Partial Regularity Results Up to the Boundary for Harmonic Maps Into a Finsler Manifold

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Partial regularity results up to the boundary for harmonic maps into a Finsler manifold

Atsushi Tachikawa

Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510, Japan Received 8 July 2008; accepted 11 May 2009

Available online 27 May 2009

Abstract

We study the energy functional for maps from a Riemannian m-manifoldM into a Finsler spaceN=(Rⁿ, F ). Under the restriction 2m4, we prove the full Hölder regularity of weakly harmonic maps (i.e., weak solutions of its Euler–Lagrange equation) fromMtoNin the case that the Finsler structureF (u, X)depends only on vectorsX, and a partial Hölder regularity of energy minimizing maps in general cases.

Résumé

Nous étudions la fonctionnelle d’énergie pour les applications d’une variété riemannienneM dans un espace de FinslerN= (Rⁿ, F ). Sous la restriction 2m4, nous prouvons la régularité de Hölder complète des applications faiblement harmoniques (i.e. solutions faibles de son équation d’Euler–Lagrange) deMàNdans le cas où la structure de FinslerF (u, X)dépend seulement des vecteursX, et nous prouvons une régularité de Hölder partielle des minimiseurs de l’énergie dans le cas général.

MSC:primary 49N60, 58E20; secondary 35B65, 53C60 Keywords:Harmonic map; Finsler manifold; Partial regularity

1. Introduction

In this paper we study regularity problems of energy minimizing and weakly harmonic maps from a Riemannian manifold into a Finsler manifold.

Since the pioneering work of J. Eells and J.H. Sampson [6] in 1964,harmonic mapsbetween Riemannian manifolds have attracted great interest of nonlinear analysts as well as geometers.Harmonic mapsbetween Riemannian manifolds are defined as follows. Let(M, g)and(N, h)be Riemannianm- andn-manifolds, respectively. Theenergy densityof a mapu:(M, g)→(N, h)is a functione(u):M→Rdefined by

e(u)(x):=1

2du(x)², x∈M,

E-mail address:tachikawa_atsushi@ma.noda.tus.ac.jp.

doi:10.1016/j.anihpc.2009.05.001

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where |du| denotes the Hilbert–Schmidt norm of du(x) ∈T_x^∗M ⊗T_u(x)N, namely, for an orthonormal basis (e₁, . . . , e_m)ofT_xM,e(u)(x)can be written as

e(u)(x)= m α=1

du(x) (e_α)²

T_u(x)N. (1.1)

For a bounded domainΩinM,the energyofuonΩ is defined by E(u, Ω):=

Ω

e(u) dμ,

wheredμstands for the volume element onM.Harmonic mapsare defined to be solutions of the Euler–Lagrange equation of the energy functional.

Finsler geometry is a natural generalization of Riemannian geometry, therefore it is very reasonable to expect to extend the notion of harmonic maps to Finsler geometry. Indeed, P. Centore [5] defined the energy and the notion of harmonicity for maps between Finsler manifolds. On the other hand, Y.-B. Shen and Y. Zhang [24] gave another definition of the energy by an integration on the sphere bundle over the source manifold. Both of these definitions of energy are extensions of the one for maps between Riemannian manifolds. We mention that their variational features are very similar at least for the case that the source manifold is Riemannian, and the results of this paper hold for both energies.

Concerning harmonic maps from Finsler manifolds into a Riemannian manifold, see, for example, X. Mo [18], X. Mo and Y. Yang [19] and recent work H. von der Mosel and S. Winklmann [26]. In [26] they obtain a priori estimates for harmonic maps with a Finsler source manifold and with small image in a Riemannian manifold.

In this paper, we discuss only the case where source manifolds are Riemannian. In the sequel, let us write shortly

“Finsler case” when the target manifold is a Finsler manifold, and “Riemannian case” when the target manifold is Riemannian. Analytic features of the energy for the Finsler case are quite different from those for the Riemannian case. Indeed let(M, g)and(P , h)be Riemannianm- andn-manifolds, and let(N, F )be a Finslern-manifold with a Finsler structureF. Using local coordinates, the energy functional for a mapu:(M, g)→(P , h)can be written as

g^αβ(x)hij(u)∂uⁱ

∂x^α

∂u^j

∂x^βdμ.

On the other hand, for a mapu:(M, g)→(N, F ), the energy should be expressed in the following form:

E_ij^αβ(x, u, Du)∂uⁱ

∂x^α

∂u^j

∂x^β dμ.

(See (2.6) below.) For the Finsler case the coefficients of the integrand depend also on the derivatives ofu, whereas they depend only onxandufor the Riemannian case. Consequently, in the Finsler case, the nonlinearity of the Euler–

Lagrange equation is much more subtle than that in the Riemannian case, and therefore it would be harder to obtain regularity results forharmonic mapsandenergy minimizing maps. One could also mention that with a Riemannian target the underlying pde decomposes into an elliptic system of diagonal form. This is another reason why life is so much harder in the Finsler setting.

For the Riemannian case, many results have been known on regularity of harmonic maps. Some of them are indeed best possible. See, for example, [15,8,11,22,23] and the references therein.

When investigating the regularity of harmonic maps into a Finsler manifold(N, F ), the difficulty arises not only from the nonlinearity but also from the singularity of the Finsler structureF (u, X). This singular character of Finsler metrics is inevitable, since it is known that if the second derivatives ofF (u, X)with respect toXare continuous at X=0, then(N, F )should be nothing but a Riemannian manifold with the metric tensor

∂²F²

∂Xⁱ∂X^j(u,0).

In view of this, we cannot assume thatF (u,·)∈C², and therefore, many known regularity results for minimizers or critical maps of general variational integrals cannot be applied directly to harmonic maps into a Finsler manifold.

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In [25], the author proves interior partial Hölder regularity of energy minimizing maps from the Euclidean spaceR^m into a Finsler space (Rⁿ, F ), under the restriction that m4, using Sobolev’s imbedding theorem and thedirect approach. Here, we call an approach getting (partial) regularity by a perturbation argument the direct approach.

(See [7, Chapter VI].)

In the present paper, under the same restriction on the dimensionmof the source manifold, we modify Campanato’s method to prove the Hölder regularity ofweakly harmonic mapsinto a Finsler space(Rⁿ, F )where the given Finsler structureF (u, X)depends only on the vector variableX. Using direct approach, we also obtain partial regularity of energy minimizing maps into a general Finsler space.

Recently, the importance of harmonic maps in Finsler geometry has been recognized and several authors inves- tigated them. For example, in [20,21], S. Nishikawa introduced harmonic maps in complex Finsler geometry and pointed out their importance.

It will be worth to consider Finsler geometric variational problems from the viewpoint of applied mathematics as well. There are many variational and evolution problems which cannot be regarded as problems from Riemannian geometry, and some of them can be considered in the context of Finsler geometry. For example, many physical and biological applications of Finsler geometry are introduced by P.L. Antonelli, R.S. Ingarten and M. Matsumoto in [1].

We mention also that G. Bellettini and M. Paolini [2] studied a problem of anisotropic motion in the context of Finsler geometry.

2. Definitions and main results

LetNbe ann-dimensionalC^∞manifold. Denote byT Nthe tangent bundle ofN, and byT^∗Nits dual. We write each point inT N as(u, X)withu∈N andX∈T_uN. The natural projectionπ:T N→N is given byπ(u, X)=u.

We put T N\0:=

(u, X)∈T N; X=0 .

T N\0 is called theslit tangent bundleofN. Theprojective sphere bundleSNofNis defined by identifying positive multiples of a vector in each fiber ofT N\0, that is,

SN=(T N\0)/∼,

where(u, X)∼(v, Y ) if and only if u=v andX=λY for someλ >0. Given (u, X)∈T N \0, we denote its equivalence class by(u,[X]). The natural projectionp:SN→Nis given byp(u,[X])=u.

In Finsler geometry, it is often convenient to handle the tensor calculus in bundles over T N,T N \0 or SN. Especially, the pull back bundlesp^∗T N→SM,π^∗T N→T N\0, and their dual bundlesp^∗T^∗N→SN,π^∗T^∗N→ T N\0 are used quite frequently.

A Finsler structureofNis a functionF:T N→ [0,∞)satisfying the following properties:

(i) Regularity:F ∈C^∞(T N\0).

(ii) Homogeneity:F (u, λX)=λF (u, X)for allλ0.

(iii) Convexity: The Hessian matrix ofF²with respect toX h_ij(u, X)

= 1

2

∂²F²(u, X)

∂Xⁱ∂X^j

is positive definite at every point(u, X)∈T N\0.

We call the pair(N, F )a Finsler manifold, andhij thefundamental tensorof (N, F ). By virtue of positive homo- geneity ofF, we have

h_ij(x, λX)=h_ij(u, X), λ >0, (2.1)

and obtain by Euler’s theorem that

F²(u, X)=h_ij(u, X)XⁱX^j. (2.2)

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Identities (2.1) and (2.2) imply that the fundamental tensor h_ij(u, X)is well defined for (u,[X])∈SN and that it defines a bundle metric on the pull-back bundlep^∗T N overSN, as well as onπ^∗T NoverT N\0.

In [5], P. Centore defined theenergyfunctional for maps between Finsler manifolds. According to his definition we define the energy densitye_C(u)of a mapufrom a Riemannian into a Finsler manifold as follows. Let(M, g)be a smooth Riemannianm-manifold and(N, F )a Finslern-manifold. LetI_xMbe the indicatrix ofgatx∈M, namely,

I_xM:=

ξ ∈T_xM; ξ _g1 .

For aC¹-mapu:M→Nwe define theenergy densityeC(u)(x)ofuatx∈Mand theenergy functionalEC(u)by eC(u)(x):=

I_xM(u^∗F )²(ξ ) dξ

IxM dξ , (2.3)

E_C(u):=

M

e_C(u)(x) dμ, (2.4)

whereu^∗F denotes the pull-back ofF byu, anddμthe measure deduced fromg.

Moreover, Y.-B. Shen and Y. Zhang [24] gave another definition of the energy of a map between Finsler manifolds by integration on the sphere bundleSM. They consider the differentialduof a mapu:M→N as a map fromSMto SN and define the energy density ofuat(x,[ξ])∈SMby

e_SZ(u) x,[ξ]

:=1

2g^αβ(x)h_ij u(x),

du_x(ξ )∂uⁱ

∂x^α(x)∂u^j

∂x^β(x).

The energyE_SZ(u)of a mapu:(M, g)→(N, h)is defined to be E_SZ(u):= 1

vol(S^m⁻¹)

SM

e_SZ(u) dμ_SM,

whereμ_SM denotes the measure onSM deduced from the Sasaki metric onT M\0. See, for details, [21,24].

As in the Riemannian case, a (weak) solutions of the Euler–Lagrange equation of the energy functionalsE_C or E_SZis called a (weakly)harmonic map.

As pointed out by S. Nishikawa [21], using an orthonormal frame for the tangent bundleT M, we can writeE_C andE_SZ in similar forms. Let us take an orthonormal frame{e_α}for the tangent bundleT M ofM, given in local coordinates by

eα=η_α^κ(x) ∂

∂x^κ, 1αm.

Using{e_α}, we identify each fiberS_xM ofSM and the indicatrixI_xMatx∈M with the unit Euclidean(m−1)- sphereS^m⁻¹and the unit Euclideanm-ballB^m, respectively. Then, by virtue of the identity

g^κν(x)=η^κ_α(x)δ^αβη_β^ν(x), we can writeEC andESZas

EC(u)=

M

1

|B^m|

B^m

hij

u(x), dux(ξ ) ξ^κξ^νdξ

η^α_κη_ν^βDαuⁱDβu^jdμ,

ESZ(u)=

M

1 2|S^m⁻¹|

S^m⁻¹

hij

u(x),

dux(ξ ) δ^κνdξ

η^α_κη_ν^βDαuⁱDβu^jdμ,

whereD_αuⁱ=∂uⁱ/∂x^α. Although the terms in parentheses are not defined at pointsxwheredu_x=0, we can define them to be arbitrary numbers without changing the values of the integrands(· · ·)η^α_κη_ν^βD_αuⁱD_βu^j,because the integrands are equal to 0, being independent on the values of h_ij whendu_x=0. So, here and in the sequel, we regard h_ij(u, X)andh_ij(u,[X])as being defined also forX=0.

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AlthoughE_SZis very similar toE_Cin appearance, there is a crucial difference between them. In general, when we consider regularity problems of minimizers for functionals defined as

G(x, u, Du) dx,

Ω⊂R^m, G(x, u, p):Ω×Rⁿ×R^mn→R ,

the ellipticity of the Hessian ∂²G(x, u, p)/∂pⁱ_α∂p_β^j plays a very important role. For Centore’s energyE_C we can deduce the ellipticity from the convexity condition (iii) on the Finsler structure. On the other hand, for E_SZ the condition (iii) does not imply the corresponding ellipticity. (See Lemma 3.1 and Remark 3.2.) For this technical reason we treat in this paper only Centore’s energy.

For(x, u, p)∈R^m×Rⁿ×R^mn, let us define the quantitiesE^αβ_ij (x, u, p)by 1

|B^m|

B^m

h_ij(u, pξ )ξ^κξ^νdξ

η_κ^α(x)η^β_ν(x)

det g_αβ(x)

. (2.5)

Since we are assuming smoothness of the manifolds,E_ij^αβ(x, u, p)are smooth inR^m×Rⁿ×(R^mn\0).

In order to consider boundary value problems on a bounded domainΩ ⊂M, let us define the energies on Ω, EC(u;Ω)by

Ω

E_ij^αβ

x, u(x), Du(x)∂uⁱ

∂x^α

∂u^j

∂x^β dx, (2.6)

whereDu=(D_αuⁱ)=(∂uⁱ/∂x^α). In the sequel, by a “weakly harmonic map” or an “energy minimizing map” we mean the corresponding notions with respect toE_Conly.

In this paper we treat only the cases in which the source manifold is a Riemannianm-manifold with 2m4 and the target manifold(N, F )=(Rⁿ, F ). Moreover, we assume that there exist positive constantsλ < Λand a concave increasing functionωwith limt→+0ω(t )=0 such that

λ|ξ|²h_ij(u, X)ξⁱξ^j=1 2

∂²F²(u, X)

∂Xⁱ∂X^j ξⁱξ^jΛ|ξ|², (2.7)

F²(u, X)−F²(v, X)ω

|u−v|²

|X|² (2.8)

hold for anyu, v∈Rⁿand(X, ξ )∈(Rⁿ\0)×Rⁿ.

Under these assumptions, the author proved interior partial C^0,α-regularity in [25]. Namely, he proved that a minimizer of Centore’s energy E_C(u;Ω) is Hölder continuous on an open subset Ω₀⊂Ω which satisfies H^m⁻²⁻^δ(Ω\Ω₀)=0 for someδ >0. Here and in the sequelH^q denote theq-dimensional Hausdorff measure.

In this paper we prove everywhere Hölder regularity of weakly harmonic maps whenF (u, X)depends only on the vectorX, and partial Hölder regularity of energy minimizing maps up to boundary for general cases. More precisely, we show the following results.

Theorem 2.1.Let (M, g) be a Riemannian m-manifold of classC³ and Ω ⊂M a bounded domain with smooth boundary∂Ω and(Rⁿ, F )a Finsler space with the Finsler structureF (u, X)which is independent onu∈Rⁿ and satisfies (2.7). Suppose that2m4 and f ∈H^1,s(Ω,Rⁿ)for somes > m. Then every weakly harmonic map v∈H^1,2(Ω,Rⁿ)with boundary valuef is in the classC^0,α up to the boundary for someα∈(0,1).

Theorem 2.2.Let (M, g) be a Riemannian m-manifold of classC³ and Ω ⊂M a bounded domain with smooth boundary ∂Ω and (Rⁿ, F ) a Finsler space with the Finsler structure F satisfying (2.7) and (2.8). Suppose that 2m4and thatf∈H^1,s(Ω,Rⁿ)for somes > m. Letu∈H^1,2(Ω,Rⁿ)be an energy minimizing map in the class

H_f^1,2 Ω,Rⁿ

:=

v∈H^1,2 Ω,Rⁿ

; v−f∈H₀^1,2

Ω,Rⁿ .

Then, there exists a relatively open subset Ω0⊂Ω such that C^0,α(Ω0,Rⁿ) for some α ∈(0,1). Moreover, H^m⁻²⁻^δ(Ω\Ω0)=0for someδ >0.

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3. Preliminary results

Ifuis a weakly harmonic map or a minimizer of the energy functional onΩ⊂M, thenuhas the same property on every coordinate neighborhood. On the other hand the regularity is a local property. So it suffices to study the problem on a domainΩ⊂R^m.

LetE_ij^αβbe defined by (2.5). Forx∈Ω,u∈Rⁿandp∈R^mnput

A(x, u, p)=E_ij^αβ(x, u, p)pⁱ_αp^j_β. (3.1)

Then, we can regardA(x, u, p)as a function defined onX=Ω×Rⁿ×R^mn. PutX=Ω×Rⁿ×(R^mn\ {0}).

Lemma 3.1.Let(M, g)is a Riemannianm-manifold of classC³,(N, F )be a Finsler space with a Finsler structure which satisfies(i)–(iii)and letA(x, u, p)be as above. ThenA(x, u, p)satisfies

A(x, u, p)∈C^1,1(X)∩C³(X). (3.2)

Moreover, there exist positive constantsλ₀< Λ₀such that

λ0|p|²A(x, u, p)Λ0|p|² for all(x, u, p)∈X, (3.3) λ0|ξ|²A

pⁱ_αp^j_β(x, u, p)ξ_αⁱξ_β^jΛ0|ξ|² for all(x, u, p, ξ )∈X×R^mn. (3.4) Here and in the sequel,A_x^γ, A_ui, A_pi

α, A

p_αⁱp^j_β, etc., denote partial derivatives.

Proof. Since (M, g)is assumed to be a Riemannian manifold of class C³, by the assumption on F, we see that A(x, u, p) is in the classC³(X). Moreover, by virtue of the 2-homogeneity of F or, equivalently, 0-homogeneity ofhimplies thatA∈C^1,1by direct calculation on its partial derivatives.

The coercivity and the boundedness (3.3) is a direct consequence of (2.7).

In order to prove (3.4) we use the special structure ofeC. Let us choose a normal coordinate system centered atx.

Theng^κν(x)=δ^κνand thereforeη_α^κ=δ_α^κ. So, we can see that E_ij^αβ(x, u, p)=

1

|B^m|

B^m

hij(u, pξ )ξ^κξ^νdξ

δ_κ^αδ_ν^β, and that

A(x, u, p)= 1

|B^m|

B^m

h_ij(u, pξ )ξ^αξ^βdξ

p_αⁱp^j_β. (3.5)

Now, mention that the 0-homogeneity ofh_ij implies the equality X^k∂hij(u, X)

∂X^k =0.

On the other hand, since 2hij(u, X)=∂²F²(u, X)/∂Xⁱ∂X^j, we see that

∂h_ij

∂X^k =∂h_ik

∂X^j =∂h_kj

∂Xⁱ. So, we have

0=∂h_ij(u, X)

∂X^k X^k=∂h_ik(u, X)

∂X^j X^k=∂h_kj(u, X)

∂Xⁱ X^k. (3.6)

From (3.6), we get the following important relation.

∂h_ij

∂X^k(u, pξ )ξ^αξ^βp_αⁱp_β^j=0. (3.7)

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Similarly, about second derivatives,

∂²h_ij

∂X^k∂X^l(u, pξ )ξ^αξ^βp_αⁱp^j_β=0 (3.8)

holds. By virtue of (3.7) and (3.8) when one calculatesA

pⁱ_αp_β^j(x, u, p) all terms which include derivatives ofE^αβ_ij vanish. So, we obtain

A_p_i

αp^j_β(x, u, p)=2E_ij^αβ(x, u, p). (3.9)

Now, (2.7) and (3.9) imply (3.4) immediately. 2

Remark 3.2.For the energyE_SZ, because of lack ofξ^κξ^ν(there isδ^κνinstead of it), (3.7) and (3.8) do not imply the equality corresponding to (3.9). Namely, when we put

A_SZ(x, u, p):=

1 2|S^m⁻¹|

S^m⁻¹

h_ij u(x),

p(ξ ) δ^κνdξ

η^α_κη_ν^βpⁱ_αp^j_β,

and calculate(ASZ)

p_αⁱp_β^j using a normal coordinate system centered atxas above, the terms including derivatives of (E_SZ)^αβ_ij :=

1 2|S^m⁻¹|

S^m−1

h_ij

u(x),[pξ] δ^αβdξ

do not vanish. So, without considering further assumptions on the derivatives ofh_ij, we cannot obtain the ellipticity.

Since we assume thatF satisfies (2.8) andΩ is bounded, there exists a concave, nondecreasing functionωwith limt→0ω(t )=0 such that

A(x, u, p)−A(y, v, p)ω

|x−y|²+ |u−v|²

|p|². (3.10)

WhenAdoes not depend onu, instead of (3.10), (2.8) and homogeneity ofF imply that A_pi

α(x, p)−A_pi

α(y, p)ω

|x−y|²

|p|. (3.11)

UsingA(x, u, p), we can express the energy functional as E_C(u;Ω):=

Ω

A(x, u, Du) dx. (3.12)

In the sequel, we use the following notation:

A^αβ_ij (x, u, p):=A

p_αⁱp_β^j(x, u, p) forp=0, (3.13)

Aˆ^αβ_ij (x, u, p):=

A^αβ_ij (x, u, p) forp=0,

δ^αβδ_ij forp=0, (3.14)

Aˇ^αβ_ij (x, u, p):=

A^αβ_ij (x, u, p) forp=0,

0 forp=0. (3.15)

For some fixedx₀∈ΩandR >0 we will write Q(x₀, R)=

x∈R^m; x^α−x₀^α< R, α=1, . . . , m , Ω(x₀, R):=Q(x₀, R)∩Ω,

and for a functionwdefined onΩ

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w_R:= −

Ω(x0,R)

w dx= 1

L^m(Ω(x0, R))

Ω(x0,R)

w dx,

whereL^mis them-dimensional Lebesgue measure.

SinceA_p^α

i(x, u, p)∈C^0,1(X), we have the following lemma.

Lemma 3.3.The following relation holds.

A_pi

α(x, u, η)−A_pi

α(x, u, ξ )= 1 0

Aˆ^αβ_ij

x, u, t η+(1−t )ξ

η^j_β−ξ_β^j

dt. (3.16)

By the chain rule for compositions of Nemitsky operators and Sobolev maps, we have the following.

Lemma 3.4.For some fixed(x0, u0)∈Ω×Rⁿ, putA⁰(p)=A(x0, u0, p)and(Aˇ⁰)^αβ_ij (p)=(Aˇ⁰)^αβ_ij (x0, u0, p). Then, forv∈H^2,2(Ω,Rⁿ), the derivatives

DγA⁰_pi α

Dv(x)

, γ=1, . . . , m,

are well defined for almost everyx∈Ωand satisfy DγA⁰_p_i

α

Dv(x)

=Aˇ⁰αβ ij

Dv(x)

DγDβv^j(x).

Proof. See Theorem 2.1 of [17]. 2

Roughly speaking, by virtue of the above lemmata, we can proceed as in the standard theory established by S. Cam- panato [3,4], M. Giaquinta and E. Giusti [8–10], etc., and get regularity results.

4. A simple case

In this section we show some fundamental estimates for weak solutions of the Euler–Lagrange equation of the following simple functional defined foru:Ω⊂R^m→Rⁿ.

A⁰(u;Ω)=

Ω

A⁰(Du) dx, (4.1)

whereA⁰is in the classC^1,1(R^mn)∩C³(R^mn\ {0})and satisfies (3.3) and (3.4).

We mention that the assumptionm4 is not used in this section. It will be used only after (5.10).

4.1. Interior estimates

Proposition 4.1.Letv∈H^1,2(Ω,Rⁿ)be a solution of

Ω

A⁰_p_i

α(Dv)D_αϕⁱdx=0 for allϕ∈H₀^1,2 Ω,Rⁿ

. (4.2)

Then there exists a positive constantε0such that for anyq∈(2,2+ε0)we havev∈H_loc^2,q(Ω,Rⁿ). Moreover, for any x∈Ω andr >0withQ(x, r)Ω we have

−

Q(x,r/2)

D²v^qdy 1/q

C

−

Q(x,r)

D²v²dy 1/2

. (4.3)

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Proof. Let(e₁, . . . , e_m)be the standard basis ofR^m. For anyϕ∈H^1,2(Ω,Rⁿ)with suppϕΩ, chooseh >0 so that dist(suppϕ, ∂Ω) > h. Then we see that

Ω

A⁰_p_i

α

Dv(x+heγ)

−A⁰_p_i

α

Dv(x) Dαϕⁱdx=0.

Writing

τ_γ^hψ (x):=1 h

ψ (x+he_γ)−ψ (x) , and using Lemma 3.3, we have

Ω

1 0

Aˆ⁰αβ ij

(1−t )Dv(x)+t Dv(x+heγ) dt

τ_γ^hDβv^j(x)Dαϕⁱ(x) dx=0. (4.4) Now, fix a subdomainDΩand a positive constanth₀withh₀<dist(D, ∂Ω)/2, and put

D:=

x∈Ω; dist(x, D) < h0 .

Take a cut-off functionη∈C₀^∞(D)so thatη≡1 onD, 0η1 and|Dη|2/ h0. Forh < h₀, putting ϕ(x)=

τ_γ^hv(x) η²(x) in (4.4), we get by (3.4)

D

τ_γ^hDv²dx C h0

D

τ_γ^hv²dx C h0

Ω

|Dv|²dx.

So,D²v∈L²_loc(Ω)and

D

D²v²dxC(D)

Ω

|Dv|²dx.

Now, since we have shown thatDv∈H_loc^1,2(Ω), by virtue of Lemma 3.4, we can deduce from (4.2) that

Ω

Aˇ⁰αβ

ij (Dv)D_γD_βvⁱD_αϕ^jdx=0 forγ=1, . . . , m,

for anyϕ∈H^1,2(Ω)with suppϕΩ. On the other hand,D²v=0 for a.e.x∈ {x∈Ω; Dv(x)=0}. So we see that

Dv(x)=0 for a.e.x∈

x∈Ω; D²v=0 . This implies by definition ofAˇ⁰in (3.15) and (3.4) that

m γ=1

Aˇ⁰αβ

ij (Dv)D_γD_βvⁱD_γD_αv^jλ₀D²v² a.e. onΩ.

Thus we can see that a Caccioppoli inequality holds forDv. Namely, we have

Q(x,r/2)

D²v²(y) dy C r²

Q(x,r)

Dv(y)−(Dv)_x,r²dy (4.5) for any cubeQ(x, r)Ω. Combining (4.5) with the Sobolev–Poincaré inequality, we get

−

Q(x,r/2)

D²v²dy 1/2

C

−

Q(x,r)

D²v²^∗dy 1/2_∗

, (4.6)

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where 2_∗ =2m/(m+2). Now, using the reverse Hölder (or Gehring-type) inequality due to Giaquinta and Modica [13], we see that there exists a constant ε₀ >0 such that, for any q ∈ (2,2+ε₀), D²v ∈L^q_loc(Ω) and

−

Q(x,r/2)

D²v^qdy 1/q

C

−

Q(x,r)

D²v²dy 1/2

holds for anyQ(x, r)Ω. 2

Remark 4.2.In general, it is hard to know the exact value ofε₀in Proposition 4.1 and we cannot expect that it is a large number. So, in the following, we proceed assuming that it is very small.

Corollary 4.3.Letv be as in Proposition4.1. Then for anyx∈Ω andρ, r with0< ρ < r <dist(x, ∂Ω)/√ 2, we have the following estimates for the case thatm3.

Q(x,ρ)

|Dv|²dyC ρ

r

2+m−2m/q

Q(x,r)

|Dv|²dy. (4.7)

Form=2we have

Q(x,ρ)

|Dv|²dyC ρ

r

2−ε

Q(x,r)

|Dv|²dy for anyε >0. (4.8)

Here the constantsCdepend only onA, manddist(x, ∂Ω).

Proof. From (4.3) and Hölder’s inequality, we see that

Q(x,ρ)

D²v²dyC ρ

r

m(1−2/q)

Q(x,r)

D²v²dy (4.9)

for 0< ρ < r <dist(x, ∂Ω). On the other hand, by Hölder and Poincaré inequalities, we have for 0< σ <dist(x, ∂Ω) and 0< t < τ <1 that

Q(x,t σ )

|Dv|²dyC

Q(x,t σ )

Dv−(Dv)_{x,τ σ}²dy+C(t σ )^m(Dv)²_{x,τ σ} C(τ σ )²

Q(x,τ σ )

D²v²dy+C t

τ m

Q(x,τ σ )

|Dv|²dy.

Combining the above inequality and (4.9), we get

Q(x,t σ )

|Dv|²dyC t

τ m

Q(x,τ σ )

|Dv|²dy+Cτ²⁺^m(1⁻^2/q)σ²

Q(x,σ )

D²v²dy.

By the above inequality, using (4.5) withr=2σ, we obtain

Q(x,t σ )

|Dv|²dyC t

τ m

Q(x,τ σ )

|Dv|²dy+Cτ²⁺^m⁻^2m/q

Q(x,2σ )

|Dv|²dy. (4.10)

Ifm3, then 2+m−2m/q < m. So, using [12, Lemma 5.12], we have

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Q(x,t σ )

|Dv|²dyC 1

τ

2+m−2m/q

Q(x,τ σ )

|Dv|²dy+

Q(x,2σ )

|Dv|²dy

t²⁺^m⁻^2m/q

C

1 τ

2+m−2m/q

+1

Q(x,2σ )

|Dv|²dy

t²⁺^m⁻^2m/q. Taking, for example,τ=1/2 we get

Q(x,t σ )

|Dv|²dyCt²⁺^m⁻^2m/q

Q(x,2σ )

|Dv|²dy

fort ∈(0,1/2). Putting 2σ =r, the above estimate implies (4.7) forρ < r/4. On the other hand, forρ∈ [r/4, r), (4.7) holds if we choose the constantCso thatC4²⁺^2m⁻^2m/q. Thus, takingCsufficiently large, we have (4.7) for anyρ∈(0, r). Whenm=2, sinceτ <1, (4.10) implies that for anyε >0

Q(x,t σ )

|Dv|²dyC t

τ 2

Q(x,τ σ )

|Dv|²dy+Cτ²⁻^ε

Q(x,2σ )

|Dv|²dy.

Now, proceeding as above, we get (4.8). 2 4.2. Boundary estimates

We can always reduce locally to the case of flat boundary, by means of a diffeomorphism which does not change the conditions on growth, convexity, etc., of the functional in question. Namely, for any fixed point p₀∈∂Ω and a sufficiently smallr >0, without loss of generality, we can assume thatp₀=(0, . . . ,0),Q(p₀, r)∩Ω ⊂R^m₊:=

{x∈R^m;x^m>0}andQ(p₀, r)∩∂Ω⊂ {x∈R^m; x^m=0}. In the sequel, we use the following notation:

Q⁺(x, r):=Q(x, r)∩R^m₊, Γ (x, r):=Q(x, r)∩

x∈R^m; x^m=0 , Π (x, r):=∂Q⁺(x, r)\Γ (x, r).

Whenx=0, we write them simply asQ⁺(r),Γ (r)andΠ (r).

By virtue of the above observation, near the boundary it is enough to consider the following problem.

Dα

A⁰_p_i

α(Dv)

=0 inQ⁺(r),

v=f onΓ (r). (4.11)

Let us first investigate the case thatf =0.

Lemma 4.4.Letz∈H^1,2(Q⁺(r),R^m)be a solution of

⎧⎪

⎨

⎪⎩

Q⁺(r)

A⁰

pⁱ_α(Dz)D_αϕⁱdx=0 for allϕ∈H₀^1,2

Q⁺(r),Rⁿ , z=0 onΓ (r).

(4.12)

Then, for anyr∈(0, r), we havez∈H^2,2(Q⁺(r),R^m). Moreover, for anyx∈Q⁺(r)∪Γ (r)andρ, σ with 0< ρ < σ < h₀:=r−r

2 , we have

Ω(x,ρ)

D²z²dy C (σ−ρ)²

Ω(x,σ ) m−1 γ=1

|D_γz|²dy, (4.13)

whereΩ(x, ρ):=Q(x, ρ)∩Q⁺(r).

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Proof. Forγ =1, . . . , m−1,η∈C₀^∞(Q(r))andh <dist(suppη, Π (r)), let ϕ=τ_γ⁻^h{(τ_γ^hz)η²}in (4.12). (Since z=0 onΓ (r), this choice ofϕis admissible.) Then we see that

Q⁺(r)

τ_γ^hA⁰

pⁱ_α(Dz)D_α τ_γ^hzⁱ

η² dx=0.

So, putting a_ij^αβ(x):=

1 0

Aˆ⁰αβ ij

t Dz(x+he_γ)+(1−t )Dz(x) dt, and using (3.16), we get

Q⁺(r)

a_ij^αβ(x)τ_γ^hD_βz^j

τ_γ^hD_αzⁱη²+2τ_γ^hzⁱηD_αη dx=0. (4.14)

For any fixedx∈Q⁺(r)∪Γ (r), letsandtbe positive numbers satisfying 0< s < t < r−maxx^γ; γ=1, . . . , m .

Choosingηso thatη∈C₀^∞(Q(x, t )),η≡1 onQ(x, s)and that|Dη|C/(t−s)in (4.14), we see with (3.4) that

Ω(x,s)

τ_γ^hDz²dy C (t−s)²

Ω(x,t )

τ_γ^hz²dy, γ=1, . . . , m−1.

So, we obtain

Ω(x,s)

|D_γDz|²dy C (t−s)²

Ω(x,t )

|D_γz|²dy, γ=1, . . . , m−1. (4.15)

Moreover, Lemma 3.4 leads to D_γA⁰_p_i

γ(Dz)=Aˇ⁰γβ

ij (Dz)D_γD_βz^j a.e. onQ⁺(r),

forγ=1, . . . , m−1. Now, we can proceed as in the proof of [4, Theorem 4.1] and obtain the assertion. 2

Proposition 4.5. Let z∈H^1,2 be as in Lemma 4.4. Then, there exists a constant ε₀>0 such that for every q∈ (2,2+ε₀)and for anyr∈(0, r), we havea∈H^2,q(Q⁺(r))and

−

Ω(x,σ )

D²z^qdy 1/q

C

−

Ω(x,2σ )

D²z²dy 1/2

, (4.16)

wherex∈Q⁺(r)and2σ < r−r.

Proof. Letxandσbe as above. IfQ(x,2σ )∩Γ (r)= ∅, then Proposition 4.1 implies (4.16). Assume thatQ(x,2σ )∩ Γ (r)= ∅. Then

H^m⁻¹

Q(x,2σ )∩Γ (r)

=(4σ )^m⁻¹.

So, remarking thatz=0 onΓ (r)(and thereforeD_γz=0 onΓ (r)forγ=1, . . . , m−1), we can apply the Sobolev–

Poincaré inequality forD_γz (γ=1, . . . , m−1)to get

Ω(x,2σ ) m−1 γ=1

|D_γz|²dy 1/2

C

Ω(x,2σ )

D²z²^∗dy 1/2_∗

. (4.17)

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Combining (4.13) with (4.17), we get

−

Ω(x,σ )

D²z²dy 1/2

C

−

Ω(x,2σ )

D²z²^∗dy 1/2_∗

. (4.18)

Now, extending D²zas 0 outsideQ⁺(r), and using the interior estimate (4.6) we see that (4.18) holds for every x∈Q⁺(r)andσ < (r−r)/2. Thus, we can use the reverse Hölder inequality due to Giaquinta and Modica [13] to obtain (4.16). 2

Now, we can proceed as in the proof of Corollary 4.3 and obtain the following.

Corollary 4.6.Letzandq >2be as in Lemma4.4and Proposition4.5. Then for anyx∈Q⁺(r)and0< ρ < σ <

(r−r)/2, we have

Ω(x,ρ)

|Dz|²dyC ρ

σ

2+m−2m/q

Ω(x,σ )

|Dz|²dy (4.19)

form3and

Ω(x,ρ)

|Dz|²dxC ρ

σ

2−ε

Ω(x,σ )

|Dz|²dx for anyε >0, (4.20)

form=2.

When the boundary conditions are generalH^1,s-functions, using Corollary 4.6, we have the following.

Corollary 4.7.Assume thatv∈H^1,2(Q⁺(r))satisfies

⎧⎪

⎨

⎪⎩

Q+(r)

A⁰_p_i

α(Dv)D_αϕⁱdx=0 ^∀ϕ∈H₀^1,2 Q⁺(r)

, v=f onΓ (r),

(4.21)

wheref is a given map in the classH^1,s(Q⁺(r))for somes > m. Letr< r,x∈Γ (r)and0< ρ < σ < (r−r)/2.

Ifm=3,4, then, for someε0>0for anyq∈(2,2+ε0), we have

Q⁺(x,ρ)

|Dv|²dxC ρ

σ

2+m−2m/q

Q⁺(x,σ )

|Dv|²dx+C

Q⁺(x,σ )

|Df|²dx. (4.22)

Ifm=2, then for everyε >0, we have

Q⁺(x,ρ)

|Dv|²dxC ρ

σ

2−ε

Q⁺(x,σ )

|Dv|²dx+C

Q⁺(x,σ )

|Df|²dx. (4.23)

Proof. Letw=v−f. Thenwsatisfies

⎧⎪

⎨

⎪⎩

Q⁺(r)

A⁰

pⁱ_α(Dw+Df )D_αϕⁱdx=0 ^∀ϕ∈H₀^1,2 Q⁺(r)

, w=0 onΓ (r).

Letx,σ be as in the statement and letz∈H^1,2(Q⁺(x, σ ))be a solution of

⎧⎪

⎪⎨

⎪⎪

⎩

Q⁺(x,σ )

A⁰(Dz) dy→minimum, z−w∈H₀^1,2

Q⁺(x, σ ) .