the Ricci flow, Surveys Diff. Geom., Volume 2 (1995), pp. 7-136
[8] Lecture Notes on Geometric Analysis, Lecture Series, vol. 6, Seoul National University, 1993 http://math.uci.edu/~pli/lecture.pdf
[9] Estimates of eigenvalues of a compact Riemannian manifold, Geometry of Laplace Operator, Proc. Symp. Pure Math., vol. XXXVI, AMS, Providence, RI, 1980, pp. 205-239
[10] On the parabolic kernel of the Schrödinger operator, Acta Math., Volume 156 (1986), pp. 153-201
[11] Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, J. Math. Pure Appl., Volume 84 (2005), pp. 1295-1361
[12] The Bochner technique and modification of the Ricci tensor, Ann. Global Anal. Geom., Volume 36 (2009), pp. 285-291
[13] A lower bound of the first eigenvalue of a closed manifold with positive Ricci curvature, Ann. Global Anal. Geom., Volume 31 (2007) no. 4, pp. 385-408
[14] Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds, J. Funct. Anal., Volume 241 (2006), pp. 374-382
[15] L. Ma, Eigenvalue estimates and L1 energy on closed manifolds, preprint, 2009.
[16] Hamilton type estimates for heat equations on manifolds | arXiv
[17] Convex eigenfunction of a drifting Laplacian operator and the fundamental gap, Pacific J. Math., Volume 240 (2009), pp. 343-361
[18] Convexity of the first eigenfunction of the drifting Laplacian operator and its applications, New York J. Math., Volume 14 (2008), pp. 393-401
[19] The entropy formula for the Ricci flow and its geometric applications, 2002 | arXiv
[20] Applications of Hessian operator in a Riemannian manifold, Indiana Univ. Math. J., Volume 26 (1977), pp. 459-472
[21] Lectures on Differential Geometry, International Press, 1994
[22] Optimal Transport, Old and New, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 2009
[23] Changyu Xia, Universal inequalities for eigenvalues of the vibration problem for a clamped plate on Riemannian manifolds, Quart. J. Math., September 4, 2009, . | DOI
- Inequalities for eigenvalues of the bi-drifting Laplacian on bounded domains in complete noncompact Riemannian manifolds and related results, Annali di Matematica Pura ed Applicata (1923 -), Volume 204 (2025) no. 1, p. 327 | DOI:10.1007/s10231-024-01486-4
- Universal Inequalities for Eigenvalues of a Clamped Plate Problem of the Drifting Laplacian, Bulletin of the Brazilian Mathematical Society, New Series, Volume 55 (2024) no. 1 | DOI:10.1007/s00574-024-00384-w
- A Reilly type integral formula and its applications, Differential Geometry and its Applications, Volume 94 (2024), p. 102136 | DOI:10.1016/j.difgeo.2024.102136
- Weighted Sobolev Type Inequalities in a Smooth Metric Measure Space, Journal of Nonlinear Mathematical Physics, Volume 31 (2024) no. 1 | DOI:10.1007/s44198-024-00168-2
- Some Gradient Estimates for Nonlinear Heat-Type Equations on Smooth Metric Measure Spaces with Compact Boundary, Journal of Nonlinear Mathematical Physics, Volume 31 (2024) no. 1 | DOI:10.1007/s44198-024-00220-1
- Manifolds with Density and the First Steklov Eigenvalue, Potential Analysis, Volume 60 (2024) no. 4, p. 1369 | DOI:10.1007/s11118-023-10091-8
- Eigenvalue Estimates on Weighted Manifolds, Results in Mathematics, Volume 79 (2024) no. 5 | DOI:10.1007/s00025-024-02214-3
- Inner Radius Estimates and Rigidity for a Finsler Measure Space with Boundary, The Journal of Geometric Analysis, Volume 34 (2024) no. 7 | DOI:10.1007/s12220-024-01638-1
- A Reilly Type Integral Formula Associated with Diffusion-Type Operators and Its Applications, Zurnal matematiceskoj fiziki, analiza, geometrii, Volume 20 (2024) no. 2, p. 250 | DOI:10.15407/mag20.02.250
- Estimates for the first eigenvalue of diffusion-type operators in weighted manifolds, Journal of Pseudo-Differential Operators and Applications, Volume 14 (2023) no. 4 | DOI:10.1007/s11868-023-00554-2
- Eigenvalue inequalities for the buckling problem of the drifting Laplacian of arbitrary order, Advances in Nonlinear Analysis, Volume 12 (2022) no. 1 | DOI:10.1515/anona-2022-0278
- Estimates for the first eigenvalues of Bi-drifted Laplacian on smooth metric measure space, Differential Geometry and its Applications, Volume 80 (2022), p. 101839 | DOI:10.1016/j.difgeo.2021.101839
- Gradient estimates for a nonlinear parabolic equation with Dirichlet boundary condition, Kodai Mathematical Journal, Volume 45 (2022) no. 1 | DOI:10.2996/kmj/kmj45106
- On the Spectra of a Family of Geometric Operators Evolving with Geometric Flows, Communications in Mathematics and Statistics, Volume 9 (2021) no. 2, p. 181 | DOI:10.1007/s40304-020-00215-6
- Estimates for eigenvalues of weighted Laplacian and weighted
-Laplacian, Hiroshima Mathematical Journal, Volume 51 (2021) no. 3 | DOI:10.32917/h2020086 - The Frankel property for self-shrinkers from the viewpoint of elliptic PDEs, Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2021 (2021) no. 773, p. 1 | DOI:10.1515/crelle-2020-0044
- Reilly's type inequality for the Laplacian associated to a density related with shrinkers for MCF, Journal of Differential Equations, Volume 272 (2021), p. 958 | DOI:10.1016/j.jde.2020.10.004
- Gradient estimates for weighted harmonic function with Dirichlet boundary condition, Nonlinear Analysis, Volume 213 (2021), p. 112498 | DOI:10.1016/j.na.2021.112498
- Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces, Open Mathematics, Volume 19 (2021) no. 1, p. 1110 | DOI:10.1515/math-2021-0100
- Sharp lower bounds for the first eigenvalues of the bi-drifting Laplacian, Differential Geometry and its Applications, Volume 68 (2020), p. 101572 | DOI:10.1016/j.difgeo.2019.101572
- Eigenvalue comparisons in Steklov eigenvalue problem and some other eigenvalue estimates, Revista Matemática Complutense, Volume 33 (2020) no. 2, p. 389 | DOI:10.1007/s13163-019-00322-1
- On the space of f-minimal surfaces with bounded f-index in weighted smooth metric spaces, manuscripta mathematica, Volume 162 (2020) no. 3-4, p. 559 | DOI:10.1007/s00229-019-01144-7
- Boundary effect of m-dimensional Bakry-Émery Ricci curvature, Analysis and Mathematical Physics, Volume 9 (2019) no. 3, p. 1319 | DOI:10.1007/s13324-018-0237-5
- Generalization of Philippin’s results for the first Robin eigenvalue and estimates for eigenvalues of the bi-drifting Laplacian, Annals of Global Analysis and Geometry, Volume 55 (2019) no. 4, p. 805 | DOI:10.1007/s10455-019-09652-1
- Some Almost-Schur type inequalities for k−Bakry-Emery Ricci tensor, Differential Geometry and its Applications, Volume 66 (2019), p. 82 | DOI:10.1016/j.difgeo.2019.05.009
- Some integral inequalities forLoperator and their applications on self-shrinkers, Journal of Mathematical Analysis and Applications, Volume 463 (2018) no. 2, p. 645 | DOI:10.1016/j.jmaa.2018.03.038
- ESTIMATES FOR THE EIGENVALUES OF THE DRIFTING LAPLACIAN ON SOME COMPLETE RICCI SOLITONS, Kyushu Journal of Mathematics, Volume 72 (2018) no. 1, p. 143 | DOI:10.2206/kyushujm.72.143
- Sobolev inequalities on a weighted Riemannian manifold of positive Bakry–Émery curvature and convex boundary, Pacific Journal of Mathematics, Volume 294 (2018) no. 2, p. 423 | DOI:10.2140/pjm.2018.294.423
- Liouville theorems, volume growth, and volume comparison for Ricci shrinkers, Pacific Journal of Mathematics, Volume 296 (2018) no. 2, p. 357 | DOI:10.2140/pjm.2018.296.357
- Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary, The Journal of Geometric Analysis, Volume 27 (2017) no. 2, p. 1680 | DOI:10.1007/s12220-016-9736-5
- Estimates for eigenvalues of a system of elliptic equations with drift and of bi-drifting Laplacian, Communications on Pure and Applied Analysis, Volume 16 (2016) no. 2, p. 475 | DOI:10.3934/cpaa.2017024
- Lower bound estimates for the first eigenvalue of the weighted p-Laplacian on smooth metric measure spaces, Differential Geometry and its Applications, Volume 45 (2016), p. 23 | DOI:10.1016/j.difgeo.2015.11.008
- Eigenvalue inequalities for the buckling problem of the drifting Laplacian on Ricci solitons, Journal of Differential Equations, Volume 260 (2016) no. 7, p. 5533 | DOI:10.1016/j.jde.2015.12.006
- Universal inequalities for eigenvalues of a system of elliptic equations of the drifting Laplacian, Monatshefte für Mathematik, Volume 181 (2016) no. 4, p. 797 | DOI:10.1007/s00605-015-0875-8
- Eigenvalues of the drifting Laplacian on complete noncompact Riemannian manifolds, Nonlinear Analysis: Theory, Methods Applications, Volume 141 (2016), p. 1 | DOI:10.1016/j.na.2016.03.017
- Sharp bounds for the first nonzero Steklov eigenvalues for
-Laplacians, TURKISH JOURNAL OF MATHEMATICS, Volume 40 (2016), p. 770 | DOI:10.3906/mat-1507-96 - A Compactness Theorem of the Space of Free Boundary f-Minimal Surfaces in Three-Dimensional Smooth Metric Measure Space with Boundary, The Journal of Geometric Analysis, Volume 26 (2016) no. 3, p. 1995 | DOI:10.1007/s12220-015-9616-4
- Estimates for Eigenvalues of the Elliptic Operator in Divergence Form on Riemannian Manifolds, Advances in Mathematical Physics, Volume 2015 (2015), p. 1 | DOI:10.1155/2015/387953
- Universal bounds for eigenvalues of the polydrifting Laplacian operator in compact domains in the
R n and S n, Annals of Global Analysis and Geometry, Volume 47 (2015) no. 4, p. 373 | DOI:10.1007/s10455-015-9450-8 - Universal inequalities of the poly-drifting Laplacian on the Gaussian and cylinder shrinking solitons, Annals of Global Analysis and Geometry, Volume 48 (2015) no. 3, p. 255 | DOI:10.1007/s10455-015-9469-x
- Isoperimetric inequalities on weighted manifolds with boundary, Doklady Mathematics, Volume 92 (2015) no. 2, p. 537 | DOI:10.1134/s1064562415050063
- f-Minimal Surface and Manifold with Positive m-Bakry–Émery Ricci Curvature, The Journal of Geometric Analysis, Volume 25 (2015) no. 1, p. 421 | DOI:10.1007/s12220-013-9434-5
- Estimates for eigenvalues of the bi-drifting Laplacian operator, Zeitschrift für angewandte Mathematik und Physik, Volume 66 (2015) no. 3, p. 703 | DOI:10.1007/s00033-014-0426-5
- Eigenvalue estimates and L 1 energy on closed manifolds, Acta Mathematica Sinica, English Series, Volume 30 (2014) no. 10, p. 1729 | DOI:10.1007/s10114-014-1726-6
- Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds, Annals of Global Analysis and Geometry, Volume 45 (2014) no. 3, p. 155 | DOI:10.1007/s10455-013-9392-y
- Eigenvalue problems on Riemannian manifolds with a modified Ricci tensor, Annals of Global Analysis and Geometry, Volume 46 (2014) no. 1, p. 63 | DOI:10.1007/s10455-014-9409-1
- Compact manifolds with positive m -Bakry–Émery Ricci tensor, Differential Geometry and its Applications, Volume 32 (2014), p. 88 | DOI:10.1016/j.difgeo.2013.10.018
- Rigidity Theorems for Diameter Estimates of Compact Manifold with Boundary, International Mathematics Research Notices (2014) | DOI:10.1093/imrn/rnu052
- Applications of some elliptic equations in Riemannian manifolds, Journal of Mathematical Analysis and Applications, Volume 409 (2014) no. 1, p. 189 | DOI:10.1016/j.jmaa.2013.07.004
- Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds, Journal of Mathematical Analysis and Applications, Volume 419 (2014) no. 1, p. 617 | DOI:10.1016/j.jmaa.2014.04.074
- Eigenvalue estimate and compactness for closedf-minimal surfaces, Pacific Journal of Mathematics, Volume 271 (2014) no. 2, p. 347 | DOI:10.2140/pjm.2014.271.347
- Inequalities for the Steklov eigenvalues, Chaos, Solitons Fractals, Volume 48 (2013), p. 61 | DOI:10.1016/j.chaos.2013.01.008
- Remarks on scalar curvature of Yamabe solitons, Annals of Global Analysis and Geometry, Volume 42 (2012) no. 2, p. 195 | DOI:10.1007/s10455-011-9308-7
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☆ The research is partially supported by the National Natural Science Foundation of China 10631020 and SRFDP 20090002110019.
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