C 1,α -Regularity of energy minimizing maps from a 2-dimentional domain into a Finsler space
aa r X i v : . [ m a t h . A P ] A ug C ,α -Regularity of energy minimizing maps froma 2-dimentional domain into a Finsler space Atsushi Tachikawa ∗ Department of Mathematics, Faculty of Science and Technology,Tokyo University of Science, Noda, Chiba, 278-8510, Japan,e-mail:tachikawa [email protected]
MSC : Primary 49N60, 58E20; Secondary 35B65, 53C60.
Key words and phrases : harmonic map, Finsler manifold, regularity
Abstract
We show C ,α -regularity for energy minimizing maps from a 2-dimensionalRiemannian manifold into a Finsler space ( R n , F ) with a Finsler structure F ( u, X ). Let N be an n -dimensional C ∞ -manifold and T N its tangent bundle. We writeeach point in
T N as ( u, X ) with u ∈ N and X ∈ T u N . We put T N \ { ( u, X ) ∈ T N ; X = 0 } .T N \ slit tangent bundle of N . A Finsler structure of N is afunction F : T N → [0 , ∞ ) with the following properties:(F-1) Regularity: F ∈ C ∞ ( T N \ Positive homogeneity: F ( u, λX ) = λF ( u, X ) for all λ ≥ . (F-3) Convexity:
The Hessian matrix of F with respect to X ( f ij ( u, X )) = (cid:18) ∂ F ( u, X ) ∂X i ∂X j (cid:19) is positive definite at every point ( u, X ) ∈ T N \ ∗ This research was partially supported by the Ministry of Education, Science, Sports andCulture, Grant-in-Aid for Scientific Research (C), 22540207
1e call the pair (
N, F ) a
Finsler manifold, and ( f ij ) the fundamental tensor of ( N, F ). Since F is positively homogeneous of degree 1, we can see that thecoefficients of the fundamental tensor are positively homogeneous of degree 0; f ij ( u, λX ) = f ij ( u, X ) , λ > . (1.1)Moreover, since F is homogeneous of degree 2, using Euler’s theorem for ho-mogeneous functions, we have F ( u, X ) = f ij ( u, X ) X i X j . (1.2)For maps between Finsler manifolds P. Centore [1] defined the energy density by using of the integral mean on the indicatrix of each point on the sourcemanifold. According to his definition we define the energy density e C ( u ) of amap u from a Riemannian into a Finsler manifold as follows. Let ( M, g ) be asmooth Riemannian m -manifold and ( N, F ) a Finsler n -manifold. Let I x M bethe indicatrix of g at x ∈ M , namely, I x M := { ξ ∈ T x M ; k ξ k g ≤ } . For a C -map u : M → N and a domain Ω ⊂ M , we define the energy density e C ( u )( x ) of u at x ∈ M and the energy on Ω E C ( u ; Ω) by e C ( u )( x ) := Z − I x M ( u ∗ F ) ( ξ ) dξ = 1 R I x M dξ Z I x M ( u ∗ F ) ( ξ ) dξ (1.3) E C ( u ; Ω) := Z Ω e C ( u )( x ) dµ. (1.4)Here and in the sequel, R − denotes the integral mean, u ∗ F the pull-back of F by u , and dµ the measure deduced from g . We call (weak) solutions of theEuler-Lagrange equation of the energy (wakly) harmonic maps. Concerning harmonic maps from a Finsler manifold into a Riemannian man-ifold, see, for example, H. von der Mosel and S. Winklmann [10].Let us take an orthonormal frame { e α } for the tangent bundle T M of M ,given in local coordinates by e α = η κα ( x ) ∂∂x κ , ≤ α ≤ m. Using { e α } , we identify each I x M at x ∈ M with the unit Euclidean m -ball B m . Then, by virtue of the identity g κν ( x ) = η κα ( x ) δ αβ η νβ ( x ) , we can write E C as E C ( u ; Ω)= Z Ω (cid:18) | B m | Z B m f ij ( u ( x ) , du x ( ξ )) ξ κ ξ ν dξ (cid:19) η ακ η βν D α u i D β u j √ gdx, (1.5)2here D α u i = ∂u i /∂x α and g = det( g αβ ). (cf. [8].) Although the termsin parentheses are not defined at points x where du x = 0, we can definethem to be arbitrary numbers without changing the values of the integrands( ..... ) η ακ η βν D α u i D β u j , because the integrands are equal to 0, being independenton the values of f ij when du x = 0. So, here and in the sequel, we regard f ij ( u, X ) as being defined also for X = 0.As in [9], let us put E αβij ( x, u, p )= (cid:18) | B m | Z B m f ij ( u ( x ) , pξ )) ξ κ ξ ν dξ (cid:19) η ακ ( x ) η βν ( x ) p g ( x ) . (1.6)Then, we can write E C ( u ; Ω) = Z Ω E αβij ( x, u, Du ) D α u i D β u j dx. (1.7)In case that m = dim( M ) = 2, the H¨older continuity of a energy minimiz-ing map is shown in [9]. For a energy minimizing map between Riemannianmanifolds, or more generally for a minimizer u of a quadratic functional Z A αβij ( x, u ) D α u i D β u j dx with smooth coefficients A αβij ( x, u ), once the H¨older continuity of u has beenshown, we see that the coefficients A αβij ( x, u ( x )) are H¨older continuous, andtherefore we can show the C ,α -regularity of u by virtue of Schauder-type es-timate. Then, inductively we get higher regularity. In contrast, if the targetmanifold is a Finsler manifold, the H¨oder continuity of u does not imply thecontinuity of the coefficients E αβij ( x, u ( x ) , Du ( x )). So, if we want to obtain C ,α -regularity of a minimizer, we have to show it directly.In differential geometric setting, usually one assumes C ∞ -regualrity on themetric as (F-1). However, to get C ,α - or C ,α -regularity for energy minimizingmaps, it is enough to emply the following conditions instead of (F-1)(F-1a) There exists a concave increasing function ω : [0 , ∞ ) → [0 , ∞ ) withlim t → +0 ω ( t ) = 0 such that | F ( u, X ) − F ( v, X ) | ≤ ω ( | u − v | ) | X | (1.8)holds for any u, v ∈ R n and X ∈ R n .(F-1b) F ( u, X ) is twice differentiable in X for every ( u, X ) ∈ T R n \ λ < Λ for which λ | ξ | ≤ f ij ( u, X ) ξ i ξ j = 12 ∂ F ( u, X ) ∂X i ∂X j ξ i ξ j ≤ Λ | ξ | (1.9)holds for any u, v ∈ R n and ( X, ξ ) ∈ ( R n \ × R n .The main result of this paper is as follows. Theorem 1.1.
Let ( M, g ) a 2-dimentional smooth Riemannian manifold, Ω ⊂ M a bounded domain with smooth boundary ∂ Ω and ( R n , F ) a Finsler spacewith the Finsler structure F satisfying (F-1a), (F-1b), (F-2) and (F-3a). Let u ∈ H , (Ω , R n ) be an energy minimizing map in the class H , φ (Ω , R n ) := { v ∈ H , (Ω , R n ) ; v − φ ∈ H , (Ω , R n ) } . Then u ∈ C ,α (Ω) ∩ C ,β (Ω) for some α ∈ (0 , and any β ∈ (0 , . In order to prove Theorem 1.1, we prepare the following higher integrability re-sults of minimizers which can be deduced easily from [7, Lemma 1] as mentionedin [9].
Lemma 2.1 ([9, Remark 5.3]) . Let ( M, g ) be a smooth Riemannian m-manifoldand Ω ⊂ M a bounded domain with smooth boundary ∂ Ω and ( R n , F ) a Finslerspace with the Finsler structure F satisfying (1.9).Suppose that φ ∈ H ,p (Ω , R n ) for some p > . Let u ∈ H , (Ω , R n ) be an energy minimizing map in the class H , φ (Ω , R n ) . Then, there exists a positive number q > such that for every q ∈ (2 , q ) , the estimate Z Ω | Du | q dx ≤ C Z Ω | Dφ | q dx (2.1) holds. Now, using several estimates which are obtained in [9], we can show the mainresult of this paper. In [9] the author supposed that A ( x, u, p ) = E αβij ( x, u, p ) p iα p jβ is in the class C , ( X ) ∩ C ( X ′ ), where X = Ω × R n × R mn and X ′ = Ω × R n × ( R mn \ { } ) . However, it is clearly superfluous to obtain C ,α -regularity of the minimizer. Infact, it is easy to see that every proof in [9] can be carried assuming on theregularity of A ( x, u, p ) only that 4i) A ( x, u, p ) is in the class C , ( X ) and twice differentiable in p at every( x, u, p ) ∈ X ′ .(ii) There exists a concave increasing function ω : [0 , ∞ ) → [0 , ∞ ) withlim t → ω ( t ) = 0 such that | A ( x, u, p ) − A ( y, v, p ) | ≤ ω ( | x − y | + | u − v | ) | p | , holds for all x, y ∈ Ω , u, v ∈ R n and p ∈ R mn \ u : Ω ⊂ M → R n minimizes the energy functional on Ω, then u minimizesit on every sub-domain of Ω. On the other hand, the regularity is a localproperty. So, it is suffices to study the regularity problem on a domain Ω ⊂ R m . Proof of Therem 1.1.
First, we show that u ∈ C ,β (Ω) for any β ∈ (0 , x ∈ Ω and
R > Q ( x, R ) := { y ∈ R m ; | y α − x α | < R, α = 1 , . . . , m } . (2.2)For x ∈ ∂ Ω we always choose local coordinates so that for sufficiently small R > Q ( x , R ) ∩ Ω ⊂ R m + = { x ∈ R m ; x m > } ,Q ( x , R ) ∩ ∂ Ω ⊂ { x ∈ R m ; x m = 0 } , and put for 0 < R < R Q + ( x , R ) := Q ( x , R ) ∩ { x ∈ R m ; x m > } . (2.3)Sometimes we write alsoΩ( x, R ) := { y ∈ Ω ; | y α − x α | < R, α = 1 , . . . , m } , (2.4)for general x ∈ Ω and
R > x is an interior point and Q ( x , r ) ⊂⊂ Ω, we havefor any δ ∈ (0 , Z Q ( x ,ρ ) | Du | dx ≤ C ((cid:16) ρr (cid:17) − δ + ˜ ω (cid:0) r + Z Q ( x , r ) | Du | dx (cid:1)) Z Q ( x , r ) | Du | dx, (2.5)where ˜ ω = ω ( q − /q for some q >
2. For a boundary point x , assuming that φ ∈ H ,s ( s > m = 2), from [9, (5.10)], we have for any δ ∈ (0 , Z Q + ( x ,ρ ) | Du | dx ≤ C ((cid:16) ρr (cid:17) − δ + ˜ ω (cid:0) r + Z Q + ( x , r ) | Du | dx (cid:1)) Z Q + ( x , r ) | Du | dx + C ( φ ) r γ , (2.6)5here γ = 2(1 − /s ) >
0. Since we are assuming that φ ∈ H , ∞ , we can take γ = 2 − ε for any ε > δ so that 2 − ε < − δ . Proceeding as in [4, pp.317–318], wecan deduce from (2.5) and (2.6) that Z Q ( x ,ρ ) | Du | dx ≤ M (cid:16) ρr (cid:17) − ε Z Q ( x ,r ) | Du | dx for x ∈ Ω , (2.7) Z Q + ( x ,ρ ) | Du | dx ≤ M (cid:16) ρr (cid:17) − ε Z Q + ( x ,r ) | Du | dx + M ρ − ε for x ∈ ∂ Ω , (2.8)for sufficiently small r > ρ ∈ (0 , r ), where M and M are constantsdepending on g, F, Ω and φ . Here, we used also the fact thatlim r → (cid:8) r + Z Ω( x , r ) | Du | dx (cid:9) = 0 (2.9)holds for any x ∈ Ω.Now, proceeding as in [4, pp.318–319], we can have that for any ε ∈ (0 , M such that Z Ω( x ,ρ ) | Du | dx ≤ ρ − ε M, (2.10)for any x ∈ Ω. So, putting 2 β = 2 − ε , by Morrey’s Dirichlet growth theorem,we see that u ∈ C ,β (Ω).Let us show C ,α -regularity of u , proceeding as in [2]. For a cube Q = Q ( x , R ) ⊂⊂ Ω, we consider the following frozen functional A defined by A ( v ) = Z Q E αβij ( x , u R , Dv ) D α v i D β v j dx, (2.11)where u R = Z − Q udx. Let v be a minimizer of A in the class { v ∈ H , ( Q ) ; v − u ∈ H , ( Q ) } . Since u ∈ H ,q for every q ∈ (2 , q ) for some q > v , we see that there exists a positive number q > q ∈ (2 , q ) there holds Z Q | Dv | q dx ≤ Z Q | Du | q dx. (2.12)6oreover, as in [9], by using of difference quotient method, we can see that v ∈ H , and that Dv satisfies a system of uniformly elliptic equations weakly.So, for any Q ( x, r ) ⊂ Q , Dv satisfies the Caccioppoli inequality, Z Q ( x,r/ | D v | dy ≤ Cr Z Q ( x,r ) | D − ( Dv ) r | dy, (2.13)and D v satisfies reverse H¨older inequalities with increasing supports due toGiaquinta-Modica (cf. [3, p.299, Theorem 3], Z − Q ( x,r/ | D v | q dy ! /q ≤ C Z − Q ( x,r ) | D v | dx ! / , (2.14)for every q ∈ (2 , q ) for some q > v ∈ C ,δ for δ = 1 − (2 /q ). Moreover,we have for ρ ∈ (0 , R/ ( ρ − − δ Z Q ( x ,ρ ) | Dv − ( Dv ) ρ | dx ) / ≤ sup Q ( x ,R/ | Dv ( x ) − Dv ( y ) || x − y | δ ≤ C k D v k L q ( Q ( x ,R/ . (2.15)For the last inequality, we used Morrey-type inequality.Combining (2.15), (2.14) and (2.13), we obtain ( ρ − − δ Z Q ( x ,ρ ) | Dv − ( Dv ) ρ | dx ) / ≤ CR q − k D v k L ( Q ( x ,R/ ≤ R − − δ Z Q ( x ,R ) | Dv − ( Dv ) R | dx ! / . (2.16)7utting w = u − v , we obtain Z Q ( x ,ρ ) | Du − ( Du ) ρ | dx ≤ Z Q ( x ,ρ ) | Du − ( Dv ) ρ | dx ≤ Z Q ( x ,ρ ) | Dv − ( Dv ) ρ | dx + Z Q ( x ,ρ ) | Dw | dx ≤ C (cid:16) ρR (cid:17) δ Z Q ( x ,R ) | Dv − ( Dv ) R | dx + C Z Q ( x ,ρ ) | Dw | dx ≤ C (cid:16) ρR (cid:17) δ Z Q ( x ,R ) | Dv − ( Du ) R | dx + C Z Q ( x ,ρ ) | Dw | dx ≤ C (cid:16) ρR (cid:17) δ Z Q ( x ,R ) | Du − ( Du ) R | dx + C Z Q ( x ,R ) | Dw | dx. (2.17)Let us estimate R | Dw | dx . Proceeding as in [9, pp.1967-1968], it is easey tosee that Z Q ( x ,R ) | Dw | dx ≤ C h Z Q ( x ,R ) ω ( | x − x | + | u − u R | ) | Du | dx + Z Q ( x ,R ) ω ( | x − x | + | v − u R | ) | Dv | dx i (2.18)=: I + II.
Using Jensen’s inequality, H¨older’s inequality and reverse H¨older ineqalitty, wecan estimate I as follows. I ≤ C (cid:16) Z Q ( x ,R ) ω q/ ( q − dx (cid:17) ( q − /q (cid:16) Z Q ( x ,R ) | Du | q (cid:17) /q ≤ C (cid:16)Z − Q ( x ,R ) ωdx ) ( q − /q R m ( q − /q (cid:16) Z Q ( x ,R ) | Du | q dx (cid:17) /q ≤ C (cid:16) ω (cid:0)Z − Q ( x ,R ) ( | x − x | + | u − u R | ) dx (cid:1)(cid:17) ( q − /q R m ( q − /q R m/q · (cid:16)Z − Q ( x ,R ) | Du | q dx (cid:17) q ≤ C (cid:16) ω (cid:0)Z − Q ( x ,R ) ( R + | u − u R | ) dx (cid:1)(cid:17) ( q − /q Z Q ( x , R ) | Du | dx. (2.19)Here we used the boundedness of ω . By virtue of (2.12), we can estimate II II ≤ C (cid:16) Z Q ( x ,R ) ω q/ ( q − dx (cid:17) ( q − /q (cid:16) Z Q ( x ,R ) | Dv | q (cid:17) /q ≤ C (cid:16) ω (cid:0)Z − Q ( x ,R ) ( R + | v − u R | ) dx (cid:1)(cid:17) ( q − /q R m (cid:16)Z − Q ( x ,R ) | Du | q dx (cid:17) /q ≤ C (cid:16) ω (cid:0) C Z − Q ( x ,R ) ( R + | u − u R | + | v − u | ) dx (cid:1)(cid:17) q − q · Z Q ( x , R ) | Du | dx. (2.20)Let us estimate the ingredients in ω . Using Sobolev’s inequality (cf. [4,p.103], we can see that for 2 ∗ = 2 m/ ( m + 2) Z − Q ( x ,R ) | u − u R | dx ≤ CR − m (cid:16) Z Q ( x ,R ) | Du | ∗ dx (cid:17) / ∗ ≤ CR − m (cid:16) Z Q ( x ,R ) / (2 − ∗ ) dx (cid:17) − ∗ (cid:16) Z Q ( x ,R ) | Du | dx (cid:17) ≤ CR − m +2 m − ∗ m (cid:16) Z Q ( x ,R ) | Du | dx (cid:17) Since we are assuming that m = 2, we have 2 ∗ = 1. Thus, the above estimatetogether with (2.10) gives for every ε ∈ (0 ,
1) the folowing estimate Z − Q ( x ,R ) | u − u R | dx ≤ C Z Q ( x ,R ) | Du | dx ≤ CR − ε . (2.21)We can see also that Z − Q ( x ,R ) | u − v | dx ≤ C Z Q ( x ,R ) (cid:0) | Du | + | Dv | (cid:1) ≤ C Z Q ( x ,R ) | Du | dx ≤ CR − ε . (2.22)Since we can assume that R ≤
1, we see that the ingredient in ω can be estimatesby CR − ε for every ε ∈ (0 , ω ( t ) ≤ Ct σ for some σ ∈ (0 , ω ( ... ) ≤ CR σ (2 − ε ) , (2.23)So, we can estimate ω ( q − /q R | Du | dx in (2.19) and (2.20) as ω ( q − /q Z Q ( x , R ) | Du | dx ≤ CR (2 − ε ) { σ ( q − /q } , (2.24)9here we used (2.10) again. Now, take ε ∈ (0 ,
1) sufficiently small so that(2 − ε ) (cid:16) σ · q − q (cid:17) > , and put γ := (2 − ε ) (cid:16) σ · q − q (cid:17) − > . (2.25)Combining (2.19), (2.20), (2.24) and (2.25), we get Z Q ( x ,R ) | Dw | dx ≤ CR γ . (2.26)Now, substituting the above inequality into (2.17), we obtain Z Q ( x ,ρ ) | Du − ( Du ) ρ | dx ≤ C (cid:16) ρR (cid:17) δ Z Q ( x ,R ) | Du − ( Du ) R | dx + CR γ .. (2.27)Using well known lemma (cf. [2, Lemma 2.2], we conclude that Z Q ( x ,ρ ) | Du − ( Du ) ρ | dx ≤ Cρ α (2.28)with α = min { δ, γ/ } for every Q ( x , ρ ) ⊂ Ω, and hence Du ∈ C α (Ω). Remark 2.2.
The perfect dominance functions treated by S.Hildebrandt and H.von der Mosel in [5, 6] have the structure similar to that of the energy density e c .So, some of their results are valid for weakly harmonic maps in 2-dimensionalcase. More precisely, for the case that F ( u, X ) is continuously differentiable in u , once the H¨oder continuity of a weakly harmonic map have shown, we canget its C ,α -regularity proceeding exactly as in the fourth section of [5]. On theother hand, in this paper, we prove C ,α -regularity using the minimality withoutassuming the differentiability of F ( u, X ) with respect to u .We should mention also that in [5] the minimality is not necessary to get C ,α -regularity for H¨older continuous weak solutions of the Euler-Lagrange equa-tion of a perfect dominance function. However, in both of [5] and this paper,the minimality is necessary to get the H¨older continuity. References [1] P. Centore. Finsler Laplacians and minimal-energy maps.
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