H型群上次Laplace方程的极大值原理和一维对称性
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摘要
本文致力于H型群上与De Giorgi猜想相联系的一维对称性的研究。在第一章,简单介绍了De Giorgi猜想的研究进展以及H型群的有关知识;在第二章,我们引入了H型群中Koranyi球上的极坐标表示,证明了H型群中次Laplace算子对径向函数的一个公式,构造并证明了算子T是一个紧算子;在第三章,我们首先证明了一个加细极大值原理,接着,利用加细极大值原理与Krein-Rutman定理证明了紧算子T具有正的特征值和特征函数;在第四章,我们结合算子T存在正的特征值与特征函数的性质,再次利用极坐标证明了一个H型群中无界域上的极大值原理;在第五章,我们利用极大值原理以及次Laplacian算子对H型群中群运算的左平移不变性,证明了H型群上与De Giorgi猜想相联系的一维对称性结果。本文将Birindelli,Prajapat在Heisenberg群上的结果推广到了更一般的H型群上,使得对De Giorgi猜想的研究进一步深化。
This paper is devoted to the study of one dimensional symmetry on H-type group ,which is related to a conjecture by De Giorgi in R~n. In Chapter 1, the history of De Giorgi' conjecture and some basic definitions on H-type group are given;In Chapter 2, the polar coordinates for the Koranyi unit sphere on H-type group are introduced . Then we prove an expression between the sub-Laplacian and the radial function, and construct a compact operator T;In Chapter 3, we provide a refined maximum principle for the sub-Laplacian L, with it and Krein-Rutman theorem to prove T has positive eigenvalue and eigenfunction;In Chapter 4, making use of the polar coordinates and the property that T has positive eigenvalue and eigenfunction, we establish a Maximum Principle on unbounded domains contained in H-type group;At last, in Chapter 5, by the fact that L is left invariant with respect to the group action 。 in H-type group and the maximum principle above, we prove one dimensional symmetry in H-type group. So, we generalize the work of Birindelli and Prajapat in Heisenberg group to H-type group, such that the study about De Giorgi' conjecture becomes deeper.
This paper is devoted to the study of one dimensional symmetry on H-type group ,which is related to a conjecture by De Giorgi in R~n. In Chapter 1, the history of De Giorgi' conjecture and some basic definitions on H-type group are given;In Chapter 2, the polar coordinates for the Koranyi unit sphere on H-type group are introduced . Then we prove an expression between the sub-Laplacian and the radial function, and construct a compact operator T;In Chapter 3, we provide a refined maximum principle for the sub-Laplacian L, with it and Krein-Rutman theorem to prove T has positive eigenvalue and eigenfunction;In Chapter 4, making use of the polar coordinates and the property that T has positive eigenvalue and eigenfunction, we establish a Maximum Principle on unbounded domains contained in H-type group;At last, in Chapter 5, by the fact that L is left invariant with respect to the group action 。 in H-type group and the maximum principle above, we prove one dimensional symmetry in H-type group. So, we generalize the work of Birindelli and Prajapat in Heisenberg group to H-type group, such that the study about De Giorgi' conjecture becomes deeper.
引文
[1] Berestycki H, Hamel F, Monneau R. One-dimensional symmetry of bounded entire solutions of some elliptic equations[J]. Duke Math. J.,2000,103(3):375-396.
[2] Ghoussoub N, Gui C. On a conjecture of De Giorgi and some related problems[J]. Math. Ann.,1998,311:481-491.
[3] Ambrosio L, Cabre. Entire solutions of semilinear elliptic equations in R~3 and a conjecture of De Giorgi[J]. J.Amer. Math. Soc.,2000,13:725-739.
[4] Du Yihong, Ma Li. Some remarks related to De Giorgi's conjecture[J]. Proc. Amer. Math.Soc.,2003,131:2415-2422.
[5] Barlow M T, Bass R F, Gui C. The Liouville property and a conjecture of De Giorgi[J]. Comm. Pure Appl. Math.,2000,53(8): 1007-1038.
[6] Birindelli I,Prajapat J. One dimensional symmetry in the Heisenberg group[J]. Annali della Scuola Normale Superiore di Pisa,2001,3:1～17.
[7] Balogh Z M,Tyson J T. Polar coordinates on Carnot groups[J]. Math. Z.,2002, 241 (4):697～730.
[8] Folland G. Subelliptic estimates and the function spaces on nilpotent Lie groups[J]. Ark. Math.,1975,13:161-207.
[9] Hormander L. Hypoelliptic second order differential equations[J]. Acta. Math., 1967,119:147-171.
[10] Kaplan A. Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms[J].Trans. Amer. Math. Soc.,1980, 258:147-153.
[11] Garofalo N, Vassilev D N. Regularity near the characteristic set in the nonlinear Dirichlet problem and conformal geometry of sub-Laplacians[J], Math. Ann.,2000, 318:453-516.
[12] Garofalo N, Vassilev D. Symmetry properties of positive entire solutions of Journal,2001,106(3):411-448.
[13] Kaplan A. Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms[J]. Trans. Amer. Math. Soc., 1980,258:147-153.
[14] Cowling M, Dooley A H, Koronyi A and Ricci F. H-type groups and Iwasawa decomposition[J]. Adv. Math., 1991,87:1-41.
[15] Birindelli I. Hopf's Lemma and Anti-maximum Principle in General Domains[J]. J.Diff.Equations, 1995,119:450～472.
[16] Birindelli I, Prajapat J. One dimensional symmetry in the Heisenberg group[J]. Annali della Scuola Normale Superiore di Pisa,2001,3:1～17.
[17] Jerison D S. Boundary regularity in the Dirichlet problem for □_b on CR manifolds[J].Comm. Pure. Appl .Math., 1983,36:143～181.
[18] Balogh Z M,Tyson J T. Polar coordinates on Carnot groups[J]. Math.Z.,2002, 241 (4):697～730.
[19] Garofalo N,Vassilev D N. Regularity near the characteristic set in the nonlinear Dirichlet problem and conformal geometry of sub-Laplacians[J]. Math.Ann.2000, 318:453～516.
[20] Jerison D S. The Poincare inequality for vector fields satisfying Hormander's condition[J]. Duke Math.J., 1986,53(2):503～523.
[21] Balogh Z M,Tyson J T. Potential Theory in Carnot groups[J]. AMS Series in Contemporary Mathematics,2003,320:15～27.
[22] Kohn J J,Nirenberg L. Non-coercive boundary value problems[J]. Comm. Pure. Appl. Math., 1965,18:443～492.
[23] Birindelli I. Hopf's Lemma and Anti-maximum Principle in General Domains[J]. J.Diff.Equations, 1995,119:450～472.
[24] Birindelli I,Prajapat J. One dimensional symmetry in the Heisenberg group[J]. Annali della Scuola Normale Superiore di Pisa,2001,3:1～17.
[25] Pinchover Y. On positive Liouville theorems and asymptotic behavior of solutions of Fuchsian type elliptic operators[J]. Ann.Inst.Henri Poincare-Analyse non-lineaire, of Fuchsian type elliptic operators[J]. Ann.Inst.Henri Poincare-Analyse non-lineaire, 1994,11:313-341.
[26] Berestycki H, Caffarelli L, Nirenberg L. Monotonicity for elliptic equations in unbounded domains[J]. Comm. Pure Appl. Math.,1997,50:1088-1111.
[27] Birindelli I, Prajapat J. One dimensional symmetry in the Heisenberg group[J]. Annali della Scuola Normale Superiore di Pisa,2001,3:1～17.
[2] Ghoussoub N, Gui C. On a conjecture of De Giorgi and some related problems[J]. Math. Ann.,1998,311:481-491.
[3] Ambrosio L, Cabre. Entire solutions of semilinear elliptic equations in R~3 and a conjecture of De Giorgi[J]. J.Amer. Math. Soc.,2000,13:725-739.
[4] Du Yihong, Ma Li. Some remarks related to De Giorgi's conjecture[J]. Proc. Amer. Math.Soc.,2003,131:2415-2422.
[5] Barlow M T, Bass R F, Gui C. The Liouville property and a conjecture of De Giorgi[J]. Comm. Pure Appl. Math.,2000,53(8): 1007-1038.
[6] Birindelli I,Prajapat J. One dimensional symmetry in the Heisenberg group[J]. Annali della Scuola Normale Superiore di Pisa,2001,3:1～17.
[7] Balogh Z M,Tyson J T. Polar coordinates on Carnot groups[J]. Math. Z.,2002, 241 (4):697～730.
[8] Folland G. Subelliptic estimates and the function spaces on nilpotent Lie groups[J]. Ark. Math.,1975,13:161-207.
[9] Hormander L. Hypoelliptic second order differential equations[J]. Acta. Math., 1967,119:147-171.
[10] Kaplan A. Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms[J].Trans. Amer. Math. Soc.,1980, 258:147-153.
[11] Garofalo N, Vassilev D N. Regularity near the characteristic set in the nonlinear Dirichlet problem and conformal geometry of sub-Laplacians[J], Math. Ann.,2000, 318:453-516.
[12] Garofalo N, Vassilev D. Symmetry properties of positive entire solutions of Journal,2001,106(3):411-448.
[13] Kaplan A. Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms[J]. Trans. Amer. Math. Soc., 1980,258:147-153.
[14] Cowling M, Dooley A H, Koronyi A and Ricci F. H-type groups and Iwasawa decomposition[J]. Adv. Math., 1991,87:1-41.
[15] Birindelli I. Hopf's Lemma and Anti-maximum Principle in General Domains[J]. J.Diff.Equations, 1995,119:450～472.
[16] Birindelli I, Prajapat J. One dimensional symmetry in the Heisenberg group[J]. Annali della Scuola Normale Superiore di Pisa,2001,3:1～17.
[17] Jerison D S. Boundary regularity in the Dirichlet problem for □_b on CR manifolds[J].Comm. Pure. Appl .Math., 1983,36:143～181.
[18] Balogh Z M,Tyson J T. Polar coordinates on Carnot groups[J]. Math.Z.,2002, 241 (4):697～730.
[19] Garofalo N,Vassilev D N. Regularity near the characteristic set in the nonlinear Dirichlet problem and conformal geometry of sub-Laplacians[J]. Math.Ann.2000, 318:453～516.
[20] Jerison D S. The Poincare inequality for vector fields satisfying Hormander's condition[J]. Duke Math.J., 1986,53(2):503～523.
[21] Balogh Z M,Tyson J T. Potential Theory in Carnot groups[J]. AMS Series in Contemporary Mathematics,2003,320:15～27.
[22] Kohn J J,Nirenberg L. Non-coercive boundary value problems[J]. Comm. Pure. Appl. Math., 1965,18:443～492.
[23] Birindelli I. Hopf's Lemma and Anti-maximum Principle in General Domains[J]. J.Diff.Equations, 1995,119:450～472.
[24] Birindelli I,Prajapat J. One dimensional symmetry in the Heisenberg group[J]. Annali della Scuola Normale Superiore di Pisa,2001,3:1～17.
[25] Pinchover Y. On positive Liouville theorems and asymptotic behavior of solutions of Fuchsian type elliptic operators[J]. Ann.Inst.Henri Poincare-Analyse non-lineaire, of Fuchsian type elliptic operators[J]. Ann.Inst.Henri Poincare-Analyse non-lineaire, 1994,11:313-341.
[26] Berestycki H, Caffarelli L, Nirenberg L. Monotonicity for elliptic equations in unbounded domains[J]. Comm. Pure Appl. Math.,1997,50:1088-1111.
[27] Birindelli I, Prajapat J. One dimensional symmetry in the Heisenberg group[J]. Annali della Scuola Normale Superiore di Pisa,2001,3:1～17.