Regularity of n/2-harmonic maps into spheres [Elektronische Ressource] / vorgelegt von Armin Schikorra

Armin Schikorra

Description

Regularity of n/2-harmonic maps into spheresVon der Fakult at fur Mathematik, Informatik und Naturwissenschaftender Rheinisch{Westf alischen Technischen Hochschule Aachenzur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigteDissertationvorgelegt vonDiplom{MathematikerArmin Schikorraaus Altenkirchen (Westerwald)Berichter: Prof. Dr. Heiko von der Mosel (RWTH Aachen)Prof. Dr. Pawe l Strzelecki (Uniwersytet Warszawski)Tag der mundlic hen Prufung: 22. September 2010Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar.AbstractIn the present thesis, we consider critical points of the functional n 2 4E (v) := vnnRn nn n n2 2evaluated at points v2 H (R ) with the constraintjvj = 1 on some domain DR . Here, H (R ) denotes2 nthe fractional Sobolev space of all functions v2L (R ) such thatn ^ 2 n2j j v ()2L (R );^where () is the Fourier transform. We extend earlier results for evenn andn = 1 to arbitrary dimensionn2Nby proving H older continuity of critical points u of E .nAs for the proof, we adapt an approach by L. Tartar, which was used originally to prove Wente’s inequality, inorder to ensure the existence of compensation phenoma appearing in the Euler-Lagrange equations.In order to localize this e ect, we establish several localization results for nonlocal operators comparable to thefractional laplacian.

Sujets

• Mathematics

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2010
• Nombre de lectures: 5
• Langue: English

Extrait

Regularity of n/2-harmonic maps into spheres

VonderFakulta¨tfu¨rMathematik,InformatikundNaturwissenschaften derRheinisch–Westfa¨lischenTechnischenHochschuleAachen zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation

vorgelegt von

Diplom–Mathematiker Armin Schikorra aus Altenkirchen (Westerwald)

Berichter: Prof. Dr. Heiko von der Mosel (RWTH Aachen) Prof.Dr.Pawe
lStrzelecki(UniwersytetWarszawski)

Tagderm¨undlichenPr¨ufung:22.September2010

DieseDissertationistaufdenInternetseitenderHochschulbibliothekonlineverf¨ugbar.

Abstract In the present thesis, we consider critical points of the functional E n (v) := ˆ  Δ 4 v  n 2 Rn evaluated at pointsv∈H n 2 (R n ) with the constraint|v|= 1 on some domainD⊂R n . Here,H n 2 (R n ) denotes the fractional Sobolev space of all functionsv∈L 2 (R n ) such that |∙| n 2 v ∧ (∙)∈L 2 (R n ), where (∙) ∧  extend earlier results for evenis the Fourier transform. Wenandn= 1 to arbitrary dimensionn∈N byprovingHo¨ldercontinuityofcriticalpointsuofE n . As for the proof, we adapt an approach by L. Tartar, which was used originally to prove Wente’s inequality, in order to ensure the existence of compensation phenoma appearing in the Euler-Lagrange equations. In order to localize this effect, we establish several localization results for nonlocal operators comparable to the fractional laplacian. Finally, in order to prove that the Euler-Lagrange equations govern the growth of Δ 4 n uwe develop a fractional version of Hodge decomposition with local estimates similar to Cauchy estimates. Then, a fractional yet localized version of the Dirichlet growth theorem which we establish in the appendix impliesH¨oldercontinuity.

Contents

1 Introduction 2 Lorentz-, Sobolev Spaces and Cutoff Functions 2.1 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Fractional Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Annuli-Cutoff Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 An Integral Definition for the Fractional Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . 3MeanValuePoincare´InequalityofFractionalOrder 3.1 On the Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 On the Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Comparison between Mean Value Polynomials on Different Sets . . . . . . . . . . . . . . . . . . . 4 Integrability and Compensation Phenomena 5 Localization Results for the Fractional Laplacian 5.1 Multiplication with Disjoint Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Equations with Disjoint Support localize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Hodge decomposition and Local Estimates ofs of Theorem Proof-harmonic Functions:1.6. . . 5.4 Products of Lower Order Operators localize well . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Fractional Product Rules for Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Local Estimates and Compensation Phenomena: Proof of Theorem1.5 7 Euler-Lagrange Equations 8 Homogeneous Norm for the Fractional Sobolev Space 8.1 Comparison Results for the Homogeneous Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Localization of the Homogeneous Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Growth Estimates: Proof of Theorem1.2 A Ingredients for the Dirichlet Growth Theorem A.1 Iteration Lemmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 A Fractional Dirichlet Growth Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 11 11 13 15 19 22 27 28 31 34 43 45 45 46 47 50 52 56 64 66 66 69 72 76 76 79

1 Introduction

1 Introduction In his seminal work [elH´90]F´e.Hko-dihetwromtapsfdtsiulinoianemsnlagureedovprinlemcinomrahrofytir B 1 (0)⊂R 2 into them-dimensional sphereS m−1 ⊂R m for arbitrarym∈N. These maps are critical points of the functional E 2 (u) := ˆ |ru| 2 ,whereu∈W 1,2 (B 1 (0),S m−1 ). B 1 (0)⊂R2 The importance of this result is the fact that harmonic maps in two dimensions are special cases of critical points of conformally invariant variational functionals, which play an important role in physics and geometry and have been studied for a long time H´lein’s approach is based on the discovery of a compensation phenomenon : e appearing in the Euler-Lagrange equations ofE 2 , using a relation between div-curl expressions and the Hardy space.ThiskindofrelationhadbeendiscoveredshortlybeforeinthespecialcaseofdeterminantsbyS.Mu¨ller [l90M¨u] and was generalized by R. Coifman, P.L. Lions, Y. Meyer and S. Semmes [CLMS93hdsineedentxleie].H´ result to the case where the sphereS m−1 general target manifold developing the so-called moving-is replaced by a frame technique which is used in order to enforce the compensation phenomenon in the Euler-Lagrange equations [H´el91inally,T].F[eR.vi`ireRiv07] was able to prove regularity for critical points of general conformally invariant functionals, thus solving a conjecture by S. Hildebrandt [Hil82]. He used an ingenious approach based on K. Uhlenbeck’s results in gauge theory [Uhl82] in order to implement div-curl expressions in the Euler-Lagrange equations,atechniquewhichcanbereinterpretedasanextensionofH´elein’smovingframemethod;see[Sch10]. FormoredetailsandreferenceswerefertoHe´lein’sbook[2l0´eH] and the extensive introduction in [Riv07] as well as [Riv09]. Naturally, it is interesting to see how these results extend to other dimensions: In the four-dimensional case, regularity can be proven for critical points of the following functional, the so-called extrinsic biharmonic maps: E 4 (u) := ˆ |Δu| 2 ,whereu∈W 2,2 (B 1 (0),R m ). B 1 (0)⊂R 4 This was done by A. Chang, L. Wang, and P. Yang [CWY99a sphere as the target manifold,] in the case of and for more general targets by P. Strzelecki [Str03], C. Wang [Wan04] and C. Scheven [Sch08]; see also T. LammandT.Rivi`ere’spaper[LR08 generally, for all even]. Moren≥similar regularity results hold, and we6 refer to the work of A. Gastel and C. Scheven [GS09] as well as the article of P. Goldstein, P. Strzelecki and A. Zatorska-Goldstein [GSZG09]. In odd dimensions non-local operators appear, and only two results for dimensionn= 1 are available. In [DLR09aLioandT],F.Dnocredlofytiuniteri`iv.RH¨veroepfenufohtanltcoiiticorcrintsalpo E ˆ  Δ 1 4 u 2 ributionsuwith finite energy andu∈S m−1 a.e. 1 (u) =  ,defined on dist R1 In [DLR10] this is extended to the setting of general target manifolds. In general, we consider forn, m∈Nand some domainD⊂R n the regularity of critical points onDof the functional E n (v) = ˆ  Δ n 4 v  2 , v∈H 2 n (R n ,R m ), v∈S m−1 a.e. inD. (1.1) Rn Here, Δ 4 n denotes the operator which acts on functionsv∈L 2 (R n ) according to n  Δ n 4 v  ∧ (ξ) =|ξ| 2 v ∧ (ξ),for almost everyξ∈R n , where () ∧ denotes the application of the Fourier transform. The spaceH n 2 (R n ) is the space of all functions v∈L 2 (R n ) such that Δ n 4 v∈L 2 (R n  us remark that the interest in energies of the type (). Let1.1) is not only motivated by above mentioned purely theoretic considerations. In fact, energies like that appear in physical models of, e.g., magnetostatic energies (cf. [KMM06 term “critical point” is defined as usual:]). The Definition 1.1(Critical Point).Letu∈H n 2 (R n ,R m ),D⊂R n . We say thatuis a critical point ofE n (∙)on Difu(x)∈S m−1 for almost everyx∈Dand d dt  t0 E(u t,ϕ ) = 0 =

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for anyϕ∈C 0 ∞ (D,R m )whereu t,ϕ ∈H n 2 (R n )is defined as u t,ϕ = ( u Π(u+tϕ)in inR D, n \D. Here,Πdenotes the orthogonal projection from a tubular neighborhood ofS m−1 intoS m−1 defined asΠ(x) = |x x | . Ifnis an even number, the domain ofE n (∙) is just the classic Sobolev spaceH n 2 (R n )≡W n 2 ,2 (R n ), for odd dimensions this is a fractional Sobolev space (see Section2.3 in). FunctionsH n 2 (R n ) can contain logarithmic singularities (cf. [Fre73]) but this space embeds continuously intoBM O(R n ), and even only slightly improved integrability or more differentiability would imply continuity. In the light of the existing results in even dimensions and in the one-dimensional case, one may expect that similar regularity results should hold for any dimension. As a first step in that direction, we establish regularity ofn/2-harmonic maps into the sphere. Theorem 1.2.For anyn≥1, critical pointsu∈H 2 n (R n )ofE n on a domainDllyH¨oldarelocantinercouous inD. Note that here – in contrast to [DLR09] – we work on general domainsD⊆R n . This is motivated by the factsthatH¨oldercontinuityisalocalpropertyandthatΔ n 4 (though it is a non-local operator) still behaves “pseudo-local”: We impose our conditions (here: being a critical point and mapping into the sphere) only in some domainD⊂R n , and still get interior regularity withinD. Let us comment on the strategy of the proof. As said before, in all even dimensions the key tool for proving regularity is the discovery ofcompensation phenomenabuilt into the respective Euler-Lagrange equation. For example, critical pointsu∈W 1,2 (D,S m−1 ) ofE 2 satisfy the following Euler-Lagrange equation [le09´H] Δu i =u i |ru| 2 ,weakly inD, for alli= 1. . . m. (1.2) For mappingsu∈W 1,2 (R 2 ,S m−1 ) this is a critical equation, as the right-hand side seems to lie only inL 1 : If we had no additional information, it would seem as if the equation admitted a logarithmic singularity (for examples see, e.g., [Riv07], [Fre73]). But, using the constraint|u| ≡1, one can rewrite the right-hand side of (1.2) as m m u i |ru| 2 =  X u i ru j −u j ru i  ∙ ru j =  X ∂ 1 B ij ∂ 2 u j −∂ 2 B ij ∂ 1 u j  j=1j=1 where theB ij are chosen such that∂ 1 B ij =u i ∂ 2 u j −u j ∂ 2 u i , and−∂ 2 B ij =u i ∂ 1 u j −u j ∂ 1 u i , a choice which is possibleduetoPoincar´e’sLemmaandbecause(1.2) implies div  u i ru j −u j ru i  = 0 for everyi, j= 1. . . m. Thus, (1.2) transforms into m Δu i =  X ∂ 1 B ij ∂ 2 u j −∂ 2 B ij ∂ 1 u j  ,(1.3) j=1 a form whose right-hand side exhibits a compensation phenomenon which in a similar way already appeared in the so-called Wente inequality [Wen69], see also [BC84], [Tar85]. In fact, the right-hand side belongs to the Hardy space (cf. [9lu¨0M], [CLMS93]) which is a proper subspace ofL 1 with enhanced potential theoretic properties.Namely,membersoftheHardyspacebehavewellwithCalder´on-Zygmundoperators,andbythis one can conclude continuity ofu. An alternative and for our purpose more viable way to describe this can be found in L. Tartar’s proof [Tar85] of Wente’s inequality: Assume we have fora, b∈L 2 (R 2 ) a solutionw∈H 1 (R 2 ) of Δw=∂ 1 a ∂ 2 b−∂ 2 a ∂ 1 bweakly inR 2 .(1.4) Taking the Fourier-Transform on both sides, this is (formally) equivalent to |ξ| 2 w ∧ (ξ) =c ˆ a ∧ (x)b ∧ (ξ−x) (x 1 (ξ 2 −x 2 )−x 2 (ξ 1 −x 1 ))dx,forξ∈R 2 .(1.5) R2 Now the compensation phenomena responsible for the higher regularity ofwcan be identified with the following inequality: 1 1 |x 1 (ξ 2 −x 2 )−x 2 (ξ 1 −x 1 )| ≤ |ξ||x| 2 |ξ−x| 2 .(1.6)

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1 Introduction

Observe, that|x|as well as|ξ−x|appear to the power 1/ these factors as Fourier multi- Interpreting2, only. pliers, this means that only “half of the gradient”, more precisely Δ 1 4 , ofaandbenters the equation, which implies that the right-hand side is a “product of lower order” operators. In fact, plugging (1.6) into (1.5), one can concludew ∧ ∈L 1 (R 2 dYanngould¨o’serytiloLnonis’auqeces–consrentzspanohesaqeeutnylbtHyj)su proven continuity ofw, because the inverse Fourier transform mapsL 1 intoC 0 . As explained earlier, (1.2) can be rewritten as (1.3) which has the form of (1.4), thus we have continuity for critical points ofE 2 , and by a bootstraping argument (see [Tom69]) one gets analyticity of these points. As in Theorem1.2we prove only interior regularity, it is natural to work with localized Euler-Lagrange equations which look as follows, see Section7: Lemma 1.3(Euler-Lagrange Equations).Letu∈H 2 n (R n )be a critical point ofE n on a domainD⊂R n . ˜ Then, for any cutoff functionη∈C 0 ∞ (D),η≡1on an open neighborhood of a ballD⊂Dandw:=ηu, we have − ˆ w i Δ n 4 w j Δ 4 n ψ ij = ˆ Δ 4 n w j H(w i , ψ ij )− ˆ a ij ψ ij ,for anyψ ij =−ψ ji ∈C 0 ∞ (D) ˜ ,(1.7) RnRnRn wherea ij ∈L 2 (R n ),i, j= 1, . . . , m, depend on the choice ofη. Here, we adopt Einstein’s summation convention. Moreover,H(∙,∙)is defined onH n 2 (R n )×H n 2 (R n )as H(a, b) := Δ n 4 (ab)−aΔ n 4 b−bΔ 4 n a,fora, b∈H 2 n (R n ).(1.8) Furthermore,u∈S m−1 onDimplies the following structure equation w i ∙Δ n 4 w i =− 1 2 H(w i , w i ) 1 +Δ 2 4 n η 2 a.e. inR n .(1.9) Similar in its spirit to [DLR09] we use that (1.7) and (1.9) together control the full growth of Δ n 4 w, though here we use a different argument applying an analogue of Hodge decomposition to show this, see below. Note moreover that as we have localized our Euler-Lagrange equation, we do not need further rewriting of the structure condition (1.9) as was done in [DLR09]. Whereas in (1.4) the compensation phenomenon stems from the structure of the right-hand side, here it comes from the leading order termH(∙,∙) appearing in (1.7) and (1.9be proved by Tartar’s approach [ can ). ThisTar85], using essentially only the following elementary “compensation inequality” similar in its spirit to (1.6) ||x−ξ| p − |ξ| p − |x| p | ≤C p ( | | x x | | p 2 p − |ξ 1 | | 2 p ξ| , +|ξ| p−1 |x|, f i i fp p∈ > 0 1 ( , ,]1 ( . .101) More precisely, we will prove in Section4 Theorem 1.4.ForHas in(1.8)andu, v∈H 2 n (R n )one has ∧ kH(u, v)k L2(Rn) ≤Ck  Δ 4 n u  ∧ k L2(Rn) k  Δ n 4 v  k L2,∞(Rn) . An equivalent compensation phenomenon was observed in the casen= 1 in [DLR09] 1  that interpreting. Note again the terms of (1.10) as Fourier multipliers, it seems as if this equation (and as a consequence Theorem1.4) estimates the operatorH(u, v) by products of lower order operators applied touandv by “products of. Here, lower order operators” we mean products of operators whose differential order is strictly between zero and n 2 and where the two operators together give an operator of order 2 n  fact, this is exactly what happens in special. In cases, e.g. if we take the casen= 4 where Δ n 4 = Δ: H(u, v) = 2ru∙ ruifn= 4. Let us remark, that by an interpolation argument similar results are also a consequence of the “Leibniz rule” for fractional order derivatives obtained by Kato and Ponce, see [KP88] and [Hof98 Another, Corollary 1.2]. case we will need to control is the case whereu=Pis a polynomial of degree less than 2 n . As (at least formally) Δ n 4 P= 0 this is to estimate Δ 4 n (P v)−PΔ 4 n v. 1 In fact, all compensation phenomena appearing in [DLR09] can be proven by our adaption of Tartar’s method using simple compensation inequalities, thus avoiding the use of paraproduct arguments (but at the expense of using the theory of Lorentz spaces).

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