偏微分方程中特征值与解的性态研究
|
摘要
本文主要研究偏微分方程特征值和解的性态.偏微分方程来自于化学、物理和生物等科学领域,具有十分强烈的实际背景.它与数学中的其他很多分支密切相关,如微分几何,复分析,调和分析等,逐渐成为数学学科中最重要的研究领域之一.特征值与解的性态问题是偏微分方程中的基本问题.因此,对于偏微分方程中的特征值和解的性态的研究不但具有科学意义,而且具有潜在的应用价值.本文的主要内容安排如下：
第1章简述有关偏微分方程特征值和解的性态的研究背景及本文的主要工作.
第2章是预备知识.介绍了本文要用到的数学术语和数学工具.
第3章,我们讨论如下Laplacian Robin特征值问题其中Ω∈Rπ(n≥2)是有界区域,αΩ是Lipschitz的,ν为边界上的单位外法向量以及0<β<+∞.本章的结论主要有两个：一个我们利用schwarz对称化方法和函数重排技巧得到了问题(1)的前两个特征值之比的一个上界估计.该估计在某个特定的限制条件之下是M.S.Ash-baugh和R.D.Benguria[4,6]证明的PPW猜测的一个推广；另一个是证明了问题(1)的第一特征函数满足Chiti型反向Holder不等式.
在第4章中,我们研究如下k-Hessian算子的特征值问题其中Ω是Rπ中光滑有界凸区域.我们推导出问题(2)的主特征值的变分公式并推导出相关的超定问题；其次,我们得到了所推导的超定问题的Serrin型对称性结果.
In this thesis, we study eigenvalues and the properties of solution for par-tial differential equations. Partial differential equations involved in a number of issues from the chemistry, physics and mathematical model of the biological field, with a strong practical background. It has become one of the most im-portant fields of study in mathematical, which interacts with many branches of mathematics, such as differential geometry, complex analysis and harmonic analysis. Eigenvalues and the properties of solution are basic problems in partial differential equations. Therefore, the study on eigenvalues and the properties of solution for partial differential equations has scientific significance and potential applications. The main contents are organized as follows:
In chapter1, we state the background of eigenvalues and the properties of solution for partial differential equations and the main work of this article.
In chapter2, we introduce some mathematical terms and tools as prelimi-naries that will be used in the following chapters.
In chapter3, Let Ω (?) Rn(n≧2) be a bounded domain with boundary (?)Ω, ν be the outward unit vector normal to (?)Ω, and0<β<+∞be a parameter. We discuss the following Robin eigenvalue problem The aim of this chapter is twofold. One is an upper bound for the ratio of the first two eigenvalues by Schwarz symmetrization method and rearrange-ment techniques, which can be used to recover the PPW conjecture proved by M.S.Ashbaugh and R.D.Benguria in [4] and [6], the other is a Chiti type reverse Holder inequality for the first eigenfunction.
In chapter4, we concern the following eigenvalue problem of k-Hessian operator where Ω is a smooth bounded convex domain in Rn. First we devote to deduce the first variational formula and some related overdetermined problems for the principle eigenvalue of problem (2), and then prove Serrin type symmetry result for our overdeter-mined problems.
第1章简述有关偏微分方程特征值和解的性态的研究背景及本文的主要工作.
第2章是预备知识.介绍了本文要用到的数学术语和数学工具.
第3章,我们讨论如下Laplacian Robin特征值问题其中Ω∈Rπ(n≥2)是有界区域,αΩ是Lipschitz的,ν为边界上的单位外法向量以及0<β<+∞.本章的结论主要有两个：一个我们利用schwarz对称化方法和函数重排技巧得到了问题(1)的前两个特征值之比的一个上界估计.该估计在某个特定的限制条件之下是M.S.Ash-baugh和R.D.Benguria[4,6]证明的PPW猜测的一个推广；另一个是证明了问题(1)的第一特征函数满足Chiti型反向Holder不等式.
在第4章中,我们研究如下k-Hessian算子的特征值问题其中Ω是Rπ中光滑有界凸区域.我们推导出问题(2)的主特征值的变分公式并推导出相关的超定问题；其次,我们得到了所推导的超定问题的Serrin型对称性结果.
In this thesis, we study eigenvalues and the properties of solution for par-tial differential equations. Partial differential equations involved in a number of issues from the chemistry, physics and mathematical model of the biological field, with a strong practical background. It has become one of the most im-portant fields of study in mathematical, which interacts with many branches of mathematics, such as differential geometry, complex analysis and harmonic analysis. Eigenvalues and the properties of solution are basic problems in partial differential equations. Therefore, the study on eigenvalues and the properties of solution for partial differential equations has scientific significance and potential applications. The main contents are organized as follows:
In chapter1, we state the background of eigenvalues and the properties of solution for partial differential equations and the main work of this article.
In chapter2, we introduce some mathematical terms and tools as prelimi-naries that will be used in the following chapters.
In chapter3, Let Ω (?) Rn(n≧2) be a bounded domain with boundary (?)Ω, ν be the outward unit vector normal to (?)Ω, and0<β<+∞be a parameter. We discuss the following Robin eigenvalue problem The aim of this chapter is twofold. One is an upper bound for the ratio of the first two eigenvalues by Schwarz symmetrization method and rearrange-ment techniques, which can be used to recover the PPW conjecture proved by M.S.Ashbaugh and R.D.Benguria in [4] and [6], the other is a Chiti type reverse Holder inequality for the first eigenfunction.
In chapter4, we concern the following eigenvalue problem of k-Hessian operator where Ω is a smooth bounded convex domain in Rn. First we devote to deduce the first variational formula and some related overdetermined problems for the principle eigenvalue of problem (2), and then prove Serrin type symmetry result for our overdeter-mined problems.
引文
[1]Adams, R.A. Sobolev spaces [M]. New York: Academic Press,1975.
[2]Alessandrini, G. A symmetry theorem for condensers [J]. Math. Methods Appl. Sci.,1992,15:315-320.
[3]Alvino A., V. Ferone. & G. Trombetti. On the properties of some nonlinear eigenvalues [J]. SIAM J. Math. Anal.,1998,29:437-451.
[4]Ashbaugh, M.S. & R.D. Benguria. Proof of the Payne-Polya-Weinberger conjec-ture [J]. Bull. Amer. Math. Soc.,1991,25:19-29.
[5]Ashbaugh, M.S. & R.D. Benguria. A second proof of the Payne-Polya-Weinberger conjeture [J]. Commun. Math. Phys.,1992,147: 181-190.
[6]Ashbaugh, M.S. & R.D. Benguria. A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions [J]. Ann. Math., 1992,135: 601-628.
[7]Ashbaugh, M.S. & R.D. Benguria. More bounds on eigenvalue ratios for Dirichlet Laplacians in n dimensions [J]. SIAM J. Math. Anal.,1993,24:1622-1651.
[8]Ashbaugh, M.S. & R.D. Benguria. Universal Bounds for the Low Eigenvalues of Neumann Laplacians in N Dimensions [J]. SIAM J. Math. Anal.,1993,24(3): 557-570.
[9]Ashbaugh, M. S. Isoperimetric and universal inequalities for eigenvalues [A]. In E.B. Davies & Yu Safalov(eds.). Spectral theory and geometry [C]. Cambridge: Cambridge Univ. Press,1999, pp.95-139.
[10]Bandle, C. Isoperimetric Inequalities and Applications [M]. Boston:Pitman, 1980.
[11]Bhattacharya, T. A proof of the Faber-Krahn inequality for the first eigenvalue of the p-Laplacian [J]. Annali di Matematica pura ed applicata, IV,1999,177: 325-343.
[12]Bossel, M.H. Membranes elastiquement liees inhomogenes ou sur une surface: une nouvelle extension du theoreme isoperimetrique de Rayleigh-Faber-Krahn [J]. Z. Angew. Math. Phys.,1988,39:733-742.
[13]Brandolini, B.& C. Trombetti. Comparison results for Hessian equations via symmetrization [J]. J. Eur. Math. Soc.,2007,9(3):561-575.
[14]Brandolini B., F. Chiacchio & C. Trombetti. Sharp estimates for eigenfunctions of a Neumann problem [J]. Comm. Partial Differential Equations,2009,34: 1317-1337.
[15]Brandolini, B., C. Nitsch, P. Salani & C. Trombetti. Serrin-type overdetermined problems:an alternative proof [J]. Arch. Ration. Mech. Anal.,2008,190:267-280.
[16]Brandolini, B., C. Nitsch & C. Trombetti. New isoperimetric estimates for solu-tions to Monge-Ampere equations [J]. Ann.Inst.H.Poincare Anal.NonLineaire, 2009,26:1265-1275.
[17]Brands, J. J. A. M. Bounds for the ratios of the first three membrane eigenvalues [J]. Arch. Rational Mech. Anal.,1964,16:265-268.
[18]Brock, F. & A. Henrot. A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative [J]. Rend. Circ. Mat. Palermo,2002,51:375-390.
[19]Bucur, D. & D. Daners. An alternative approach to the Faber-Krahn inequality for Robin problems [J]. Calc. Var. Partial Differential Equations,2010,37:75-86.
[20]Bucur, D. & A. Giacomini. A Variational Approach to the Isoperimetric Inequal-ity for the Robin Eigenvalue Problem [J]. Arch. Rational Mech. Anal.,2010, 198(3):927-961.
[21]Buttazzo, G. & B. Kawohl. Overdetermined boundary value problems for the oo-Laplacian [J]. Int. Math. Res. Not.,2011,237-247.
[22]Caffarelli, L., L. Nirenberg & J. Spruck. The Dirichlet problem for nonlinear second order elliptic equations III. Functions of the eigenvalues of the Hessian [J]. Acda Math.,1985,155(3-4):261-301.
[23]Chavel, I. Eigenvalues in Riemannian Geometry [M]. New York:Academic Press, 1984.
[24]Chiti, G. A reverse Holder inequality for the eigenfunctions of linear second order elliptic operators [J]. J. Appl. Math, and Phys. (ZAMP),1982,33:143-148.
[25]Chiti, G. An isoperimetric inequality for the eigenfunctions of linear second order elliptic operators [J]. Boll. Un. Mat. Ital.,1982,1-A (6):145-151.
[26]Chiti, G. A bound for the ratio of the first two eigenvalues of a membrane [J]. SIAM J. Math. Anal.,1983,14:1163-1167.
[27]Choulli, M. & A. Henrot, Use of the domain derivative to prove symmetry results in partial differential equations [J]. Math. Nachr.,1998,192:91-103.
[28]Cianchi, A. & P. Salani. Overdetermined anisotropic elliptic problems [J]. Math. Ann.,2009,345:859-881.
[29]Colesanti, A. & M. Fimiani. The Minkowski problem for the torsional rigidity [J]. Indiana Univ. Math. J..2010,59:1013-1039.
[30]Colesanti, A., P.Cuoghi & P.Salani. Brunn-Minkowski inequalities for two func-tionals involving the p-Laplace operator [J]. Appl.Anal.,2006,85:45-66.
[31]Dai, Q.Y. & Y.X. Fu. Faber-Krahn inequality for Robin problem involving p-Laplacian [J]. Acda Math. Appl. Sin. (English Series),2011,27:13-28.
[32]Dai, Q.Y., R.C. He & H.X. Hu. Isoperimetric inequalities and sharp estimate for the positive solution of sublinear elliptic equations [J]. arXiv:1003.3768vl.
[33]Daners, D. A Faber-Krahn inequality for the Robin problems in any space di-mension [J]. Math. Ann,2006,335:767-785.
[34]Daners, D. & J. Kennedy. Uniqueness in the Faber-Krahn inequality for Robin problems [J]. SIAM J. Math. Anal.,2007,39:1191-1207.
[35]Damascelli, L. & F. Pacella. Monotonicity and symmetry results for p-Laplace equations and applications [J]. Adv. Diff. Equ.,2000,5:1179-1200.
[36]de Vries, H. L. On the upper bound for the ratio of the first two membrane eigenvalues [J]. Z. Naturforsch.,1967,22A:152-153.
[37]Evans, L.C. Partial Differential Equations:Second Edition[M]. American Math-ematical Society,2010.
[38]Faber, G. & Beweis. dass unter allen homogenen Membranen von gleicher Flache und gleicher Spannung die kreisformige den tiefsten Grundton gibt, Sitzungsber [J]. Bayr. Akad. Wiss. Miinchen, Math.-Phys. Kl,1923,169-127.
[39]Farina, A. & B. Kawohl. Remarks on an overdetermined boundary value prob-lem[J]. Calc. Var. Partial Differential Equations,2008,89:351-357.
[40]Fragala, I., F. Gazzola & B. Kawohl. Overdetermined problems with possibly degenerate ellipticity, a geometric approach [J]. Math. Z.,2006,254:117-132.
[41]Fragala, I., F. Gazzola & B. Kawohl. Overdetermined problems for the oo-Laplacian and web functions [J]. preprint,2011.
[42]Fragala, I. Symmetry results for overdetermined problems on convex domains via Brunn-Minkowski inequalities [J]. J. Math. Pures Appl.,2012,97:55-65.
[43]Garofalo, N. & E. Sartori. Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a priori estimates [J]. Adv. Diff. Equ.,1999,4:137-161.
[44]Garofalo, N. & J.L. Lewis. A symmetry result related to some overdetermined boundary value problems [J]. Amer. J. Math.,1989,111:9-33.
[45]Gazzola, F. No geometric approach for general overdetermined elliptic problems with nonconstant source [J]. Le Matematiche,2005,60:259-268.
[46]Gilbarg, D. & N.S. Trudinger. Elliptic partial differential eqautions of second order[M]. Berlin:Springer,1998.
[47]Giorgi, T. & R.G.Smits. Monotonicity results for the principal eigenvalue of the generalized robin problem [J]. Illinois J. Math.,2005,49(4):1133-1143.
[48]Greco, A. Radial symmetry and uniqueness for an overdetermined problem [J]. Math. Methods Appl. Sci.,2001,24:103-115.
[49]Hardy, G. H., J. E. Littlewood & G. Polya. Some simple inequalities satisfied by convex functions [J]. Messenger Math.,1929,58:145-152.
[50]Hardy, G. H., J. E. Littlewood & G. Polya. Inequalities, second edition [M]. Cambridge: Cambridge University Press,1952.
[51]Henrot, A. Minimization problems for eigenvalues of the Laplacian [J]. J.Evol. Equ.,2003,3:443-461.
[52]Henrot, A. & E.Oudet. Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions [J]. Arch. Rational Mech.Anal.,2003,169 :73-87.
[53]Henrot, A. Extremum problems for eigenvalues of elliptic operators [M]. Basel: Birkhauser Verlag,2006.
[54]Henrot, A. & G. Philippin. Some overdetermined boundary value problems with elliptical free boundaries [J]. SIAM J. Math. Anal.,1998,29:309-320.
[55]Jerison, D. A Minkowski problem for electrostatic capacity [J]. Acta.Math,1996, 176:1-47.
[56]Kawohl, B. Rearrangements and convexity of Level sets in PDES [M]. Heidelberg: Springer-Verlag,1985.
[57]Kesavan, S. Symmetrization and Applicantions. Series in Anylysis. Vol.3 [M]. World Scientific Publishing Co. Pte. Ltd,2006.
[58]Krahn, E. Uber eine von Rayleigh formulierte Minimaleigenschaft des Kreises [J]. Math. Ann.,1925,94:97-100.
[59]Kroger, P. Upper bounds for the Neumann eigenvalues on a bounded domains in Euclidean space [J]. J.Funct. Anal.,1992,106:353-357.
[60]Kuttler J. R. & V. G. Sigillito. Eigenvalues of the Laplacian in two dimensions [J]. SIAM Rev.,1984,26:163-193.
[61]Lieb E. H. & M. Loss. Analysis, second edition, Graduate Studies in Mathemat-ics, vol.14 [M]. Providence: Amer. Math. Soc.,2001 (see, in particular, Chapter 3).
[62]Liu, P., X.N. Ma & L. Xu. A Brunn-Minkowski inequality for the Hessian eigen-value in three-dimensional convex domain [J]. Adv. Math.,2010,225:1616-1633.
[63]Mendez, O. & W. Reichel. Electrostatic characterization of spheres [J]. Forum Math.,2000,12:223-245.
[64]Osserman, R. The isoperimetric inequality [J]. Bull. Amer. Math. Soc,1978, 84:1182-1238.
[65]Payne, L.E., G.Polya & H.F.Weinberger. Sur le quotient de devx frequences propres consecutives [J]. Comptes Rendus Acad. Sci. Paris,1955,241:917-919.
[66]Payne, L.E., G.Polya & H.F.Weinberger. On the ratio of consecutive eigenvalues [J].J.Math, and Phys.,1956,35:289-298.
[67]Payne, L.E. & M.E.Rayner. An isoperimetric inequality for the first eigenfunc-tion in the fixed membrane problem [J]. Z. Angew. Math. Phys.,1972,23:13-15.
[68]Payne, L.E. & I.Stakgold. On the mean value of the fundamental mode in the fixed membrane problem [J]. Appl. Anal.,1973,3:295-303.
[69]Payne, L.E. & P.W.Schaefer. Eigenvalue and eigenfunction inequalities for the elastically supported membrane [J]. Z.Angew.Math.Phys,2001,52:888-895.
[70]Payne, L.E. & P.W. Schaefer. Duality theorems in some overdetermined bound-ary value problems [J]. Math. Methods Appl. Sci.,1989,11:805-819.
[71]Pockels, F. Uber die partielle Differentialgleichung △μ+k2u= 0 und deren Auftreten in der mathematischen Physik (with a forward by F. Klein) [M]. Leipzig: Teubner,1891.
[72]Polya G. & G. Szego. Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, vol.27 [M]. Princeton: Princeton University Press,1951.
[73]Prajapat, J. Serrin's result for domains with a corner or cusp [J]. Duke Math. J.,1998,91:29-31.
[74]Reichel, W. Radial symmetry for elliptic boundary value problems on exterior domains [J]. Arch. Ration. Mech. Anal.,1997,137,381-394.
[75]Reilly, R.C. On the Hessian of a function and the curvatures of its graph, Michi-gan Math. J. [J].1973,20:373-383.
[76]Salani, P. A Brunn-Minkowski inequality for the Monge-Ampere eigenvalue [J]. Adv.Math.,2005,194:67-86.
[77]Salani, P. Convexity of solutions and Brunn-Minkowski inequalities for Hessian equations in R3 [J]. Adv.Math.,2012,229:1924-1948.
[78]Salani, P. Equazioni Hessiane e k-convessita(Phd Thesis.) [D]. Firenze: Universita di. Firenze,1997.
[79]Schneider, R. Convex Bodies:The Brunn-Minkowski Theory [M]. Cambridge Univ. Press,1993.
[80]Serrin, J. A symmetry problem in potential theory [J]. Arch. Ration. Mech. Anal.,1971,43:304-318.
[81]Sirakov, B. Symmetry for exterior elliptic problems and two conjectures in poten-tial theory [J]. Ann. Inst. H. Poincare Anal. Non Lineaire,2001,18:135-156.
[82]Szego, G. Inequalities for certain eigenvalues of a membrane of given area [J]. J. Bational Mech. Anal,1954,3:343-356.
[83]Talenti, G. Elliptic equations and rearrangements [J]. Ann. Scuola Norm. Sup. Pisa.,1976,3:697-718.
[84]Talenti, G. On isoperimetric theorems of mathematical physics [A]. In P. M. Gru-ber & J. M. Wills(eds.). Handbook of Convex Geometry, vol.B[C]. Amsterdam: North-Holland,1993, pp.1131-1147.
[85]Talenti, G. Inequalities in rearrangement invariant function spaces [A]. In M. Krbec, A. Kufner, B. Opic, & J. Rakosnik(eds.). Nonlinear Analysis. Func-tion Spaces and Applications,. vol.5[C]. Prague:Prometheus Publishing House, 1994, pp.177-230.
[86]Takahashi, F. & A.Uegaki. A Payne-Rayner Type Inequality for the Robin Prob-lem on Arbitrary Minimal Surfaces in Rn [J]. Results Math.,2011,59:107-114.
[87]Therese, M. & K.Jobin. Isoperimetric monotonicity and Isoperimetric inequali-ties of Payne-Rayner type for the first eigenfunction of the Helmholtz problem [J]. J. Appl.Math. Phys.,1981,32:625-646.
[88]Thompson, C.J. on the ratio of conseccutive eigenvalues in N-dimensions [J]. Stud.Appl.Math.,1969,48:281-283.
[89]Trudinger, N.S. On new isoperimetric inequalities and symmetrization [J]. J. Reine Angew. Math.,1997,488:203-220.
[90]Trudinger, N.S. & X-J.Wang. Hessian measures I, Dedicated to Olga Ladyzhen-skaya. Topol [J]. Methods Nonlinear Anal.,1997,10(2):225-239.
[91]Trudinger, N.S. & X-J. Wang. Hessian measures Ⅱ [J]. Ann.Math.,1999,15(2): 579-604.
[92]Tso, K. Remarks on critical exponents for Hessian operators [J]. Ann. Inst. H. Poincare.Anal. Non Lineaire,1990,7:113-122.
[93]Tso,K.On symmetrization and Hessian Equations[J].J.Anal.Math.,1989, 52:94-106.
[94]Vogel,A.L.symmetry and regularity for general regions having solutions to certain overdetermined boundary value problems[J].Atti semin.Mat.Fis.Univ. Modena.,1992,40:443-484.
[95]Wang, Q.L. & C.Y. Xia. Isoperimetric bounds for the first eigenvalue of the Laplacian [J]. Z.Angew. Math. Phys., 2010,61:171-175.
[96]Wang,X.J.A class of fully nonlinear elliptic equations and related functionals [J]. Indiana Univ.Math. J., 1994,43(1):25-54.
[97]Weinberger,H.F.An isoperimetric inequality for the n—dimensional free mem-brane problem[J].J.Rational Mech.Anal.,1956,5:633-636.
[98]Weinberger,H.Remark on the preceding paper of Serrin[J].Arch.Ration.Mech. Anal.,1971,43:319-320.
[99]M.Willem.Minimax Theorems[M].Birkhauser,1996.
[100]伍卓群,尹景学＆王春鹏,椭圆与抛物型方程引论[M].北京:科学出版社, 2006.
[2]Alessandrini, G. A symmetry theorem for condensers [J]. Math. Methods Appl. Sci.,1992,15:315-320.
[3]Alvino A., V. Ferone. & G. Trombetti. On the properties of some nonlinear eigenvalues [J]. SIAM J. Math. Anal.,1998,29:437-451.
[4]Ashbaugh, M.S. & R.D. Benguria. Proof of the Payne-Polya-Weinberger conjec-ture [J]. Bull. Amer. Math. Soc.,1991,25:19-29.
[5]Ashbaugh, M.S. & R.D. Benguria. A second proof of the Payne-Polya-Weinberger conjeture [J]. Commun. Math. Phys.,1992,147: 181-190.
[6]Ashbaugh, M.S. & R.D. Benguria. A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions [J]. Ann. Math., 1992,135: 601-628.
[7]Ashbaugh, M.S. & R.D. Benguria. More bounds on eigenvalue ratios for Dirichlet Laplacians in n dimensions [J]. SIAM J. Math. Anal.,1993,24:1622-1651.
[8]Ashbaugh, M.S. & R.D. Benguria. Universal Bounds for the Low Eigenvalues of Neumann Laplacians in N Dimensions [J]. SIAM J. Math. Anal.,1993,24(3): 557-570.
[9]Ashbaugh, M. S. Isoperimetric and universal inequalities for eigenvalues [A]. In E.B. Davies & Yu Safalov(eds.). Spectral theory and geometry [C]. Cambridge: Cambridge Univ. Press,1999, pp.95-139.
[10]Bandle, C. Isoperimetric Inequalities and Applications [M]. Boston:Pitman, 1980.
[11]Bhattacharya, T. A proof of the Faber-Krahn inequality for the first eigenvalue of the p-Laplacian [J]. Annali di Matematica pura ed applicata, IV,1999,177: 325-343.
[12]Bossel, M.H. Membranes elastiquement liees inhomogenes ou sur une surface: une nouvelle extension du theoreme isoperimetrique de Rayleigh-Faber-Krahn [J]. Z. Angew. Math. Phys.,1988,39:733-742.
[13]Brandolini, B.& C. Trombetti. Comparison results for Hessian equations via symmetrization [J]. J. Eur. Math. Soc.,2007,9(3):561-575.
[14]Brandolini B., F. Chiacchio & C. Trombetti. Sharp estimates for eigenfunctions of a Neumann problem [J]. Comm. Partial Differential Equations,2009,34: 1317-1337.
[15]Brandolini, B., C. Nitsch, P. Salani & C. Trombetti. Serrin-type overdetermined problems:an alternative proof [J]. Arch. Ration. Mech. Anal.,2008,190:267-280.
[16]Brandolini, B., C. Nitsch & C. Trombetti. New isoperimetric estimates for solu-tions to Monge-Ampere equations [J]. Ann.Inst.H.Poincare Anal.NonLineaire, 2009,26:1265-1275.
[17]Brands, J. J. A. M. Bounds for the ratios of the first three membrane eigenvalues [J]. Arch. Rational Mech. Anal.,1964,16:265-268.
[18]Brock, F. & A. Henrot. A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative [J]. Rend. Circ. Mat. Palermo,2002,51:375-390.
[19]Bucur, D. & D. Daners. An alternative approach to the Faber-Krahn inequality for Robin problems [J]. Calc. Var. Partial Differential Equations,2010,37:75-86.
[20]Bucur, D. & A. Giacomini. A Variational Approach to the Isoperimetric Inequal-ity for the Robin Eigenvalue Problem [J]. Arch. Rational Mech. Anal.,2010, 198(3):927-961.
[21]Buttazzo, G. & B. Kawohl. Overdetermined boundary value problems for the oo-Laplacian [J]. Int. Math. Res. Not.,2011,237-247.
[22]Caffarelli, L., L. Nirenberg & J. Spruck. The Dirichlet problem for nonlinear second order elliptic equations III. Functions of the eigenvalues of the Hessian [J]. Acda Math.,1985,155(3-4):261-301.
[23]Chavel, I. Eigenvalues in Riemannian Geometry [M]. New York:Academic Press, 1984.
[24]Chiti, G. A reverse Holder inequality for the eigenfunctions of linear second order elliptic operators [J]. J. Appl. Math, and Phys. (ZAMP),1982,33:143-148.
[25]Chiti, G. An isoperimetric inequality for the eigenfunctions of linear second order elliptic operators [J]. Boll. Un. Mat. Ital.,1982,1-A (6):145-151.
[26]Chiti, G. A bound for the ratio of the first two eigenvalues of a membrane [J]. SIAM J. Math. Anal.,1983,14:1163-1167.
[27]Choulli, M. & A. Henrot, Use of the domain derivative to prove symmetry results in partial differential equations [J]. Math. Nachr.,1998,192:91-103.
[28]Cianchi, A. & P. Salani. Overdetermined anisotropic elliptic problems [J]. Math. Ann.,2009,345:859-881.
[29]Colesanti, A. & M. Fimiani. The Minkowski problem for the torsional rigidity [J]. Indiana Univ. Math. J..2010,59:1013-1039.
[30]Colesanti, A., P.Cuoghi & P.Salani. Brunn-Minkowski inequalities for two func-tionals involving the p-Laplace operator [J]. Appl.Anal.,2006,85:45-66.
[31]Dai, Q.Y. & Y.X. Fu. Faber-Krahn inequality for Robin problem involving p-Laplacian [J]. Acda Math. Appl. Sin. (English Series),2011,27:13-28.
[32]Dai, Q.Y., R.C. He & H.X. Hu. Isoperimetric inequalities and sharp estimate for the positive solution of sublinear elliptic equations [J]. arXiv:1003.3768vl.
[33]Daners, D. A Faber-Krahn inequality for the Robin problems in any space di-mension [J]. Math. Ann,2006,335:767-785.
[34]Daners, D. & J. Kennedy. Uniqueness in the Faber-Krahn inequality for Robin problems [J]. SIAM J. Math. Anal.,2007,39:1191-1207.
[35]Damascelli, L. & F. Pacella. Monotonicity and symmetry results for p-Laplace equations and applications [J]. Adv. Diff. Equ.,2000,5:1179-1200.
[36]de Vries, H. L. On the upper bound for the ratio of the first two membrane eigenvalues [J]. Z. Naturforsch.,1967,22A:152-153.
[37]Evans, L.C. Partial Differential Equations:Second Edition[M]. American Math-ematical Society,2010.
[38]Faber, G. & Beweis. dass unter allen homogenen Membranen von gleicher Flache und gleicher Spannung die kreisformige den tiefsten Grundton gibt, Sitzungsber [J]. Bayr. Akad. Wiss. Miinchen, Math.-Phys. Kl,1923,169-127.
[39]Farina, A. & B. Kawohl. Remarks on an overdetermined boundary value prob-lem[J]. Calc. Var. Partial Differential Equations,2008,89:351-357.
[40]Fragala, I., F. Gazzola & B. Kawohl. Overdetermined problems with possibly degenerate ellipticity, a geometric approach [J]. Math. Z.,2006,254:117-132.
[41]Fragala, I., F. Gazzola & B. Kawohl. Overdetermined problems for the oo-Laplacian and web functions [J]. preprint,2011.
[42]Fragala, I. Symmetry results for overdetermined problems on convex domains via Brunn-Minkowski inequalities [J]. J. Math. Pures Appl.,2012,97:55-65.
[43]Garofalo, N. & E. Sartori. Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a priori estimates [J]. Adv. Diff. Equ.,1999,4:137-161.
[44]Garofalo, N. & J.L. Lewis. A symmetry result related to some overdetermined boundary value problems [J]. Amer. J. Math.,1989,111:9-33.
[45]Gazzola, F. No geometric approach for general overdetermined elliptic problems with nonconstant source [J]. Le Matematiche,2005,60:259-268.
[46]Gilbarg, D. & N.S. Trudinger. Elliptic partial differential eqautions of second order[M]. Berlin:Springer,1998.
[47]Giorgi, T. & R.G.Smits. Monotonicity results for the principal eigenvalue of the generalized robin problem [J]. Illinois J. Math.,2005,49(4):1133-1143.
[48]Greco, A. Radial symmetry and uniqueness for an overdetermined problem [J]. Math. Methods Appl. Sci.,2001,24:103-115.
[49]Hardy, G. H., J. E. Littlewood & G. Polya. Some simple inequalities satisfied by convex functions [J]. Messenger Math.,1929,58:145-152.
[50]Hardy, G. H., J. E. Littlewood & G. Polya. Inequalities, second edition [M]. Cambridge: Cambridge University Press,1952.
[51]Henrot, A. Minimization problems for eigenvalues of the Laplacian [J]. J.Evol. Equ.,2003,3:443-461.
[52]Henrot, A. & E.Oudet. Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions [J]. Arch. Rational Mech.Anal.,2003,169 :73-87.
[53]Henrot, A. Extremum problems for eigenvalues of elliptic operators [M]. Basel: Birkhauser Verlag,2006.
[54]Henrot, A. & G. Philippin. Some overdetermined boundary value problems with elliptical free boundaries [J]. SIAM J. Math. Anal.,1998,29:309-320.
[55]Jerison, D. A Minkowski problem for electrostatic capacity [J]. Acta.Math,1996, 176:1-47.
[56]Kawohl, B. Rearrangements and convexity of Level sets in PDES [M]. Heidelberg: Springer-Verlag,1985.
[57]Kesavan, S. Symmetrization and Applicantions. Series in Anylysis. Vol.3 [M]. World Scientific Publishing Co. Pte. Ltd,2006.
[58]Krahn, E. Uber eine von Rayleigh formulierte Minimaleigenschaft des Kreises [J]. Math. Ann.,1925,94:97-100.
[59]Kroger, P. Upper bounds for the Neumann eigenvalues on a bounded domains in Euclidean space [J]. J.Funct. Anal.,1992,106:353-357.
[60]Kuttler J. R. & V. G. Sigillito. Eigenvalues of the Laplacian in two dimensions [J]. SIAM Rev.,1984,26:163-193.
[61]Lieb E. H. & M. Loss. Analysis, second edition, Graduate Studies in Mathemat-ics, vol.14 [M]. Providence: Amer. Math. Soc.,2001 (see, in particular, Chapter 3).
[62]Liu, P., X.N. Ma & L. Xu. A Brunn-Minkowski inequality for the Hessian eigen-value in three-dimensional convex domain [J]. Adv. Math.,2010,225:1616-1633.
[63]Mendez, O. & W. Reichel. Electrostatic characterization of spheres [J]. Forum Math.,2000,12:223-245.
[64]Osserman, R. The isoperimetric inequality [J]. Bull. Amer. Math. Soc,1978, 84:1182-1238.
[65]Payne, L.E., G.Polya & H.F.Weinberger. Sur le quotient de devx frequences propres consecutives [J]. Comptes Rendus Acad. Sci. Paris,1955,241:917-919.
[66]Payne, L.E., G.Polya & H.F.Weinberger. On the ratio of consecutive eigenvalues [J].J.Math, and Phys.,1956,35:289-298.
[67]Payne, L.E. & M.E.Rayner. An isoperimetric inequality for the first eigenfunc-tion in the fixed membrane problem [J]. Z. Angew. Math. Phys.,1972,23:13-15.
[68]Payne, L.E. & I.Stakgold. On the mean value of the fundamental mode in the fixed membrane problem [J]. Appl. Anal.,1973,3:295-303.
[69]Payne, L.E. & P.W.Schaefer. Eigenvalue and eigenfunction inequalities for the elastically supported membrane [J]. Z.Angew.Math.Phys,2001,52:888-895.
[70]Payne, L.E. & P.W. Schaefer. Duality theorems in some overdetermined bound-ary value problems [J]. Math. Methods Appl. Sci.,1989,11:805-819.
[71]Pockels, F. Uber die partielle Differentialgleichung △μ+k2u= 0 und deren Auftreten in der mathematischen Physik (with a forward by F. Klein) [M]. Leipzig: Teubner,1891.
[72]Polya G. & G. Szego. Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, vol.27 [M]. Princeton: Princeton University Press,1951.
[73]Prajapat, J. Serrin's result for domains with a corner or cusp [J]. Duke Math. J.,1998,91:29-31.
[74]Reichel, W. Radial symmetry for elliptic boundary value problems on exterior domains [J]. Arch. Ration. Mech. Anal.,1997,137,381-394.
[75]Reilly, R.C. On the Hessian of a function and the curvatures of its graph, Michi-gan Math. J. [J].1973,20:373-383.
[76]Salani, P. A Brunn-Minkowski inequality for the Monge-Ampere eigenvalue [J]. Adv.Math.,2005,194:67-86.
[77]Salani, P. Convexity of solutions and Brunn-Minkowski inequalities for Hessian equations in R3 [J]. Adv.Math.,2012,229:1924-1948.
[78]Salani, P. Equazioni Hessiane e k-convessita(Phd Thesis.) [D]. Firenze: Universita di. Firenze,1997.
[79]Schneider, R. Convex Bodies:The Brunn-Minkowski Theory [M]. Cambridge Univ. Press,1993.
[80]Serrin, J. A symmetry problem in potential theory [J]. Arch. Ration. Mech. Anal.,1971,43:304-318.
[81]Sirakov, B. Symmetry for exterior elliptic problems and two conjectures in poten-tial theory [J]. Ann. Inst. H. Poincare Anal. Non Lineaire,2001,18:135-156.
[82]Szego, G. Inequalities for certain eigenvalues of a membrane of given area [J]. J. Bational Mech. Anal,1954,3:343-356.
[83]Talenti, G. Elliptic equations and rearrangements [J]. Ann. Scuola Norm. Sup. Pisa.,1976,3:697-718.
[84]Talenti, G. On isoperimetric theorems of mathematical physics [A]. In P. M. Gru-ber & J. M. Wills(eds.). Handbook of Convex Geometry, vol.B[C]. Amsterdam: North-Holland,1993, pp.1131-1147.
[85]Talenti, G. Inequalities in rearrangement invariant function spaces [A]. In M. Krbec, A. Kufner, B. Opic, & J. Rakosnik(eds.). Nonlinear Analysis. Func-tion Spaces and Applications,. vol.5[C]. Prague:Prometheus Publishing House, 1994, pp.177-230.
[86]Takahashi, F. & A.Uegaki. A Payne-Rayner Type Inequality for the Robin Prob-lem on Arbitrary Minimal Surfaces in Rn [J]. Results Math.,2011,59:107-114.
[87]Therese, M. & K.Jobin. Isoperimetric monotonicity and Isoperimetric inequali-ties of Payne-Rayner type for the first eigenfunction of the Helmholtz problem [J]. J. Appl.Math. Phys.,1981,32:625-646.
[88]Thompson, C.J. on the ratio of conseccutive eigenvalues in N-dimensions [J]. Stud.Appl.Math.,1969,48:281-283.
[89]Trudinger, N.S. On new isoperimetric inequalities and symmetrization [J]. J. Reine Angew. Math.,1997,488:203-220.
[90]Trudinger, N.S. & X-J.Wang. Hessian measures I, Dedicated to Olga Ladyzhen-skaya. Topol [J]. Methods Nonlinear Anal.,1997,10(2):225-239.
[91]Trudinger, N.S. & X-J. Wang. Hessian measures Ⅱ [J]. Ann.Math.,1999,15(2): 579-604.
[92]Tso, K. Remarks on critical exponents for Hessian operators [J]. Ann. Inst. H. Poincare.Anal. Non Lineaire,1990,7:113-122.
[93]Tso,K.On symmetrization and Hessian Equations[J].J.Anal.Math.,1989, 52:94-106.
[94]Vogel,A.L.symmetry and regularity for general regions having solutions to certain overdetermined boundary value problems[J].Atti semin.Mat.Fis.Univ. Modena.,1992,40:443-484.
[95]Wang, Q.L. & C.Y. Xia. Isoperimetric bounds for the first eigenvalue of the Laplacian [J]. Z.Angew. Math. Phys., 2010,61:171-175.
[96]Wang,X.J.A class of fully nonlinear elliptic equations and related functionals [J]. Indiana Univ.Math. J., 1994,43(1):25-54.
[97]Weinberger,H.F.An isoperimetric inequality for the n—dimensional free mem-brane problem[J].J.Rational Mech.Anal.,1956,5:633-636.
[98]Weinberger,H.Remark on the preceding paper of Serrin[J].Arch.Ration.Mech. Anal.,1971,43:319-320.
[99]M.Willem.Minimax Theorems[M].Birkhauser,1996.
[100]伍卓群,尹景学＆王春鹏,椭圆与抛物型方程引论[M].北京:科学出版社, 2006.