A NOTE FOR $W^{1,P}(V)$ AND $W_{0}^{1,P}(V)$ ON A LOCALLY FINITE GRAPH

doi:10.1007/s10473-025-0517-8

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (5): 2135-2141.doi: 10.1007/s10473-025-0517-8

A NOTE FOR $W^{1,P}(V)$ AND $W_{0}^{1,P}(V)$ ON A LOCALLY FINITE GRAPH

Yulu TIAN^*, Liang ZHAO

School of Mathematical Sciences Key Laboratory of Mathematics and Complex Systems of MOE, Beijing Normal University, Beijing 100875, China

收稿日期:2024-02-21 出版日期:2025-09-25 发布日期:2025-10-14

A NOTE FOR $W^{1,P}(V)$ AND $W_{0}^{1,P}(V)$ ON A LOCALLY FINITE GRAPH

Yulu TIAN^*, Liang ZHAO

School of Mathematical Sciences Key Laboratory of Mathematics and Complex Systems of MOE, Beijing Normal University, Beijing 100875, China

Received:2024-02-21 Online:2025-09-25 Published:2025-10-14
Contact: ^*Yulu Tian, E-mail: tianyl@mail.bnu.edu.cn
About author:Liang Zhao, E-mail: liangzhao@bnu.edu.cn
Supported by:
Tian's research was supported by the National Key R and D Program of China (2020YFA0713100), the National Natural Science Foundation of China (12271039) and the Open Project Program (K202303) of Key Laboratory of Mathematics and Complex Systems, Beijing Normal University.

摘要/Abstract

摘要： In this paper, we investigate the Sobolev spaces $W^{1,p}(V)$ and $W_{0}^{1,p}(V)$ on a locally finite graph $G=(V,E)$, which are fundamental tools when we apply the variational methods to partial differential equations on graphs. As a key contribution of this note, we show that in general, $W^{1,p}(V)\neq W_0^{1,p}(V)$ on locally finite graphs, which is different from the situation on Euclidean space $\mathbb{R}^N$.

关键词: Sobolev space, locally finite graph, Cheeger constant

Abstract: In this paper, we investigate the Sobolev spaces $W^{1,p}(V)$ and $W_{0}^{1,p}(V)$ on a locally finite graph $G=(V,E)$, which are fundamental tools when we apply the variational methods to partial differential equations on graphs. As a key contribution of this note, we show that in general, $W^{1,p}(V)\neq W_0^{1,p}(V)$ on locally finite graphs, which is different from the situation on Euclidean space $\mathbb{R}^N$.

Key words: Sobolev space, locally finite graph, Cheeger constant

中图分类号:

46E39

Yulu TIAN, Liang ZHAO. A NOTE FOR $W^{1,P}(V)$ AND $W_{0}^{1,P}(V)$ ON A LOCALLY FINITE GRAPH[J]. 数学物理学报(英文版), 2025, 45(5): 2135-2141.

Yulu TIAN, Liang ZHAO. A NOTE FOR $W^{1,P}(V)$ AND $W_{0}^{1,P}(V)$ ON A LOCALLY FINITE GRAPH[J]. Acta mathematica scientia,Series B, 2025, 45(5): 2135-2141.

参考文献

[1] Aubin T. Espaces de Sobolev sur les variétés riemanniennes. Bull Sci Math, 1976, 100(2): 149-173
[2] Bauer F, Keller M, Wojciechowski R K. Cheeger inequalities for unbounded graph Laplacians. J Eur Math Soc, 2015, 17(2): 259-271
[3] Chang X J, Wang R, Yan D K. Ground states for logarithmic Schrödinger equations on locally finite graphs. J Geom Anal, 2023, 33(7): 211
[4] Ge H B. A $p$-th Yamabe equation on graph. Proc Amer Math Soc, 2018, 146(5): 2219-2224
[5] Ge H B. The $p$-th Kazdan-Warner equation on graphs. Commun Contemp Math, 2020, 22(6): Art 1950052
[6] Grigor'yan A, Lin Y, Yang Y Y. Kazdan-Warner equation on graph. Calc Var Partial Differential Equations, 2016, 55(4): 1-13
[7] Grigor'yan A, Lin Y, Yang Y Y. Yamabe type equations on graphs. J Differential Equations, 2016, 261(9): 4924-4943
[8] Grigor'yan A, Lin Y, Yang Y Y. Existence of positive solutions to some nonlinear equations on locally finite graphs. Sci China Math, 2017, 60(7): 1311-1324
[9] Han X L, Shao M Q. $p$-Laplacian equations on locally finite graphs. Acta Math Sin (Engl Ser), 2021, 37(11): 1645-1678
[10] Han X L, Shao M Q, Zhao L. Existence and convergence of solutions for nonlinear biharmonic equations on graphs. J Differential Equations, 2020, 268(7): 3936-3961
[11] Hebey E.Sobolev Spaces on Riemannian Manifolds. Berlin: Springer-Verlag, 1996
[12] Hou S B, Sun J M. Existence of solutions to Chern-Simons-Higgs equations on graphs. Calc Var Partial Differential Equations, 2022, 61(4): Art 139
[13] Hua B B, Li R W, Wang L D. A class of semilinear elliptic equations on groups of polynomial growth. J Differential Equations, 2023, 363: 327-349
[14] Hua B B, Xu W D. Existence of ground state solutions to some nonlinear Schrödinger equations on lattice graphs. Calc Var Partial Differential Equations, 2023, 62(4): Art 127
[15] Keller M, Lenz D. Dirichlet forms and stochastic completeness of graphs and subgraphs. J Reine Angew Math, 2012, 666: 189-223
[16] Keller M, Lenz D, Wojciechowski R K.Graphs and Discrete Dirichlet Spaces. Cham: Springer, 2021
[17] Keller M, Mugnolo D. General Cheeger inequalities for $p$-Laplacians on graphs. Nonlinear Anal, 2016, 147(4): 80-95
[18] Keller M, Schwarz M. The Kazdan-Warner equation on canonically compactifiable graphs. Calc Var Partial Differential Equations, 2018, 57(2): 1-18
[19] Ni C C, Lin Y Y, Luo F, Gao J. Community detection on networks with Ricci flow. Sci Rep, 2019, 9(1): Art 9984
[20] Shao M Q, Yang Y Y, Zhao L. Sobolev spaces on locally finite graphs. Proc Amer Math Soc, 2025, 153(2): 693-708
[21] Sun L L, Wang L Q. Brouwer degree for Kazdan-Warner equations on a connected finite graph. Adv Math, 2022, 404(2): Art 108422
[22] Zhang N, Zhao L. Convergence of ground state solutions for nonlinear Schrödinger equations on graphs. Sci China Math, 2018, 61(8): 1481-1494
[23] Zhang X X, Lin A J. Positive solutions of $p$-th Yamabe type equations on infinite graphs. Proc Amer Math Soc, 2019, 147(4): 1421-1427

A NOTE FOR $W^{1,P}(V)$ AND $W_{0}^{1,P}(V)$ ON A LOCALLY FINITE GRAPH

A NOTE FOR $W^{1,P}(V)$ AND $W_{0}^{1,P}(V)$ ON A LOCALLY FINITE GRAPH

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 0

[1]	Yun-Ho KIM, Taek-Jun JEONG, Jun-Yeob, SHIM. ON KIRCHHOFF-HARDY TYPE PROBLEMS INVOLVING DOUBLE PHASE OPERATORS[J]. 数学物理学报(英文版), 2025, 45(5): 1814-1854.
[2]	Min ZHU, Hongshuai DAI. ASYMPTOTIC BEHAVIOR OF STOCHASTIC ANISOTROPIC NAVIER-STOKES MODELS[J]. 数学物理学报(英文版), 2025, 45(5): 2264-2278.
[3]	Lucas DA SILVA, Marco A. S. SOUTO. A GENERALIZED CHOQUARD EQUATION WITH WEIGHTED ANISOTROPIC STEIN-WEISS POTENTIAL ON A NONREFLEXIVE ORLICZ-SOBOLEV SPACES[J]. 数学物理学报(英文版), 2025, 45(2): 569-601.
[4]	Yong CHEN, Young Joo LEE. SUMS OF DUAL TOEPLITZ PRODUCTS ON THE ORTHOGONAL COMPLEMENTS OF FOCK-SOBOLEV SPACES[J]. 数学物理学报(英文版), 2024, 44(3): 810-822.
[5]	Jin Tao, Zhenyu Yang, Wen Yuan. JOHN-NIRENBERG-Q SPACES VIA CONGRUENT CUBES^*[J]. 数学物理学报(英文版), 2023, 43(2): 686-718.
[6]	Guiqiao XU, Yongping Liu, Dandan GUO. OPTIMAL BIRKHOFF INTERPOLATION AND BIRKHOFF NUMBERS IN SOME FUNCTION SPACES^*[J]. 数学物理学报(英文版), 2023, 43(1): 125-142.
[7]	李思然. A NEW PROOF OF GAFFNEY’S INEQUALITY FOR DIFFERENTIAL FORMS ON MANIFOLDS-WITH-BOUNDARY: THE VARIATIONAL APPROACH À LA KOZONO–YANAGISAWA[J]. 数学物理学报(英文版), 2022, 42(4): 1427-1452.
[8]	邓方文, 欧阳才衡, 彭茹. SOME QUESTIONS REGARDING VERIFICATION OF CARLESON MEASURES[J]. 数学物理学报(英文版), 2021, 41(6): 2136-2148.
[9]	李丹, 李俊峰, 肖杰. AN UPBOUND OF HAUSDORFF'S DIMENSION OF THE DIVERGENCE SET OF THE FRACTIONAL SCHRÖDINGER OPERATOR ON H^s($\mathbb{R}^n$)[J]. 数学物理学报(英文版), 2021, 41(4): 1223-1249.
[10]	Abdelkrim MOUSSAOUI, Jean VELIN. EXISTENCE AND BOUNDEDNESS OF SOLUTIONS FOR SYSTEMS OF QUASILINEAR ELLIPTIC EQUATIONS[J]. 数学物理学报(英文版), 2021, 41(2): 397-412.
[11]	郭常予, 向长林. REGULARITY OF P-HARMONIC MAPPINGS INTO NPC SPACES[J]. 数学物理学报(英文版), 2021, 41(2): 633-645.
[12]	戴济能, 周静云. GLEASON'S PROBLEM ON FOCK-SOBOLEV SPACES[J]. 数学物理学报(英文版), 2021, 41(1): 337-348.
[13]	Jae-Myoung KIM, Yun-Ho KIM, Jongrak LEE. RADIALLY SYMMETRIC SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING NONHOMOGENEOUS OPERATORS IN AN ORLICZ-SOBOLEV SPACE SETTING[J]. 数学物理学报(英文版), 2020, 40(6): 1679-1699.
[14]	陶文宇, 陈艳萍, 李纪黎. GRADIENT ESTIMATES FOR THE COMMUTATOR WITH FRACTIONAL DIFFERENTIATION FOR SECOND ORDER ELLIPTIC OPERATORS[J]. 数学物理学报(英文版), 2019, 39(5): 1255-1264.
[15]	林勇, 吴艺婷. BLOW-UP PROBLEMS FOR NONLINEAR PARABOLIC EQUATIONS ON LOCALLY FINITE GRAPHS[J]. 数学物理学报(英文版), 2018, 38(3): 843-856.