Heisenberg群上几类偏微分方程解的性质
|
摘要
次黎曼流形(Sub-Riemannian manifold),粗略地讲,就是被赋予了一个分布及此分布上的一个纤维内积的流形,当考虑的分布为整个切丛时,次黎曼流形就成为黎曼流形.次黎曼流形研究的主要来源之一是控制论；同时,次黎曼流形也被用来研究力学中的非完整系统.近些年来,众多学者对次黎曼流形作了大量研究,内容涉及分析、方程、代数、几何等领域.而Heisenberg群是一类最简单的、非平凡的次黎曼流形；同时,由于平移映射族和伸缩映射族的存在,Heisenberg群具有丰富的内蕴结构,而且其几何结构与欧氏空间有本质的区别,因此开展Heisenberg群上的几何和分析问题的研究具有很重要的理论意义和实际价值.本论文主要研究三方面内容：一是讨论Heisenberg群上H-调和函数的增长性；二是讨论Heisenberg群上一类含有奇异位势项的散度型变系数次椭圆方程弱解的唯一延拓性；三是研究Heisenberg群上次p-Laplace方程及抛物型次p-Laplace方程的粘性解的渐近平均值公式.
在H-调和函数增长性方面,我们首先讨论Heisenberg群上H-调和函数的Almgren频率的性质,得到了H-调和函数的Almgren频率与其在原点消失阶的内在关系;然后证明了H-调和函数一种新的Liouville型定理：全空间上频率有界的H-调和函数必是多项式；最后应用H-调和多项式的正交性讨论其局部增长性,证明当频率为常数时,一类具有旋转不变性的H-调和函数是齐次多项式.
在次椭圆方程解的唯一延拓性方面,我们讨论Heisenberg群上一类含有奇异位势项的散度型变系数次椭圆方程弱解的唯一延拓性.首先定义这类方程弱解的频率函数,证明频率函数的单调性,然后利用频率函数单调性证明双倍条件,最后证明当方程系数、位势函数满足一定条件时解具有唯一延拓性.
在渐近平均值公式方面,我们首先证明Heisenberg群上函数水平最大(最小)增长方向与水平导数之间的关系,这个关系正是Heisenberg群区别于欧氏空间的集中表现；之后研究次p-Laplace方程粘性解的渐近平均值公式,证明了粘性解和渐近平均值公式的等价性,并举反例说明这类渐近平均值公式在非渐近情形下是不成立的；最后我们讨论Heisenberg群上抛物型次p-Laplace方程,证明抛物型次p-Laplace方程粘性解的一个等价定理,在其基础上用渐近平均值公式刻画抛物型次p-Laplace方程的粘性解.
A sub-Riemannian manifold, roughly speaking, is a smooth manifold associated with a distribution and a fibre-inner product on it. When the distribution is the whole tangent bun-dle, then the sub-Riemannian manifold reduces to be a Riemannian manifold. Sub-Riemannian structures arise, for instance, when dealing with non-holonomic systems in mechanics and in control theory. In recent years, there are a lot of investigations on sub-Riemannian manifolds which have strong relationships with many fields such as analysis, PDE, algebra and geometry. The Heisenberg group is the most important and simplest model of sub-Riemannian manifolds. Moreover, because of the existence of the translation and dilation, the Heisenberg group has rich intrinsic structure. Therefore, the research on the Heisenberg group has great significance in theory and application. This thesis deals with problems in three aspects:one is to discuss the growth of H-harmonic functions on the Heisenberg group; the second, we study unique contin-uation property of solutions of sub-elliptic equations with singular potential on the Heisenberg group; finally, we discuss asymptotic mean value formulae for the viscosity solutions to sub-p-Laplace equations and to sub-p-Laplace parabolic equations on the Heisenberg group.
In the first aspect, the properties of Almgren's frequency for H-harmonic functions are discussed. The relationship between frequency and the vanishing order at the origin is obtained. A new Liouville type Theorem is proved, that is, an H-harmonic function on the Heisenberg group with bounded frequency is a polynomial. Finally, we show that a class of H-harmonic functions are homogeneous polynomials provided that the frequency of such a function is equal to some constant.
In the second aspect, the Almgren's frequency for solutions of a class of sub-elliptic equa-tions with singular potential on the Heisenberg group is introduced. The monotonicity property of the frequency is established and a doubling condition is obtained. Consequently, a quantita-tive proof of the unique continuation property for these equations is given.
In the third aspect, we first characterize the directions of horizontal maximum (minima) of a function in terms of the horizontal gradient. Secondly, we characterize sub-p-harmonic functions on the Heisenberg group in terms of an asymptotic mean value property; Moreover, we construct an example to show that these formulae do not hold in non-asymptotic sense. Fi-nally, we derive two equivalent definitions of the viscosity solutions to sub-p-Laplace parabolic equations on the Heisenberg group, and characterize the viscosity solutions of sub-p-Laplace parabolic equations in terms of an asymptotic mean value formula.
在H-调和函数增长性方面,我们首先讨论Heisenberg群上H-调和函数的Almgren频率的性质,得到了H-调和函数的Almgren频率与其在原点消失阶的内在关系;然后证明了H-调和函数一种新的Liouville型定理：全空间上频率有界的H-调和函数必是多项式；最后应用H-调和多项式的正交性讨论其局部增长性,证明当频率为常数时,一类具有旋转不变性的H-调和函数是齐次多项式.
在次椭圆方程解的唯一延拓性方面,我们讨论Heisenberg群上一类含有奇异位势项的散度型变系数次椭圆方程弱解的唯一延拓性.首先定义这类方程弱解的频率函数,证明频率函数的单调性,然后利用频率函数单调性证明双倍条件,最后证明当方程系数、位势函数满足一定条件时解具有唯一延拓性.
在渐近平均值公式方面,我们首先证明Heisenberg群上函数水平最大(最小)增长方向与水平导数之间的关系,这个关系正是Heisenberg群区别于欧氏空间的集中表现；之后研究次p-Laplace方程粘性解的渐近平均值公式,证明了粘性解和渐近平均值公式的等价性,并举反例说明这类渐近平均值公式在非渐近情形下是不成立的；最后我们讨论Heisenberg群上抛物型次p-Laplace方程,证明抛物型次p-Laplace方程粘性解的一个等价定理,在其基础上用渐近平均值公式刻画抛物型次p-Laplace方程的粘性解.
A sub-Riemannian manifold, roughly speaking, is a smooth manifold associated with a distribution and a fibre-inner product on it. When the distribution is the whole tangent bun-dle, then the sub-Riemannian manifold reduces to be a Riemannian manifold. Sub-Riemannian structures arise, for instance, when dealing with non-holonomic systems in mechanics and in control theory. In recent years, there are a lot of investigations on sub-Riemannian manifolds which have strong relationships with many fields such as analysis, PDE, algebra and geometry. The Heisenberg group is the most important and simplest model of sub-Riemannian manifolds. Moreover, because of the existence of the translation and dilation, the Heisenberg group has rich intrinsic structure. Therefore, the research on the Heisenberg group has great significance in theory and application. This thesis deals with problems in three aspects:one is to discuss the growth of H-harmonic functions on the Heisenberg group; the second, we study unique contin-uation property of solutions of sub-elliptic equations with singular potential on the Heisenberg group; finally, we discuss asymptotic mean value formulae for the viscosity solutions to sub-p-Laplace equations and to sub-p-Laplace parabolic equations on the Heisenberg group.
In the first aspect, the properties of Almgren's frequency for H-harmonic functions are discussed. The relationship between frequency and the vanishing order at the origin is obtained. A new Liouville type Theorem is proved, that is, an H-harmonic function on the Heisenberg group with bounded frequency is a polynomial. Finally, we show that a class of H-harmonic functions are homogeneous polynomials provided that the frequency of such a function is equal to some constant.
In the second aspect, the Almgren's frequency for solutions of a class of sub-elliptic equa-tions with singular potential on the Heisenberg group is introduced. The monotonicity property of the frequency is established and a doubling condition is obtained. Consequently, a quantita-tive proof of the unique continuation property for these equations is given.
In the third aspect, we first characterize the directions of horizontal maximum (minima) of a function in terms of the horizontal gradient. Secondly, we characterize sub-p-harmonic functions on the Heisenberg group in terms of an asymptotic mean value property; Moreover, we construct an example to show that these formulae do not hold in non-asymptotic sense. Fi-nally, we derive two equivalent definitions of the viscosity solutions to sub-p-Laplace parabolic equations on the Heisenberg group, and characterize the viscosity solutions of sub-p-Laplace parabolic equations in terms of an asymptotic mean value formula.
引文
[1]G.K. Alexopoulos, Sub-Laplacians with drift on Lie groups of polynomial volume growth, (Vol.739) Mem. Amer. Math. Soc.,2002.
[2]F.J.Jr. Almgren, Dirichlet's problem for muptiple valued functions and the regular-ity of mass minimizing integral currents, in "Minimal Submanifolds and Geodesies", (M.Obata,Ed.) 1-6, North Holland, Amsterdam,1979.
[3]G. Arena, A.O. Caruso, and A. Causa, Taylor formula on step two Carnot groups, Rev. Mat. Iberoamericana,26(1)(2010),239-259.
[4]N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl.,36(9)(1957),235-249.
[5]N. Aronszajn, A. Krzywicki and J. Szarski, A unique continuation theorems for exterior differential forms on Riemannian manifolds, Arkiv for Matematik,4(1962),417-435.
[6]S. Axler, P. Bourdon and W. Ramey, Harmonic function theory (second edition), (GTM137), Springer,2004.
[7]H. Bahouri, Non prolongement unique des solutions d'ope rateurs, Ann. Inst. Fourier, Grenoble,36(4)(1986),137-155.
[8]W. Bechner, On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math. Soc., 129(2001),1233-1246.
[9]A. Bellaiche, The tangent space in sub-Riemannian geometry. Sub-Riemannian geome-try, Progr. Math.,144(1996),1-78.
[10]T. Bieske, On oo-harmonic functions on the Heisenberg group, Comm. Partial Differen-tial Equations,27(3,4)(2002),727-761.
[11]T. Bieske, Equivalence of weak and viscosity solutions to the p-Laplace equation in the Heisenberg group, Annales Academiae Scientiarum Fennicae Mathematica, 31(2006),363-379.
[12]A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified lie groups and potential theory for their sub-laplacians, Springer-Verlag Berlin Heidelberg, New York,2007.
[13]J.M. Bony, Principe du maximum, integalite de Harnack et unicite d u probleme de Cauchy pour les operateurs elliptique degeneres, Ann. Inst. Fourier, Grenoble, 1(119)(1969),277-304.
[14]L. Capogna, D. Danielli and N. Garofalo, An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations, 18(1993),1765-1794.
[15]L. Capogna, Regularity of quasi-linear equations in the Heisenberg group, Comm. Pure Appl. Math.,50(1997),867-889.
[16]L. Capogna, D. Danielli and N. Garofalo, Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math.,118(1996),1153-1196.
[17]L. Capogna, N. Garofalo, Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Homander type, J. Eur. Math. Soc.,5(2003),1-40.
[18]T. Carleman, Sur un probleme d'unicite pur les systemes d'equations aux derivees par-tielles a deux variables independantes, Ark. Mat., Astr. Fys.,26(17)(1939),1-9.
[19]Y. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of gener-alized mean curvature flow equations, J. Differential Geom.,33(1991),749-786.
[20]M.G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc.,27(1)(1992),1-67.
[21]J. Cygan, Subadditivity of homogeneous norms on certain nilpotent Lie groups, Proc. Amer. Math. Soc.,83(1)(1981),69-70.
[22]E. Dibenedetto, CI+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal.,7(1983),827-850.
[23]A. Domokos, JJ. Manfredi, Subelliptic Cordes estimates, Proc. Amer. Math. Soc., 133(2005),1047-1056.
[24]A. Domokos, Differentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg group, J. Differential Equations,204(2004),439-470.
[25]C.F. Dunkl, An addition theorem for Heisenberg harmonics, in coeference on harmonic analysis in honor of Antoni Zygmund. Wadsworth International, (1982),688-705.
[26]L.C. Evans, J. Spruck, Motion of level sets by mean curvature I, J. Differential Geom., 33(1991),635-681.
[27]H. Federer, Geometric measure theory, Springer, Berlin,Heidelberg, New York,1969.
[28]G.B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc, 79(1973),373-376.
[29]G.B. Folland, E. M. Stein, Hardy spaces on homogeneus groups, Princeton University Press and University of Tokyo Press,1982.
[30]N. Garofalo, E. Lanconelli, Frequency functions on the Heisenberg group, the uncer-tainty principle and unique continuation, Ann. Inst. Fourier,44(2)(1990),313-356.
[31]N. Garofalo, F.-H. Lin, Monotonicity properties of variational integrals,Ap weights and unique continuation, Indiana Univ. Math. J.,35(1986),245-267.
[32]N. Garofalo, F.-H. Lin, Unique continuation for elliptic operators:a geometric-variational approach, Comm. Pure Appl. Math.,40(3)(1987),347-366.
[33]N. Garofalo, D. Vassilev, Strong unique continuation properties of generalized Baouendi-Grushin operators, Comm. Partial Differential Equations,32(2007),643-663.
[34]N. Garofalo, Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension. J. Differential Equations,104(l)(1993),117-146.
[35]N. Garofalo, Z.W. Shen, Carleman estimates for a subelliptic operator and unique con-tinuation, Ann. Inst. Fourier,44(1)(1994),129-166.
[36]N. Garofalo, Gradient bounds for the horizontal p-Laplacian on a Carnot group and some applications, Manuscripta Math.,130(2009),375-385.
[37]B. Gaveau, Principe de moindre action, propagation de la chaleur et estimees sous ellip-tiques sur certains groups nilpotents, Acta Math.,139(1977),95-153.
[38]PC. Greiner, Spherical harmonics on the Heisenberg group, Canad. Math. Bull., 23(4)(1980),383-396.
[39]V.V. Grushin, On a class of hypoelliptic operators, Math. USSR Sbornik, 12(3)(1970),458-476.
[40]V.V. Grushin, On a class of hypoelliptic pseudodifferential operators degenerate on sub-manifold, Math. USSR Sbornik,13(2)(1971),155-186.
[41]Q. Han, Singular stes to elliptic equations, Indiana Univ. Math. J.,43(1994),983-1002.
[42]Q. Han, Singular sets of harmonic functions in R2 and their complexifications in C2, Indiana Univ. Math. J.,53(5)(2004),1365-1380.
[43]Q. Han, F.-H. Lin, Nodal sets of solutions of elliptic differential equations, Books avail-able on Han's homepage,2010.
[44]J. Heinonen, I. Holopainen, Quasiregular maps on Carnot groups, J. Geom. Anal., 7(1997),109-148.
[45]E. Holmgren, Uber Systeme von linearen partiellen Differentialgleichungen", Ofversigt af Kongl. Vetenskaps-Academien Forhandlinger,58(1901),91-103.
[46]H. Hormander, Hypoelliptic second-order differential equations, Acta Math., 119(1967),147-171.
[47]D. Jerison, C.E. Kenig, Unique continuation and absence of positive eigenvalues for Schroinger operators, Ann. of Math.,121(3) (1985),463-494.
[48]P. Juutinen, B. Kawohl, On the evolution governed by the infinity Laplacian, Math. Ann., 335(2006),819-851.
[49]H. Koch, D. Tataru, Carleman estimates and unique continuation for second order elliptic equations with nonsmooth coefficients, Comm. Pure Appl. Math.,54(3)(2001),339-360.
[50]J.L. Lewis, Regularity of the Derivatives of Solutions to Certain Degenerate Elliptic Equations, Indiana Univ. Math. J.,32(6)(1983),849-858.
[51]F.-H. Lin, Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., XLIV(1991),287-308.
[52]JJ. Manfredi, M. Parviainen and J.D. Rossi, An asymptotic mean value characterization for p-harmonic functions, Proc. Amer. Math. Soc.,138(3)(2010),881-889.
[53]J.J. Manfredi, M. Parviainen and J.D. Rossi, An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games, SIAM J. Math. Anal.,42(5)(2010),2058-2081.
[54]J.J. Manfredi, M. Parviainen and J.D. Rossi, Dynamic programming principle for tug-of-war games with noise, ESAIM:Control, Optimisation and Calculus of Variations, 18(1)(2012),81-90.
[55]S. Marchi, C1,α local regularity for the solutions of the p-Laplacian on the Heisenberg group for 2≤ p ≤ 1+√5, J. for Analysis and its Applications,20(2001),617-636.
[56]S. Marchi, Cha local regularity for the solutions of the p-Laplacian on the Heisenberg group for 1+1/√5< p≤ 2, Comm. Math. Univ. Carolinae,44 (2003),33-56.
[57]T. Matsuzawa, Gevrey hypoellipticity for Grushin operators, Publ, RIMS, Kyoto Univ., 33(1997),775-799.
[58]K. Miller, Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Holder continuous coefficients, Arch. Rational Mech. Anal.,54(1974),105-117.
[59]G. Mingione, A. Zatorska-Goldstein and X. Zhong, Gardient regularity for elliptic equa-tions in the Heisenberg group, Advances in Math.,222(2009),62-129.
[60]R. Monti, D. Morbidelli, Kelvin tranform for Grushin operators and critical semilinear equations, Duke Math. J.,131(1)(2006),167-202.
[61]A. Nagel, E.M. Stein and S. Wainger, Balls and metrics defined by vector fields I:basic properties, Acta Math.,155(1985),103-147.
[62]S. Ohnuma, K. Sato, Singular degenerate parabolic equations with applications to the p-Laplace diffusion equation, Comm. Partial Differential Equations,22(1997),381-411.
[63]Y. Peres, S. Sheffield, Tug-of-war with noise:a game theoretic view of the p-Laplacian, Duke Math. J.,145(2008),91-120.
[64]Y. Peres, O. Schramm, S. Sheffield, and D. B. Wilson, Tug-of-war and the infinity Lapla-cian, J. Amer. Math. Soc.,22(2009),167-210.
[65]E.M. Stein, Harmonic Analysis, Princeton University Press, Princeton, NJ,1993.
[66]G. Szego, Orthogonal polynomials, Amer. Math. Soc. Colloquium Publication,23, Amer. Math. Soc, PRovidence, R.I.1939.
[67]K.H. Tan, X.-P. Yang, Sub-Riemannian objects and hypersurfaces of sub-Riemannian manifolds, Bull. Austral Math. Soc.,70(2)(2004),177-198.
[68]L. Tian, X.-P. Yang, Nodal sets and horizontal singular sets of H-harmonic functions on the Heisenberg group, Preprint.
[69]J.L. Wang, P.C. Niu, Unique Continuation Properties for Generalized Baouendi-Grushin Operators with Singular Weights, Acta Mathematica Sinica, English Series, 27(8)(2011),1637-1644.
[70]C.-J. Xu, Subelliptic variational problems, Bull. Soc. Math. France,118(1990),147-169.
[2]F.J.Jr. Almgren, Dirichlet's problem for muptiple valued functions and the regular-ity of mass minimizing integral currents, in "Minimal Submanifolds and Geodesies", (M.Obata,Ed.) 1-6, North Holland, Amsterdam,1979.
[3]G. Arena, A.O. Caruso, and A. Causa, Taylor formula on step two Carnot groups, Rev. Mat. Iberoamericana,26(1)(2010),239-259.
[4]N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl.,36(9)(1957),235-249.
[5]N. Aronszajn, A. Krzywicki and J. Szarski, A unique continuation theorems for exterior differential forms on Riemannian manifolds, Arkiv for Matematik,4(1962),417-435.
[6]S. Axler, P. Bourdon and W. Ramey, Harmonic function theory (second edition), (GTM137), Springer,2004.
[7]H. Bahouri, Non prolongement unique des solutions d'ope rateurs, Ann. Inst. Fourier, Grenoble,36(4)(1986),137-155.
[8]W. Bechner, On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math. Soc., 129(2001),1233-1246.
[9]A. Bellaiche, The tangent space in sub-Riemannian geometry. Sub-Riemannian geome-try, Progr. Math.,144(1996),1-78.
[10]T. Bieske, On oo-harmonic functions on the Heisenberg group, Comm. Partial Differen-tial Equations,27(3,4)(2002),727-761.
[11]T. Bieske, Equivalence of weak and viscosity solutions to the p-Laplace equation in the Heisenberg group, Annales Academiae Scientiarum Fennicae Mathematica, 31(2006),363-379.
[12]A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified lie groups and potential theory for their sub-laplacians, Springer-Verlag Berlin Heidelberg, New York,2007.
[13]J.M. Bony, Principe du maximum, integalite de Harnack et unicite d u probleme de Cauchy pour les operateurs elliptique degeneres, Ann. Inst. Fourier, Grenoble, 1(119)(1969),277-304.
[14]L. Capogna, D. Danielli and N. Garofalo, An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations, 18(1993),1765-1794.
[15]L. Capogna, Regularity of quasi-linear equations in the Heisenberg group, Comm. Pure Appl. Math.,50(1997),867-889.
[16]L. Capogna, D. Danielli and N. Garofalo, Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math.,118(1996),1153-1196.
[17]L. Capogna, N. Garofalo, Regularity of minimizers of the calculus of variations in Carnot groups via hypoellipticity of systems of Homander type, J. Eur. Math. Soc.,5(2003),1-40.
[18]T. Carleman, Sur un probleme d'unicite pur les systemes d'equations aux derivees par-tielles a deux variables independantes, Ark. Mat., Astr. Fys.,26(17)(1939),1-9.
[19]Y. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of gener-alized mean curvature flow equations, J. Differential Geom.,33(1991),749-786.
[20]M.G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc.,27(1)(1992),1-67.
[21]J. Cygan, Subadditivity of homogeneous norms on certain nilpotent Lie groups, Proc. Amer. Math. Soc.,83(1)(1981),69-70.
[22]E. Dibenedetto, CI+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal.,7(1983),827-850.
[23]A. Domokos, JJ. Manfredi, Subelliptic Cordes estimates, Proc. Amer. Math. Soc., 133(2005),1047-1056.
[24]A. Domokos, Differentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg group, J. Differential Equations,204(2004),439-470.
[25]C.F. Dunkl, An addition theorem for Heisenberg harmonics, in coeference on harmonic analysis in honor of Antoni Zygmund. Wadsworth International, (1982),688-705.
[26]L.C. Evans, J. Spruck, Motion of level sets by mean curvature I, J. Differential Geom., 33(1991),635-681.
[27]H. Federer, Geometric measure theory, Springer, Berlin,Heidelberg, New York,1969.
[28]G.B. Folland, A fundamental solution for a subelliptic operator, Bull. Amer. Math. Soc, 79(1973),373-376.
[29]G.B. Folland, E. M. Stein, Hardy spaces on homogeneus groups, Princeton University Press and University of Tokyo Press,1982.
[30]N. Garofalo, E. Lanconelli, Frequency functions on the Heisenberg group, the uncer-tainty principle and unique continuation, Ann. Inst. Fourier,44(2)(1990),313-356.
[31]N. Garofalo, F.-H. Lin, Monotonicity properties of variational integrals,Ap weights and unique continuation, Indiana Univ. Math. J.,35(1986),245-267.
[32]N. Garofalo, F.-H. Lin, Unique continuation for elliptic operators:a geometric-variational approach, Comm. Pure Appl. Math.,40(3)(1987),347-366.
[33]N. Garofalo, D. Vassilev, Strong unique continuation properties of generalized Baouendi-Grushin operators, Comm. Partial Differential Equations,32(2007),643-663.
[34]N. Garofalo, Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension. J. Differential Equations,104(l)(1993),117-146.
[35]N. Garofalo, Z.W. Shen, Carleman estimates for a subelliptic operator and unique con-tinuation, Ann. Inst. Fourier,44(1)(1994),129-166.
[36]N. Garofalo, Gradient bounds for the horizontal p-Laplacian on a Carnot group and some applications, Manuscripta Math.,130(2009),375-385.
[37]B. Gaveau, Principe de moindre action, propagation de la chaleur et estimees sous ellip-tiques sur certains groups nilpotents, Acta Math.,139(1977),95-153.
[38]PC. Greiner, Spherical harmonics on the Heisenberg group, Canad. Math. Bull., 23(4)(1980),383-396.
[39]V.V. Grushin, On a class of hypoelliptic operators, Math. USSR Sbornik, 12(3)(1970),458-476.
[40]V.V. Grushin, On a class of hypoelliptic pseudodifferential operators degenerate on sub-manifold, Math. USSR Sbornik,13(2)(1971),155-186.
[41]Q. Han, Singular stes to elliptic equations, Indiana Univ. Math. J.,43(1994),983-1002.
[42]Q. Han, Singular sets of harmonic functions in R2 and their complexifications in C2, Indiana Univ. Math. J.,53(5)(2004),1365-1380.
[43]Q. Han, F.-H. Lin, Nodal sets of solutions of elliptic differential equations, Books avail-able on Han's homepage,2010.
[44]J. Heinonen, I. Holopainen, Quasiregular maps on Carnot groups, J. Geom. Anal., 7(1997),109-148.
[45]E. Holmgren, Uber Systeme von linearen partiellen Differentialgleichungen", Ofversigt af Kongl. Vetenskaps-Academien Forhandlinger,58(1901),91-103.
[46]H. Hormander, Hypoelliptic second-order differential equations, Acta Math., 119(1967),147-171.
[47]D. Jerison, C.E. Kenig, Unique continuation and absence of positive eigenvalues for Schroinger operators, Ann. of Math.,121(3) (1985),463-494.
[48]P. Juutinen, B. Kawohl, On the evolution governed by the infinity Laplacian, Math. Ann., 335(2006),819-851.
[49]H. Koch, D. Tataru, Carleman estimates and unique continuation for second order elliptic equations with nonsmooth coefficients, Comm. Pure Appl. Math.,54(3)(2001),339-360.
[50]J.L. Lewis, Regularity of the Derivatives of Solutions to Certain Degenerate Elliptic Equations, Indiana Univ. Math. J.,32(6)(1983),849-858.
[51]F.-H. Lin, Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., XLIV(1991),287-308.
[52]JJ. Manfredi, M. Parviainen and J.D. Rossi, An asymptotic mean value characterization for p-harmonic functions, Proc. Amer. Math. Soc.,138(3)(2010),881-889.
[53]J.J. Manfredi, M. Parviainen and J.D. Rossi, An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games, SIAM J. Math. Anal.,42(5)(2010),2058-2081.
[54]J.J. Manfredi, M. Parviainen and J.D. Rossi, Dynamic programming principle for tug-of-war games with noise, ESAIM:Control, Optimisation and Calculus of Variations, 18(1)(2012),81-90.
[55]S. Marchi, C1,α local regularity for the solutions of the p-Laplacian on the Heisenberg group for 2≤ p ≤ 1+√5, J. for Analysis and its Applications,20(2001),617-636.
[56]S. Marchi, Cha local regularity for the solutions of the p-Laplacian on the Heisenberg group for 1+1/√5< p≤ 2, Comm. Math. Univ. Carolinae,44 (2003),33-56.
[57]T. Matsuzawa, Gevrey hypoellipticity for Grushin operators, Publ, RIMS, Kyoto Univ., 33(1997),775-799.
[58]K. Miller, Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Holder continuous coefficients, Arch. Rational Mech. Anal.,54(1974),105-117.
[59]G. Mingione, A. Zatorska-Goldstein and X. Zhong, Gardient regularity for elliptic equa-tions in the Heisenberg group, Advances in Math.,222(2009),62-129.
[60]R. Monti, D. Morbidelli, Kelvin tranform for Grushin operators and critical semilinear equations, Duke Math. J.,131(1)(2006),167-202.
[61]A. Nagel, E.M. Stein and S. Wainger, Balls and metrics defined by vector fields I:basic properties, Acta Math.,155(1985),103-147.
[62]S. Ohnuma, K. Sato, Singular degenerate parabolic equations with applications to the p-Laplace diffusion equation, Comm. Partial Differential Equations,22(1997),381-411.
[63]Y. Peres, S. Sheffield, Tug-of-war with noise:a game theoretic view of the p-Laplacian, Duke Math. J.,145(2008),91-120.
[64]Y. Peres, O. Schramm, S. Sheffield, and D. B. Wilson, Tug-of-war and the infinity Lapla-cian, J. Amer. Math. Soc.,22(2009),167-210.
[65]E.M. Stein, Harmonic Analysis, Princeton University Press, Princeton, NJ,1993.
[66]G. Szego, Orthogonal polynomials, Amer. Math. Soc. Colloquium Publication,23, Amer. Math. Soc, PRovidence, R.I.1939.
[67]K.H. Tan, X.-P. Yang, Sub-Riemannian objects and hypersurfaces of sub-Riemannian manifolds, Bull. Austral Math. Soc.,70(2)(2004),177-198.
[68]L. Tian, X.-P. Yang, Nodal sets and horizontal singular sets of H-harmonic functions on the Heisenberg group, Preprint.
[69]J.L. Wang, P.C. Niu, Unique Continuation Properties for Generalized Baouendi-Grushin Operators with Singular Weights, Acta Mathematica Sinica, English Series, 27(8)(2011),1637-1644.
[70]C.-J. Xu, Subelliptic variational problems, Bull. Soc. Math. France,118(1990),147-169.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |