On Bernstein-type theorems

Size: px

Start display at page:

Download "On Bernstein-type theorems"

Dayna Montgomery
5 years ago
Views:

1 On Bernstein-type theorems Peter Lewintan arxiv: v [math.dg] 2 Jan 209 July 8, 203 In memory of Franki Dillen We summarize results concerning the Bernstein property of differential equations. In this short overview we will look at entire solutions of partial differential equations of second order. We say a solution to be entire if it is defined over the entire plane (R 2 ) or over the entire space (R n ). As we will see, some differential equations possess only linear functions as entire solutions, i.e., in these cases the linearity of an entire solution follows from its mere existence, without any boundedness conditions. If a partial differential equation has only affine linear functions as entire solutions, we say that it has the Bernstein property, according to a celebrated result of S. N. Bernstein [B], whichstatesthatevery C 2 -solutionof theminimalsurface equation ( + u y 2 )u xx 2u x u y u xy + ( + u x 2 )u yy = 0 over theentireplane R 2 isnecessarilyaffinelinear. We start with the following operator introduced in[zt]: L γ,ε [u] := ( ) ( ) 2ε + (γ + )u 2 2 x + (γ )u y u xx +4u x u y u xy + 2ε + (γ )u 2 2 x + (γ + )u y u yy with γ, ε R and consider the equation L γ,ε [u] = 0 over R 2. Without loss of generalitywe can choose ε { ; 0; }, for we can obtainentire C 2 -solutions of L γ,ε [u] = 0 (with ε 0) via an appropriatescaling of thesolutions of L γ,± [u] = 0 and vice versa. Ourequation L γ,ε [u] = 0isellipticif εγ > 0and γ. Westartwiththiscaseandconsider other cases later: peter.lewintan@uni-due.de, University ofduisburg-essen, Germany.

2 First we choose γ = ε =, so that L, [u] = 0 corresponds to the familiar minimal surface equation over R 2. As we have already mentioned, it has the Bernstein property. The extension of this result to higher dimensions is well-known: The Bernstein theorem was extended to n 7, i.e., each entire C 2 -solution of the minimal surface equation Du div = 0 over R n + Du 2 hastobeaffine linear,cf. [dg]for n = 3, [Al]for n = 4and [Sms] for n 7. Surprisingly, the Bernstein theorem fails for dimensions n 8 as there exist entire nonlinear solutions to the minimal surface equation, cf. [BdGG]. However, under suitable growth conditions on the solution u or its gradient Du, one can prove Bernstein-type theorems in every dimension n N, cf. e.g. [BdGM], [Mo], [CNS], [Ni], [EH]. Let us now turn to higher codimension k >, so that instead of the minimal surface equation, we consider a system of partial differential equations, the so-called minimal surface system. Already in the simplest case of dimension n = 2 and codimension k = 2 we obtain several entire non-linearsolutions f C 2 (R 2, R 2 )of thecorrespondingsystem ( + f y 2 )f xx 2f x f y f xy + ( + f x 2 )f yy = 0, for everyholomorphicfunction f : C C, regardedasamap R 2 R 2, solvesit. If k >,the boundedness of the gradient is a sufficient condition for a Bernstein-type theorem only when n 3, cf. [CO]and[Fis]. Ananalogousresult for n 4iswrong, asfollowsfrom thelawson- Osserman cone[lo].undersufficiently strongassumptions, onecanstillachievethelinearity of solutions, cf. e.g. [HJW], [JX], [Wg], [JXY]. Returning to the original Bernstein theorem, we can extend it to further differential equations. Such classes of elliptic differential equations (over R 2 ), entire C 2 -solutions of which are necessarily affine linear, were given e.g. in[be],[f],[je], [Si], [Si2]. The minimal surface equation is included in all these classes. Equations of minimal surface type possess the Bernstein property for n 7, cf. [Si], and we need additional growth conditions in other dimensions, cf. [Win]. For an elaborated account of the minimal surface case we refer to the monograph [DHT]. Now we choose γ = ε =, so that L, [u] = 0 corresponds to the wrong minimal surface equation ( + u x 2 )u xx + 2u x u y u xy + ( + u y 2 )u yy = 0. In [Si2] Simon posed the question whether this equation has the Bernstein property. We can answer in two different ways: Bytheseparationansatz u(x, y) = g(x)+h(y)weconstructentirenon-linear C 2 -solutions of this equation explicitly. In a similar manner we can determine further (not necessarily elliptic) differential equations without the Bernstein property, cf. [Lew]. This cone is anexamplefor anon-analytic Lipschitzsolutionin higher codimensions. 2

3 WeuseanexplicitcriterionofJ.C.C.NitscheandJ.A.Nitsche[NN],whichensuresthe existenceofentirenon-linear C 2 -solutionsofcertainellipticdifferentialequationsincluding the wrong minimal surface equation. The Nitsche criterion states: The Euler-Lagrange equation arising from the regular variational integral R 2 F( Du 2 )dx hasentirenon-linear C 2 -solutionsif theintegral + wλ(w) 2 + wλ(w) dw w with λ(w) := 2 F (w) F (w) diverges. Withthiscriterionwecantreatalltheothercombinationsof γ and εintheellipticcase,more precisely: In the elliptic case (εγ > 0 and γ ) we obtain our equation L γ,ε [u] = 0 as Euler- Lagrange equation of the functional F γ,ε (u) := F γ,ε ( Du 2 )dx R 2 by setting w := Du 2 and F γ,ε (w) := In all thesecaseswe gain The integral { (2 ε + γ w) γ γ, for (γ >, ε > 0) or(γ, ε < 0), e w 2ε, for (γ =, ε > 0). λ γ,ε (w) = 2 F γ,ε (w) F γ,ε (w) = 2 2ε + (γ )w. + wλ γ,ε (w) 2 + wλ γ,ε (w) dw w = 2 ( 2ε + γw + w diverges for all admissible γ. By the Nitsche criterion, in all these cases there exist entire non-linear C 2 -solutions of the corresponding Euler-Lagrange equation, i.e. for γ, ε > 0 and γ <, ε < 0 resp. theequation L γ,ε [u] = 0doesnothave thebernstein property. If we now let εtendto zerowith γ >, theintegrandconverges to F γ,0 (w) = ( γ w) γ γ. ) dw 3

4 By thesubstitution p = 2γ (for γ ) we obtain γ F γ,0 ( Du 2 ) = c(p) p Du p. Thus, we canassociateour functional F γ,0 withthefunctional F p (u) := Du p dx where p = p R 2 2γ γ. The minimizers of the latter functional are the so-called p-harmonic functions, the solutions of p u := div( Du p 2 Du) = 0 over R 2. Wecanregardsolutionsof L γ,0 [u] = 0assolutionsof p u = 0. Itiscommontointroducethe p-harmonicfunctionsintheweaksenseandnotassolutionsof L γ,0 [u] = 0. Asfarastheauthor is aware, it is not clearwhether entire C 2 -solutions of L γ,0 [u] = 0 with γ > are necessarily affine linear. However, under suitable growth conditions, cf. [KSZ], each entire p-harmonic function is affine linear. This statement also holds true in higher dimensions. For γ = weget p = and theequation L,0 [u] = 0, i.e. which corresponds to u y 2 u xx 2u x u y u xy + u x 2 u yy = 0, ( ) Du u := div = 0. Du The equation L,0 [u] = 0doesnot have thebernstein property,for therearesolutionsof the form u(x, y) = g(x), withanarbitrary g C 2 (R, R), or u(x, y) = e x+y. In a more interesting case γ tends to and p tends to +. Indeed, the equation L,0 [u] = 0 correspondstotheequationof theso-called -harmonic function over R 2 u x 2 u xx + 2u x u y u xy + u y 2 u yy = 0. The latter has the Bernstein property, cf. [Ar]. One can extend this result to higher dimensions if additional regularity be assumed, more precisely: Eachentire C 4 -solutionof n u := u xj u xk u xj x k = 0 j,k= over R n is necessarily affine linear,cf. [Yu]. It is not clear whether C 2 -regularity suffices here. If so, the equation u = 0 would have the Bernstein property in all dimensions, in contrast to the minimal surface equation. Concerning all the other combinations of ε and γ, we state the following: 4

5 For ε = γ = 0wehaveentirenon-linear C 2 -solutions: u(x, y) = x 2 +y 2 solves L 0,0 [u] = 0. Also,for ε > 0and γ = ourequationdoesnothavethebernsteinproperty,for L, [u] = 0 admits solutions of the form u(x, y) = x + h(y), with an arbitrary h C 2 (R, R). However, L, [u] = 0correspondstothemaximalsurfaceequation ( u y 2 )u xx + 2u x u y u xy + ( u x 2 )u yy = 0, andunderconstraint Du C 0 < itsentiresolutionsareaffinelinear. Thisisalsovalidinhigher dimensions, cf. [Ca] and [CY]. For ε < 0 and γ = our equation L γ,ε [u] = 0 arises in the study of isentropic irrotational steady plane flows, cf. [CF]. References [Al] F. J. Jr. Almgren, Some interior regularity theorems for minimal surfaces and an extension of Bernstein s theorem, Annals of Mathematics(2) 84 (966), [Ar] G. Aronsson, On the partial differential equation u x 2 u xx + 2u xu yu xy + u y 2 u yy = 0, Arkiv för Matematik 7 (968), [B] S. N. Bernstein, Sur un théorème de géométrie et son application aux équations aux dérivées partielles du type elliptique, Comm. Soc. Math. de Kharkov 2éme sér. 2(95 97), no. 5, 38 45(French); German transl.: Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus, Mathematische Zeitschrift 26(927), no., [Be] L. Bers, Non-linear elliptic equations without non-linear entire solutions, Journal of Rational Mechanics and Analysis 3(954), [BdGG] E. Bombieri, E. de Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Inventiones Mathematicae 7(969), [BdGM] E. Bombieri, E. de Giorgi and M. Miranda, Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche, Archive for Rational Mechanics and Analysis 32 (969), (Italian). [Ca] E. Calabi, Examples of Bernstein problems for some nonlinear equations, Global Analysis (Proceedings of Symposia inpuremathematics), VolXV, [CNS] L. A. Caffarelli, L. Nirenberg and J. Spruck, On a form of Bernstein s theorem, Analyse mathématique et applications (988), [CO] S.-S. Chern and R. Osserman, Complete minimal surfaces in euclidean n-space, Journal d Analyse Mathématique 9 (967),5 34. [CF] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, N. Y.948. [CY] S. Y. Cheng and S. T. Yau, Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Annals of Mathematics (2) 04(976),no. 3, [dg] E. de Giorgi, Una estensione del teorema di Bernstein, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 9(965),no.,79 85 (Italian). [DHT] U. Dierkes, S. Hildebrandt and A. Tromba, Global Analysis of Minimal Surfaces, 2nd ed., Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences], 34, Springer, Heidelberg, 200. [EH] K. Ecker and G. Huisken, A Bernstein result for minimal graphs of controlled growth, Journal of Differential Geometry3 (990),no. 2, [F] R. Finn, On a problem of type, with application to elliptic partial differential equations, Journal of Rational Mechanics and Analysis 3(954),

6 [Fis] D. Fischer-Colbrie, Some rigidity theorems for minimal submanifolds of the sphere, Acta Mathematica 45 (980),no. 2, [Je] H. Jenkins, On quasi-linear elliptic equations which arise from variational problems, Journal of Mathematics and Mechanics 0(96), [HJW] S. Hildebrandt, J. Jost and K. O. Widman, Harmonic mappings and minimal submanifolds, Inventiones Mathematicae 62 (980), [JX] J. Jost and Y. L. Xin, Bernstein type theorems for higher codimension, Calculus of Variations and Partial Differential Equations9(999),no. 4, [JXY] J. Jost, Y. L. Xin and L. Yang, The geometry of Grassmannian manifolds and Bernstein type theorems for higher codimension, arxiv: [KSZ] T. Kilpelinen, H. Shahgholian and X. Zhong, Growth estimates through scaling for quasilinear partial differential equations, Annales Academiæ Scientiarium Fennicæ. Mathematica 32(2007), no. 2, [LO] H. B. Lawson, Jr., and R. Osserman, Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system,actamathematica 39 (977),no. 2, 7. [Lew] P. Lewintan, The Wrong Minimal Surface Equation does not have the Bernstein property, Analysis. International Mathematical Journal of Analysis and its Applications 3(20), no. 4, [Mo] J. Moser, On Harnack s theorem for elliptic differential equations, Communications on Pure and Applied Mathematics 4(96), [Ni] J. C. C. Nitsche, Lectures on minimal surfaces, Vol. : Introduction, fundamentals, geometry and basic boundary value problems, Cambridge University Press, Cambridge, 989. [NN] J. C. C. Nitsche and J. A. Nitsche, Ein Kriterium für die Existenz nicht-linearer ganzer Lösungen elliptischer Differentialgleichungen, Archiv der Mathematik 0(959), (German). [Si] L. M. Simon, On some extensions of Bernstein s theorem, Mathematische Zeitschrift 54 (977), no. 3, [Si2] L. M. Simon, Asymptotics for exterior solutions of quasilinear elliptic equations, Geometry from the Pacific Rim (997), [Sms] J. Simons, Minimal varieties in riemannian manifolds, Annals of Mathematic (2) 88 (968), [Wg] M.-T. Wang, On graphic Bernstein type results in higher codimension, Transactions of the American Mathematical Society355 (2002),no., [Win] S. Winklmann, A Bernstein result for entire F-minimal graphs, Analysis. International Mathematical Journal of Analysis and its Applications27 (2007),no. 4, [Yu] Y. Yu, A remark on C 2 infinity-harmonicfunctions, Electronic Journal of Differential Equations (2006), no. 22, 4. [ZT] I. A. Zorina and V. G. Tkachev, On entire solutions of quasilinear equations with a quadratic principal part, Vestnik Samarskogo Gosudarstvennogo Universiteta. Estestvennonauchnaya Seriya(2008), no. 3, 08 23(Russian). 6

Bernstein's Theorem MARIA ATHANANASSENAS

Bernstein's Theorem MARIA ATHANANASSENAS Abstract: We present a proof of Bernstein's theorem for minimal surfaces which makes use of major techniques from geometric measure theory. 1. THE PROBLEM Bernstein's