### Paper Menu >>

### Journal Menu >>

Advances in Pure Mathematics, 2011, 1, 99-104 doi:10.4236/apm.2011.13022 Published Online May 2011 (http://www.scirp.org/journal/apm) Copyright © 2011 SciRes. APM Liouville Type Theor ems for Lichner owicz Equations and Ginzburg-Landau Equation: Survey Li Ma1 Department of Mat hematics, Henan Normal University, Xinxiang, China Department of Mat he m at i c al Sci e nces, Tsinghua Universi ty, Beijing, China E-mail: lma@math.tsinghua.edu.cn Received April 11, 2011; revised May 10, 2011; accepted May 12, 2011 Abstract In this survey paper, we firstly review some existence aspects of Lichnerowicz equation and Ginz- burg-Landau equations. We then discuss the uniform bounds for both equations in Rn. In the last part of this report, we consider the Liouville type theorems for Lichnerowicz equation and Ginzburg-Landau equations in Rn via two approaches from the use of maximum principle and the monotonicity formula. Keywords: Liouville Theorems, Ancient Solution, Ginzburg-Landau Equation, Lichenrowicz Equation 1. Introduction This article is based on the lecture given at March 3rd, 2011 in the international conference “Recent Advances in Nonlinear Partial Differential Equations: Part I” held at the Chinese University of Hong Kong. The initial-value problem of gen eral relativity consists in the resolution of a coupled system of three linear equ- ations and a quasilinear equation which determines the conformity factor, on an initial Riemannian manifold , M g. In the case where there are sources, the qua- si-linear equation can be written as 735 8=0 guRuVu Quu (1.1) where >0u is the unknown on the Riemannian mani- fold , M g, R is the scalar curvature, V, Q, are functions derived from the Ricci curvature of , M g. (1.1) is called the Lichnerowicz equation on , M g. Let 2=2 2nn and S be the best So- bolev constant. One interesting result derived from mountain pass lemma is below. Theorem 1.1 Assume that , n M g is compact, 3n. Consider the following Lichnerowicz equation 21 21 =u uBuAu with >0A and >0 maxMB. Assume that there is a positive function >0 and a constant >0Cn such that 1 222 1,>0 max n HMM M Cn AB SB Then there is a positive solution to Lichnerowicz equa- tion. Hebey-Pacard-Pollack [11] have applied the mountain pass lemma to the perturbation functionals to get positive approximation solutions and have proved the conver- gence of a subsequence to a positive solution. The Ginzburg-Landau (GL) model is proposed in 50’s in the context of super-co nductivity th eory an d its ener gy density is 2 22 11 =1. 24 eu uu Here :nk uR R. The stationary E-L equation for GL model is 2 =1.uuu Yanyan Li and Z. C. Han (1995) have also studied p-laplacian GL models. Other models with term 2 2 1m u, 1m, is also of interesting. The Lichnerowicz equation and the Ginzburg-Landau equations are important models in mathematical physics. The existence of solutions to both can be obtained via variational methods, the monotone method (also called sub-super solution method or barrier method), and per- 1The research is partially supported by the National Natural Science Foundation of China 10631020 and SRFDP 2009 0 0 02110019. L. MA Copyright © 2011 SciRes. APM 100 turbation methods. There are huge literatures about G-L models [2,5, 10,14,15,20], but there is not much works about the Lichnerowicz equation [4,12,18]. For the existence re- sults of the Lichnerowicz equation, one may see the works [7], [8], and [11]. For the existence results of the Ginzburg-Landau equation, one may find more refer- ences from the work [22]. Our main topic for both equations is about the Liou- ville type results, which are closely related to compact- ness theorems of the solution spaces. The heat flow me- thod to both equations will be an interesting topic for studying. 2. Classical Liouville Type The or e m s a n d Keller-Osserman Theory The famous Liouville theory says that any non-negative harmonic function is constant. One can prove this by Harnack inequality or differential gradient Harnack es- timate. People may extend this to non-negative p-harmonic function or to non-negative solution to the elliptic equa- tion , ==0,in. n ij ij ij LuauR We now recall the famous Keller-Osserman theory obtained around 1957. Given a domain . Consider the differential inequality ,in n ufuR where f t is a positive, continuous, and monotone increasing function for 0 tt satisfying the Osgood condition 12 00 dd<. tfs st Then any twice continuously differentiable function u can not satisfy >0u on the whole space and ufu outside of some ball. As an application of above theory, J. B. Keller and R. Osserman consider the non-existence result for the Gaussian curvature equation on the plane 2 =,in. u uKxe R Professor. Ni, Lin, W. Ding, W. Chen and C. Li, etc., have obtained a lot interesting existence results to this problem. More results and references may be found in my work [17]. It is also interesting to study the non-existence of non- trivial non-negative solutions or energy solutions to 1 =,in. pn uKxu uR where 2n and >1p. Another application of the Keller and Osserman theory (H. Brezis, 1984) we have that for 2 0n vCR sat- isfying ,>1,in pn vvpR we have =0v. In fact, H. Brezis getting upper bound of solutions by using the following boundary blow-up super-solution on the ball R Bp, 2 2 = RCR ux Rxp with =12p and a suitable constant C inde- pendent of >0R. Then for any fixed point n pR, sending R, we get =0v. If the Laplacian is replaced by p-Laplacian, Du and Guo (2002) can ex tend the Keller and Osserman theorem to this case. For more, one may see the works of A. Fa- rina, A. Ratto, M. Rigoli. Parallel result to Keller and Osserman can be done for non-negative ancient solutions to the parabolic inequal- ity: ,>1,in ,0. pn tvvp R See also the interesting work of J. Serrin [24] for more Liouville type theorems about elliptic and parabolic equ- ations. We have the following Liouville type Theorem. Theorem 2.1 (L. Ma, 2010 [16]) Let >0u in (1.1) with =n M R, =0Q, =1V, =1 . Then =1u. For positive solutions to the general equation 2 =, qq uu u on n R with >1q, we have the same result. However, H. Brezis [3] proves that for 0,1q, the same result is not true, but we always have 1u. Similar result is also true for M being a complete Riemannian manifold , M g with its Ricci curvature bounded from blow. Here is the argument of Theorem 2.1. Let f s 2 =qq uu for >0q. For any fixed n x R and >0 , consider the new function 2 =.uy uyyx Note that uyu y as y. Then the minimum of it can be achieved at some point z. Then, =,uz ux ux which implies that uz ux. Using the monotonicity of f, i .e. 0f, we have f uz fux L. MA Copyright © 2011 SciRes. APM 101 At this point z, we have 220Du z , 0=2= 2.uzuzn fuzn Then we have 02 2fuzn fuxnfux as 0 . Recall that =qp fuuu or some >0q and >0p. Then we have 1u. Assume now that >1q. =10vu . Then =1. q vfvv Using the Keller-Osserman theory we then conclude that =0,..=1.vieu This completes the argument of Theorem 2.1. We now give som e remark. Set 3 =fuu u for 0u, which is the special case of Ginzurg-Landau equation. We then conclude from the argument above that =1u. In general, we know the below. Theorem 2.2 (Du, Ma [9], 2001) Assume that 2, n uCRR such that 3 =uuu on n R. Then we have 1u. In the papers of Du and Ma (2001-2003), more general logistic equations have been studied. It is there that we are interested in a problem related to Di Giorgi conjec- ture, which has been completely solved by Del Pino, J. Wei, etc., Savin, C. Gui, Ambrosio and Cabre, etc. Interestingly, Du and Guo can obtaine the below. Theorem 2.3 (Du, Guo, 2002) Assume that 2, n uCRR such that 3 =,1<, puu up on n R. Then we have 1u. There is a similar De Giorgi conjecture related to the equation above (see the work of L. Caffarelli, etc., A Gradient Bound for Entire Solutions of Quasi-Linear Equations and Its Consequences, Communications on Pure and Applied Mathematics, Volume 47, Issue 11, November 1994, Pages: 1457-1473 ). It would be interesting to study the following evolu- tion equation 3=0, in(0,) tp uuuuT with suitable initial and boundary conditions. The results above can be extended to nonlinear heat equations. Set =qp fuu u for some >0q and >0p. Consider ancient solutions to the following parabolic equation =,>0,in ,0. n tufuu R (2.1) For any fixed ,,0 n xR and >0 , con- sider the new function 22 ,= , ,, n uytuyyxt yt R Note that ,uyt as yt . Then the minimum of it can be achieved at some point ,zt . Then at this point 0 tu , which implies that 22 0,2=,2, tuzt nfuztn which is less than 2 ,2 ,fux nfux as 0 . Hence 1u (and one may also show that =1u). Theorem 2.4 Let >0u be an ancient solution to (2.1). Then 1u. 3. Results for Ginzburg-Landau Equations Our uniform bound result (due to H. Brezis) is Theorem 3.1 Any smooth solution to GL model is bounded in the sense that 1ux. H. Brezis uses the Kato inequality to prove Theorem 3.1. We shall report here his argument. My argument is different but it is also based on the maximum principle. Before I give the proof, let’s recall the famous Kato inequality. Assume that 1 loc uL and 1 loc uL . Firstly we may assume that u is smooth. Note that for >0 , we have 22 22 3 =. () u uuu uu The latter is bigger than 2. uu u Hence we have for any 2 0 0C we have 2 2. u uu u Sending 0 , we then get sign ,uuu i.e., sign .uuu We have the parabolic version of the Kato inequality. L. MA Copyright © 2011 SciRes. APM 102 Consider =,uuxt with in addition 1 tloc uL. Using 2 2 =, t t uu uu we have 2 2. tt u uuu u Then, letting 0 , we have ||sign . tt uuuu With the understanding above, we have that for any non-negative ancient solution to ,>1,in ,0 pn t vvvpR is trivial. The argument of this fact is almost the same as Bre- zis’s argument (1984). Therefore, we have the following extension of H. Bre- zis’ theorem. Theorem 3.2 Any ancient solution :, n uR 0k R to 2 =1 t uu uu must have 1u. Going back to the parabolic version of Lichnerowicz equation on manifold with non-negative Ricci curvature, we have the following result. Proposition 3.1 Any positive ancient solution to =qp t uuuu with >1q and >0p must be =1u. We now give the proof o f T h eorem 3.1. Brezis’s argument: Let 2 =1Wu . Then we have 22 sign1 .Wu u Note that 2222 =2 221.uuu uuu Then we have 22 22 21sign12 12.WuuuWW W Using the Keller-Osserman theory we then have 2 =0,..1.Wieu In my proof of this bound, I have used the original barrier functions use d by Kel ler -O sserman. It is interesting to know if one can extend the result above to p-Laplacian G-L model. In the following, we present the monotnonicity for- mula method to the Liouville type theorems [1,10,19]. It is our intention here to generalize the Liouville type results to a large class of solutions of a more general non-linear equations/systems 2 =0,in,,, nnk uWuRuC RR (3.1) where W is the gradient of the smooth function Wu on k R and 0Wu. We shall use an idea from Professor Hesheng Hu (1980) who introduced it for harmonic maps. Please see the book of Y. L. Xin [25] for results of harmonic maps. It is also interesting to know if one can extend this kind of result to p-laplacian case. Theorem 3.3 Assume that 0Wu is a non-trivial smooth function. Let :nk uR R, 2n, be a smooth solution to the Ginzburg-Landau system (3.1). Assume that there are a positive constant 0>0R and a positive function r on 0,R such that 0 22<. lim BB RRR ru Wu and 0d= . R rr r Then u is a constant. For any smooth mapping from the Riemannian mani- fold , M g to k R, we define the Stress-Energy tensor (see the paper of Baird-Eells) by 2 1 =dd, 2 u Suguu which is 1 =2 ijjj ikik Suuguu in local frames j e on M . By direct computation we know that div=,. ug Sudu A consequence of this formula is that if u is harmonic, then div 0 u S . Let X be a smooth vector field on M . Define the tensor X by ,= ,. ijej i X ee g Xe Then we hav e 2 1div=div, 2 ,, jj u uX duXue du XuSX Take any compact domain DM with smooth L. MA Copyright © 2011 SciRes. APM 103 boundary D. Choose the local frame j e such that = n e be the unit outward normal to the boundary. Then we hav e 2 1,d, 2 =div, . D uu D uX uXu SXS X (3.2) Below we let =n M R. To explain the main idea of the proof, we start with the simple case when =0Wu and =1r. That is, =0u and u has finite energy. We take in (3.2) =0 R DB , =r X r. Note that = ij ij X . Then we have 2 2 ,= ==, 2 uijiju n SXSXtrS u ,= X R and 2 d,= r uX uRu on 0 R B. Hence 2 0 2 ,= 2 u BR n SX u (3.3) and by (3.2), we have 222 12 =. 22 r BB RR n Ruu u Then we have the following Liouville theorem for harmonic functions. Theorem 3.4 Let :n uR R be a harmonic fun ction with slowly energy divergence, i.e., there are a positive constant 0>0R and a positive function r on 0,R such that 2 0 <. lim BB RRR ru and 0d= . R rr r Then u is a constant. Here is the idea of proof: Assume that u is not a constant. Then there are some positive constants >0C and 0>0R such that 0 2 0 22>0. 2 BR nuC By this we know that 2 BR RuC (3.4) for any 0 RR. Hence we have 00 0 22 d, RR BB RBR RR R r uuC r as R, which gives a contradiction. In fact, the proof of Theorem 3.4 goes below. As sume u is not a constant. By (3.4) we know that 22 0 0 0d, R BBR B RR r R R rur u r Cr r as R, which gives a contradiction. This completes the proof of Theorem 3.4. We now turn to the proof of Theorem 3.3. Assume that Wu is non-trivial. In this case div=, =, u Su duWudu and div=,=. uXX SXWu uWu Again we take =0 R DB . Then by (3.2) we have 2 1,, 2 =,. D Xu D uX duXu WuSX (3.5) Simplifying this identity we can derive th e following 2 1. 2 BR RuWuC Then using the argument above we obtain Theorem 3.3. We remark that the results above can be extended to complete Riemannian manif olds with bounded Ricci cu r- vature. We remark that for a large class of elliptic equations/ systems (also for parabolic equation/system), the Liou- ville type theorems are equivalent to a local uniform bound of solutions. For this direction, one may see the recent works of P. Polacik, P. Souplet, P. Quittner, H. Zou, etc. However, it is open for Lichnerowicz equation on compact Riemannian manifolds (see also in [13]). We also make a remark below. In the study of elliptic systems, one may use the Pohozaev type identity (which is a sister of monotonicity formula) and the interpolation inequalities to derive a contraction mapping property about p L norm of the solution. From the contraction mapping property one then get the solution trivial (and the Liouville type theorem). One may see the works of Chen-Li [6], Souplet, etc. [21,23], for th is kind of results for the Lane-Emden conjecture. We would like to thank Professor H. Brezis, who (see also [3]) has informed me that in the statements of Theorems 1 and 2 in [16], the power >1p should be >2p. Actually, we have used >2p in the proof of Theorem 1.1 in [16]. 4. References [1] N. D. Alikakos, “Some Basic Facts on the System L. MA Copyright © 2011 SciRes. APM 104 =0uWu ,” Procceedings of the American Mathe- matical Society, Vol. 139, No. 1, January 2011, pp. 153-162. doi:10.1090/S0002-9939-2010-10453-7 [2] F. Bethuel, P. Gravejat and J.-C. Saut, “Traveling Waves for the Gross-Pitaevskii Equation II,” Communications in Mathematical Physics, Vol. 285, No. 2, 2009, pp. 567-651. doi:10.1007/s00220-008-0614-2 [3] H. Brezis, “Comments on Two Notes by L. Ma and X. Xu,” Comptes Rendus Mathematique, 2011. [4] A. Chaljub-Simon and Y. Choquet-Bruhat, “Global Solu- tions of the Lichnerowicz Equation in General Relativity on an Asymptotically Euclidean Complete Manifold,” General Relativity and Gravitation, Vol. 12, No. 2, 1980, pp. 175-185. doi:10.1007/BF00756471 [5] F. Bethuel, H. Brezis and F. Helein, “Ginzburg-Landau Vortices,” Birkhauser, Basel and Boston, 1994. [6] W. X. Chen and C. M. Li, “An Integral System and the Lane-Emden Conjecture,” Discrete Continuous Dynami- cal Systems, Vol. 24, No. 4, 2009, pp. 1167-1184. doi:10.3934/dcds.2009.24.1167 [7] Y. Choquet-Bruhat, J. Isenberg and D. Pollack, “The Einstein-Scalar Field Constraints on Asymptotically Euc- lidean Manifolds,” Chinese Annals of Mathematics, Se- ries B, Vol. 27, No. 1, 2006, pp. 31-52. doi:10.1007/s11401-005-0280-z [8] Y. Choquet-Bruhat, J. Isenberg and D. Pollack, “The Constraint Equations for the Einstein-Scalar Field System on Compact Manifolds,” Classical and Quantum Gravity, Vol. 24, No. 4, 2007, pp. 809-828. doi:10.1088/0264-9381/24/4/004 [9] Y. H. Du and L. Ma, “Logistic Equations on n R by a Squeezing Method Involving Boundary Blow-up Solu- tions,” Journal of London Mathematical Society, Vol. 64, No. 2, 2001, pp. 107-124. doi:10.1017/S0024610701002289 [10] A. Farina. “Finite-Energy Solutions, Quantization Effects and Liouville-Type Results for a Variant of the Ginzburg -Landau Systems in RK,” Comptes rendus de l'Académie des Sciences, Série 1, Mathématique, Vol. 325, No. 5, 1997, pp. 487-491 [11] E. Hebey, F. Pacard and D. Pollack, “A Variational Analysis of Einstein-Scalar Field Lichnerowicz Equations on Compact Riemannian Manifolds,” Communications in Mathematical Physics, Vol. 278, No. 1, 2008, pp. 117-132. doi:10.1007/s00220-007-0377-1 [12] E. Hebey, “Existence, Stability and Instability for Ein- stein-Scalar Field Lichnerowicz Equations,” Two hours lectures, Institute for Advanced Study, Princeton, October 2008. [13] J. Davila, “Global Regularity for a Singular Equation and Local Minimizers of a Nondifferentiable Functional,” Communications in Contemporary Mathematics, Vol. 6, No. 1, 2004, pp. 165-193. doi:10.1142/S0219199704001240 [14] F. H. Lin, “Static and Moving Vortices in Ginz- burg-Landau Theories,” In: T. N. Knoxville, Ed., Nonli- near Partial Differential Equations in Geometry and Physics, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Basel, Vol. 29, 1997, pp. 71-111. [15] F. H. Lin and J. C. Wei, “Traveling Wave Solutions of the Schrodinger Map Equation,” Communications on Pure and Applied Mathematics, Vol. 63, No. 12, 2010, pp. 1585-1621. doi:10.1002/cpa.20338 [16] L. Ma, “Liouville Type Theorem and Uniform Bound for the Lichnerowicz Equation and the Ginzburg-Landau Equation,” Comptes Rendus Mathematique, Vol. 348, No. 17, 2010, pp. 993-996. doi:10.1016/j.crma.2010.07.031 [17] L. Ma, “Three Remarks on Mean Field Equations,” Pa- cific Journal of Mathematics, Vol. 242, No. 1, 2009, pp. 167-171. doi:10.2140/pjm.2009.242.167 [18] L. Ma and X. W. Xu, “Uniform Bound and a Non-Existence Result for Lichnerowicz Equation in the Whole N-Space,” Comptes Rendus Mathematique, Vol. 347, No. 13-14, 2009, pp. 805-808. doi:10.1016/j.crma.2009.04.017 [19] L. Modica, “Monotonicity of the Energy for Entire Solu- tions of Semilinear Elliptic Equations,” In: F. Colombini, A. Marino and L. Modica, Eds., Partial Differential Equ- ations and the Calculus of Variations, Birkhauser, Boston, Vol. 2, 1989, pp. 843-850. [20] E. Sandier and S. Serfaty., “Vortices in the Magnetic Ginzburg-Landau Model,” Birkhauser, Basel, 1997. [21] P. Souplet, “The Proof of the Lane-Emden Conjecture in Four Space Dimensions,” Advances in Mathematics, Vol. 221, No. 5, 2009, pp. 1409-1427. doi:10.1016/j.aim.2009.02.014 [22] M. del Pino, etc., “Variational Reduction for Ginz- burg-Landau Vortices,” Journal of Functional Analysis, Vol. 239, No. 2, 2006, pp. 497-541. doi:10.1016/j.jfa.2006.07.006 [23] P. Polacik, P. Souplet and P. Quittner, “Singularity and Decay Estimates in Superlinear Problems via Liou- ville-Type Theorems, Part 1: Elliptic Equations and Sys- tems,” Duke Mathematical Journal, Vol. 139, No. 3, 2007, pp. 555-579. doi:10.1215/S0012-7094-07-13935-8 [24] J. Serrin, “Entire Solutions of Nonlinear Poisson Equa- tions,” Proceedings of London Mathematical Society, Vol. s3-24, No. 2, 1972, pp. 343-366. [25] Y. L. Xin, “Geometry of Harmonic Maps. Series: Pro- gress in Nonlinear Differential Equations and their Ap- plications,” Birkhauser Boston, Inc., Boston, Vol. 23, 1996, pp. 241. |