非光滑泛函的局部C1(Ω) -极小对W1, p(x)(Ω) -极小

数学物理学报 ›› 2013, Vol. 33 ›› Issue (3): 424-430.

非光滑泛函的局部C¹(Ω) -极小对W^{1, p(x)}(Ω) -极小

代国伟|达婷

西北师范大学数学与统计学院兰州 730070

收稿日期:2011-11-02 修回日期:2012-11-08 出版日期:2013-06-25 发布日期:2013-06-25
基金资助:
国家自然科学基金(11261052, 11101335)资助

Local C¹(Ω) -minimizers Versus W^{1, p(x)}(Ω)-minimizers of Nonsmooth Functionals

DAI Guo-Wei, DA Ting

Department of Mathematics, Northwest Normal University, Gansu Lanzhou 730070

Received:2011-11-02 Revised:2012-11-08 Online:2013-06-25 Published:2013-06-25
Supported by:
国家自然科学基金(11261052, 11101335)资助

摘要/Abstract

摘要：

研究如下的非可微泛函
J(u)=∫_Ω1/p(x)(|∨u|^p^(x)+|u|^p^(x))dx+∫_Ω j₁(x, u)dx+∫_∂Ωj₂(x, γu)dσ,
其中p(x)∈ C^0, β(Ω), β∈(0,1), 1<p^-≤p⁺<+∞, j₁: Ω×R→R 和j₂: ∂Ω×R→R 是局部Lipschitz 函数. 该文证明了J 的局部C¹(Ω) -极小一定是J 的局部W^{1, p(x)}(Ω) -极小.

关键词: 非光滑泛函, 局部极小, 变指数

Abstract:

In this paper, we study not necessarily differentiable functionals of the form
J(u)=∫_Ω1/p(x)(|∨u|^p^(x)+|u|^p^(x))dx+∫_Ω j₁(x, u)dx+∫_∂Ωj₂(x, γu)dσ,
where p(x)∈ C^0, β(Ω), with β∈(0,1), and 1<p^-≤p⁺<+∞, j₁: Ω×R→R as well as j₂: ∂Ω×R→R are locally Lipschitz functions. We prove that local C¹(Ω)-minimizers of J must be local W^{1, p(x)}(Ω)-minimizers of J.

Key words: Nonsmooth functionals, Local minimizers, Variable exponent

中图分类号:

35R70

代国伟, 达婷. 非光滑泛函的局部C¹(Ω) -极小对W^{1, p(x)}(Ω) -极小[J]. 数学物理学报, 2013, 33(3): 424-430.

DAI Guo-Wei, DA Ting. Local C¹(Ω) -minimizers Versus W^{1, p(x)}(Ω)-minimizers of Nonsmooth Functionals[J]. Acta mathematica scientia,Series A, 2013, 33(3): 424-430.

参考文献

[1] Adams R A. Sobolev Spaces//Pure and Applied Mathematics, vol 65. Academic Press (A subsidiary of Harcourt Brace Jovanovich, Publishers), New York: London, 1975

[2] Azorero J G, Manfredi J J, Alonso I P. Sobolev versus H\"{o}lder local minimizer and global multiplicity for some
quasilinear elliptic equations. Commun Contemp Math, 2000, 2: 385--404

[3] Brezis H, Nirenberg L. H1 versus C¹ local minimizers. C R Acad Sci Paris S\'{e}r I Math, 1993, 317: 465--472

[4] Clarke F H. A new approach to Lagrange multipliers. Math Oper Res, 1976, 1(2): 165--174

[5] Clarke F H. Optimization and Nonsmooth Analysis. New York: Wiley, 1983

[6] Chang K C. Variational methods for nondifferentiable functionals and their applications to partial differential equations. Math J Anal Appl, 1981, 80: 102--129

[7] Fan X L. On the sub-supersolution methods for p(x)-Laplacian equations.J Math Anal Appl, 2007, 330: 665--682

[8] Fan X L. Global C¹,^α regularity for variable exponent elliptic equations in divergence form. J Differential Equations, 2007, 235: 397--417

[9] Fan X L. Boundary trace embedding theorems for variable exponent Sobolev spaces. J Math Anal Appl, 2008, 339: 1395--1412

[10] Fan X L, Zhao D. A class of de giorgi type and H\"{o}lder continuity. Nonlinear Anal, 1996, 36: 295--318

[11] Fan X L, Zhao D. On the spaces L^p^(x) and W^{m, p}^(x). J Math Anal Appl, 2001, 263: 424--446

[12] L. Gasi\'{e}ski and N.S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Chapman and Hall/CRC, Boca Raton (2005)

[13] Guo Z M, Zhang Z T. W^1, ^p versus C¹ local minimizers and multiplicity results for quasilinear elliptic equations. J Math Anal Appl, 2003, 286: 32--50

[14] Winkert P. Local C¹-minimizers versus local W^{1, p}-minimizers of nonsmooth functionals. Nonlinear Analysis, 2010, 72: 4298--4303

非光滑泛函的局部C¹(Ω) -极小对W^{1, p(x)}(Ω) -极小

Local C¹(Ω) -minimizers Versus W^{1, p(x)}(Ω)-minimizers of Nonsmooth Functionals

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 7

Metrics

本文评价

推荐阅读 10

[1]	王立伟, 束立生. 本征平方函数在变指数Herz及Herz-Hardy空间上的有界性[J]. 数学物理学报, 2018, 38(4): 716-727.
[2]	黄正刚. 一种新的广义次梯度及其性质[J]. 数学物理学报, 2015, 35(5): 927-935.
[3]	佟玉霞, 郑神州, 于海燕. 变指数A-调和方程弱解的梯度的局部Hölder连续性[J]. 数学物理学报, 2015, 35(4): 656-667.
[4]	王华, 贺艺军. 一类具变指数和正能量的半线性双曲方程解的爆破[J]. 数学物理学报, 2015, 35(2): 288-293.
[5]	宋文晶, 郭斌, 高文杰. 具变指数的弱耦合抛物方程组解的爆破和全局有界性[J]. 数学物理学报, 2013, 33(2): 285-291.
[6]	付永强, 张夏. R^N上一类p(x)-Laplacian 方程的无穷多解问题[J]. 数学物理学报, 2010, 30(2): 465-471.
[7]	朱文兴. 总体优化一类双参数填充函数算法的改进[J]. 数学物理学报, 1999, 19(5): 550-558.