非线性锥退化 Laplace 方程的比较原理——献给陈化教授70寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (2): 503-517.

非线性锥退化 Laplace 方程的比较原理——献给陈化教授70寿辰

魏雅薇^*(), 张梦楠()

南开大学数学科学学院天津 300071

收稿日期:2025-12-08 修回日期:2026-02-05 出版日期:2026-04-26 发布日期:2026-04-27
通讯作者: 魏雅薇 E-mail:weiyawei@nankai.edu.cn;1120220030@mail.nankai.edu.cn
作者简介:张梦楠, Email：1120220030@mail.nankai.edu.cn
基金资助:
国家自然科学基金(12271269)

Comparison Principle for Nonlinear Cone Degenerate Laplace Equations

Yawei Wei^*(), Mengnan Zhang()

School of Mathematical Sciences, Nankai University, Tianjin 300071

Received:2025-12-08 Revised:2026-02-05 Online:2026-04-26 Published:2026-04-27
Contact: Yawei Wei E-mail:weiyawei@nankai.edu.cn;1120220030@mail.nankai.edu.cn
Supported by:
NSFC(12271269)

摘要/Abstract

摘要：

该文研究以下一类由锥退化 Laplace 算子导出的, 以锥微积分为研究背景的非散度型非线性椭圆方程粘性解的比较原理 $$t^{-2}{\rm div}_ \mathbb{B}(\nabla_ \mathbb{B}u) +t^{-2}(n-2)(t\partial_t u)+h(u)=f(t,x) \ \ \ \ (t,x) \in \mathbb{B}.$$ 通过一个特殊的辅助函数, 在 $f,h$ 满足一定条件下建立了方程粘性解的比较原理, 并由此得到了相应的 Dirichlet 问题粘性解的唯一性.

关键词: 比较原理, 锥型奇点, 退化椭圆方程

Abstract:

This paper concerns a class of non-divergence non-linear elliptic equations driven by the cone degenerate Laplacian motivated by cone calculus, as follows $$t^{-2}{\rm div}_ \mathbb{B}(\nabla_ \mathbb{B}u) +t^{-2}(n-2)(t\partial_t u)+h(u)=f(t,x) \ \ \ \ (t,x) \in \mathbb{B}.$$ Using a special auxiliary function, we establish the comparison principle for the viscosity solutions under some assumptions on $f$ and $h$, and then obtain the uniqueness of the viscosity solutions for the corresponding Dirichlet problem.

Key words: comparison principle, conical singularity, degenerate elliptic equations

中图分类号:

O175.25

魏雅薇, 张梦楠. 非线性锥退化 Laplace 方程的比较原理——献给陈化教授70寿辰[J]. 数学物理学报, 2026, 46(2): 503-517.

Yawei Wei, Mengnan Zhang. Comparison Principle for Nonlinear Cone Degenerate Laplace Equations[J]. Acta mathematica scientia,Series A, 2026, 46(2): 503-517.

参考文献 26

[1]	Barles G, Busca J. Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term. Communications in Partial Differential Equations, 2001, 26(11/12): 2323-2337
[2]	Bayraktar E, Ekren I, Zhang X. Comparison of viscosity solutions for a class of second-order PDEs on the Wasserstein space. Communications in Partial Differential Equations, 2025, 50(4): 570-613
[3]	Bieske T, Forrest Z. Existence and uniqueness of viscosity solutions to the infinity Laplacian relative to a class of Grushin-type vector fields. Constructive Mathematical Analysis, 2023, 6(2): 77-89
[4]	Birindelli I, Demengel F. Comparison principle and Liouville type results for singular fully nonlinear operators. Annales de la Faculté des Sciences de Toulouse. Mathématiques, 2004, 13(2): 261-287
[5]	Birindelli I, Demengel F. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure and Applied Analysis, 2007, 6(2): 335-366
[6]	Caffarelli L A, Crandall M G, Kocan M, Swic ech A. On viscosity solutions of fully nonlinear equations with measurable ingredients. Communications on Pure and Applied Mathematics, 1996, 49(4): 365-397
[7]	Chen H, Hu J T, Liu X C, et al. Existence results for cone degenerate $p$-Laplace equations. Communications in Contemporary Mathematics, 2025. DOI:10.1142/S0219199725500932
[8]	Chen H, Liu X C, Wei Y W. Existence theorem for a class of semilinear totally characteristic elliptic equations with critical cone Sobolev exponents. Annals of Global Analysis and Geometry, 2011, 39(1): 27-43
[9]	Chen H, Liu X C, Wei Y W. Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on a manifold with conical singularities. Calculus of Variations and Partial Differential Equations, 2012, 43(3/4): 463-484
[10]	Chen H, Liu X C, Wei Y W. Multiple solutions for semilinear totally characteristic elliptic equations with subcritical or critical cone Sobolev exponents. Journal of Differential Equations, 2012, 252(7): 4200-4228
[11]	Chen H, Wei Y W. Existence of the eigenvalues for the cone degenerate $p$-Laplacian. Chinese Annals of Mathematics Series B, 2021, 42(2): 217-236
[12]	Chen H, Wei Y W. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure and Applied Analysis, 2021, 20(7/8): 2505-2518
[13]	Chen H, Luo P, Tian S Y. Positive solutions for asymptotically linear cone-degenerate elliptic equations. Chinese Annals of Mathematics Series B, 2022, 43(5): 685-718
[14]	Crandall M G, Ishii H, Lions P L. User's guide to viscosity solutions of second order partial differential equations. Bulletin: American Mathematical Society, 1992, 27(1): 1-67
[15]	Crandall M G, Lions P L. Viscosity solutions of Hamilton-Jacobi equations. Transactions of the American Mathematical Society, 1983, 277(1): 1-42
[16]	Egorov Y V, Schulze B W. Pseudo-Differential Operators, Singularities, Applications. Basel: Birkhäuser Verlag, 1997
[17]	Ishii H. Perron's method for Hamilton-Jacobi equations. Duke Mathematical Journal, 1987, 55(2): 369-384
[18]	Ishii H. On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs. Communications on Pure and Applied Mathematics, 1989, 42(1): 15-45
[19]	Ishii H, Lions P L. Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. Journal of Differential equations, 1990, 83(1): 26-78
[20]	Jimenez C, Marigonda A, Quincampoix M. Optimal control of multiagent systems in the {W}asserstein space. Calculus of Variations and Partial Differential Equations, 2020, 59(2): Art 58
[21]	Kawohl B, Kutev N. Comparison principle for viscosity solutions of fully nonlinear, degenerate elliptic equations. Communications in Partial Differential Equations, 2007, 32(7-9): 1209-1224
[22]	Li Y Y. Degenerate conformally invariant fully nonlinear elliptic equations. Archive for Rational Mechanics and Analysis, 2007, 186(1): 25-51
[23]	Li Y Y, Nguyen L, Wang B. Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations. Calculus of Variations and Partial Differential Equations, 2018, 57(4): Art 96
[24]	Li Y Y, Wang B. Strong comparison principles for some nonlinear degenerate elliptic equations. Acta Mathematica Scientia, 2018, 38B(5): 1583-1590
[25]	Schrohe E. Introduction to the Analysis on Manifolds with Conical Singularities. Cham: Birkhäuser, 2024
[26]	Schulze B W. Boundary Value Problems and Singular Pseudo-Differential Operators. Chichester: John Wiley & Sons, 1998