Pointwise curvature estimates for <span class="mathjax-formula">$F$</span>-stable hypersurfaces

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Pointwise curvature estimates for F -stable hypersurfaces

Estimations ponctuelles de la courbure des hypersurfaces F -stables

Sven Winklmann

Universität Duisburg-Essen, Institut für Mathematik, 47048 Duisburg, Germany

Received 2 July 2004; accepted 6 October 2004 Available online 7 April 2005

Abstract

We consider immersed hypersurfaces in euclideanRⁿ⁺¹which are stable with respect to an elliptic parametric functional of the formF(X)=

MF (N )dµ. We prove a pointwise curvature estimate provided thatn5 andF is sufficiently close to the area integrand. This extends the pointwise curvature estimates of Schoen, Simon and Yau [Acta Math. 134 (1975) 275] for stable minimal hypersurfaces inRⁿ⁺¹and of Simon [Math. Z. 154 (1977) 265] for minimizers ofF. Our result follows from an integral curvature estimate and a generalized Simons inequality that were established recently [Calc. Var. Partial Differential Equations (2004), DOI: 10.1007/S00526-004-0306-5], together with a Moser type iteration argument.

2005 Elsevier SAS. All rights reserved.

Résumé

Nous considérons des hypersurfaces dans l’espace euclidienRⁿ⁺¹qui sont stables par rapport à une fonctionnelle paramé- trique elliptique de la formeF(X)=

MF (N )dµ. Nous démontrons une estimation ponctuelle de la courbure sous l’hypothèse n5 et Fest suffisamment proche de l’intégrande aire. Ce résultat étend l’estimation ponctuelle de la courbure obtenue par Schoen, Simon et Yau [Acta Math. 134 (1975) 275] pour les hypersurfaces minimales stables deRⁿ⁺¹et par Simon [Math. Z.

154 (1977) 265] pour les minimiseurs deF. Une estimation intégrale de la courbure, une inégalité de Simons généralisée, établie récemment dans [Calc. Var. Partial Differential Equations (2004), DOI : 10.1007/S00526-004-0306-5], ainsi qu’un argument itératif de type Moser nous permet d’obtenir cette estimation ponctuelle.

2005 Elsevier SAS. All rights reserved.

MSC: 53C42; 49Q10; 35J60

1. Introduction

Ifuis a solution of the two-dimensional minimal surface equation defined over the diskBR(x₀):= {x∈R²:

|x−x₀|< R}, then the principal curvaturesκ₁,κ₂of the corresponding graph can be estimated by

E-mail address: winklmann@math.uni-duisburg.de (S. Winklmann).

0294-1449/$ – see front matter 2005 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpc.2004.10.005

(κ₁²+κ₂²)(x₀) C R²

with a universal constantC. This is the well-known curvature estimate of Heinz [11] which appears as a quantitative version of Bernstein’s celebrated theorem [1]. In fact, by lettingR→ ∞one deduces that every entire solution of the minimal surface equation inR²has to be an affine linear function.

There are several important extensions and variants of these results, see e.g. [7] and [10]. In particular, Schoen [18] has proved an analogue of Heinz’s estimate for stable minimal surfaces inR³, and again this estimate implies a Bernstein result.

In higher dimensions Schoen, Simon and Yau [19] have obtained an integral curvature estimate for stable mini- mal hypersurfaces immersed in a Riemannian manifoldNⁿ⁺¹, and in the particular case whereN is the euclidean Rⁿ⁺¹this estimate leads to a pointwise curvature estimate up ton5. Improvements forn6 have been made by Simon [21] and Schoen, Simon [20] using regularity theory for minimal hypersurfaces and methods from geometric measure theory, respectively.

In this paper we consider immersed hypersurfacesX:Mⁿ→Rⁿ⁺¹with Gauß mappingNand induced surface measureµwhich are stable with respect to a parametric functional of the form

F(X)=

M

F (N )dµ.

The integrandF is of classC^∞(Rⁿ⁺¹\ {0})and satisfies the homogeneity condition

F (t z)=t F (z) for allz∈Sⁿ, t >0. (1)

Moreover, throughout the paperF is assumed to be elliptic, i.e.

F_zz(z)= ∂²F

∂z^α∂z^β(z)

α,β=1,...,n+1

:z^⊥→z^⊥

is a positive definite endomorphism for allz∈Sⁿ, or equivalently λ(F ):= inf

z∈Sⁿ, V∈z^⊥\{0}

(∂²F /∂z^α∂z^β)(z)V^αV^β

|V|² >0. (2)

Clearly,Fgeneralizes the area functional A(X)=

M

dµ

which is obtained in caseF (z)=A(z):= |z|is the area integrand.

In a previous paper [25] we have shown that an integral curvature estimate forF-stable hypersurfaces can be proved, wheneverFis sufficiently close to the area functional. To be precise, define the norm

GC^k:=sup

z∈Sⁿ |I|k

∂^|I|G

∂z^I (z) ²1/2

for an arbitrary integrand G∈ C^∞(Rⁿ⁺¹\ {0}). It was proven in [25, Theorem 1.1] that given n2 and p∈(4,4+√

8/n), there exists a constantδ(n, p) >0 with the property that ifF is an elliptic integrand satis- fyingF −AC⁴< δand ifXis anF-stable hypersurface, then the integral curvature estimate

M

|S|^pϕ^pdµC(n, p, F )

M

|∇ϕ|^pdµ (3)

holds for all nonnegative testfunctionsϕ∈C_c^∞(M). Here,|S|stands for the length of the Weingarten operator, i.e.

|S|²=κ₁²+ · · · +κ_n², whereκ₁, . . . , κ_ndenote the principal curvatures ofX.

In this paper we wish to discuss a pointwise curvature estimate forF-stable hypersurfaces. To this end, define δ(n):=sup

δ(n, p): 4< p <4+

8/n, p > n for 2n5. Moreover, let us use the notation

BR(x₀):=

x∈M: r(x) < R , R >0,

wheneverr:M→Ris a Lipschitz function withr(x₀)=0 and|∇r|1µ-a.e.. Our main result is the following:

Theorem 1.1. Let 2n5 and letF be an elliptic integrand satisfyingF −AC⁴ < δ(n). SupposeX is a stable hypersurface for the parametric functional

F(X)=

M

F (N )dµ.

IfBR(x₀)Mfor some pointx₀∈Mand radiusR >0, and ifµ(BR(x₀))KRⁿ, then we have sup

Bθ R(x₀)

|S|²C(n, F, K, θ )

R² (4)

for allθ∈(0,1).

In particular, if we choose r(x)=d(x, x₀), then we obtain the curvature estimate on the usual open balls B_{θ R}(x₀), and lettingR→ ∞we infer the Bernstein result [25]:

Corollary 1.2. Let 2n5 and letF be an elliptic integrand withF−AC⁴< δ(n). SupposeXis a complete connectedF-stable hypersurface that satisfies the growth condition

µ B_R(x₀)

KRⁿ

for some pointx₀∈Mand some sequenceR→ ∞. ThenX(M)is a hyperplane.

Observe that in caseF =Ais the area integrand, Theorem 1.1 yields the pointwise curvature estimate of Schoen, Simon and Yau [19, Theorem 3], withRⁿ⁺¹as the ambient manifold.

Moreover, we remark that curvature estimates and Bernstein type results for two-dimensional parametric func- tionals have been obtained by Jenkins [12], White [23,24], Lin [13], Sauvigny [17], Fröhlich [8,9], Räwer [16]

and Clarenz [2]. All these results are strictly two-dimensional. Simon [22] has obtained a pointwise estimate for minimizers ofF up to dimensionn6 provided thatF−AC³ is sufficiently small. It is unknown, whether our results can be improved to hold up ton6.

The paper is organized as follows. In Section 2 we recall some preliminary results taken from [25], including a generalized Simons inequality and an equivalent form of the integral curvature estimate. In Section 3 we then use these results to establish anL^p-estimate for the curvature. Here, we proceed similarly as in Dierkes [5,6], where a class of singular parametric functionals of the type

Eα(X)=

M

|X_n+1|^αdµ, α >0,

is considered. Finally, in Section 4 we use the L^p-estimate together with a Moser type iteration argument to prove (4).

2. Preliminaries

In this section we recall some preliminary results from [25] concerning the geometry of parametric functionals.

Let X:Mⁿ→Rⁿ⁺¹ be a smooth immersion of ann-dimensional oriented manifold without boundary into euclideanRⁿ⁺¹. LetN:M→Sⁿstand for the corresponding Gauß mapping and denote byµthe induced measure with respect to the pull backgof the euclidean metric. Consider the parametric functional

F(X)=

M

F (N )dµ

with an elliptic integrandF ∈C^∞(Rⁿ⁺¹\ {0})satisfying the homogeneity condition (1) and the ellipticity condi- tion (2). We say thatXisF-stationary if

δF(X, ϕ):= d

dεF(X+εϕN )

ε=0=0

for allϕ∈C_c^∞(M)and we say thatXisF-stable, if in addition δ²F(X, ϕ):= d²

dε²F(X+εϕN )

ε=00 for allϕ∈C_c^∞(M).

In order to give a geometric description ofF-stationary hypersurfaces, we need to recall the notion ofF-mean curvature as introduced by Räwer [16] and Clarenz [2]: LetAF be the symmetric, positive definite endomorphism- field given by

A_F :=dX⁻¹◦F_zz(N )◦dX,

and letS:= −dX⁻¹◦dN denote the Weingarten operator. Then S_F :=A_F◦S

is calledF-Weingarten operator and H_F:=tr(S_F)

is theF-mean curvature ofX. Now, according to [16] and [2] the first variation ofFis given by δF(X, ϕ)= −

M

H_Fϕdµ.

Hence,XisF-stationary if and only if itsF-mean curvature vanishes. Moreover, for anF-stationary hypersurface the second variation is given by

δ²F(X, ϕ)=

M

g(AF∇ϕ,∇ϕ)−tr(AFS²)ϕ² dµ,

where∇ϕdenotes the gradient ofϕwith respect tog. Consequently,XisF-stable if and only ifH_F =0 and

M

tr(A_FS²)ϕ²dµ

M

g(A_F∇ϕ,∇ϕ)dµ (5)

for allϕ∈C_c^∞(M). For further details see also [3] and [4]. Also note that ifF is the area integrand, thenA_F=id and so the definitions coincide with their classical analogues.

Now, as in [25], let us introduce as an additional geometric quantity the abstract metric g_F(v, w):=g

A⁻_F¹(v), w

, (v, w∈T M)

which can be seen as an n-dimensional analogue of Sauvigny’s [17] weighted first fundamental form. In [25, Propositions 2.2 and 2.3] we have shown the estimates

c(F )|S||S_F|FC(F )|S|, (6)

c(F )|∇h|| ∇h|F C(F )|∇h| (7)

and

c(F )dµdµ_FC(F )dµ (8)

for suitable positive constantsc(F )andC(F ), where∇ϕ,|T|F andµ_F denote the gradient ofϕ, the length of the tensorT and the measure, all taken with respect tog_F. In particular, these estimates imply the equivalence of the integral curvature estimate (3) to an estimate of the form

M

|S_F|^p_Fϕ^pdµ_F C(n, p, F )

M

|∇ϕ|^p_Fdµ_F (9)

for all nonnegative functionsϕ∈C_c^∞(M).

Now, it was shown in [25, Proposition 2.5] that the stability inequality (5) implies

M

|S_F|²Fϕ²dµ_FC(F )

M

|∇ϕ|²Fdµ_F (10)

for all ϕ∈C_c^∞(M), where C(F )is a positive constant that tends to 1 asF −AC² →0. Moreover, we have shown that anyF-stationary hypersurfaceXsatisfies the generalized Simons inequality

1

2_F|S_F|²_F 1−η

1+θ

1+2 n

|∇|S_F|F|²_F− 1

λ(F )+C(η, θ )ε(F )

|S_F|⁴_F (11)

for allθ >0 andη∈(0,1]with a nonnegative constantε(F )that tends to 0 asF −AC⁴→0, cp. [25, Theo- rem 4.2]. Here, of course,F denotes the Laplace-Beltrami operator with respect togF.

Using (10) and (11), we have shown that if p∈(4,4+√

8/n)and if F is an elliptic integrand satisfying F −AC⁴< δ(n, p), then (9), and hence (3), holds for anyF-stable hypersurfaceX.

In the next section, we do use the generalized Simons inequality (11) and the integral curvature estimate (9) to establish anL^p-estimate for the curvature|SF|F. To accomplish this, we will need the following version of the Michael–Simon Sobolev inequality [14]:

Lemma 2.1. LetF be an elliptic integrand and letX:M→Rⁿ⁺¹be an immersed hypersurface. If 1p < n, then we have

M

|h|^nnp−pdµ_F ^n−p

np C(n, p, F )

M

|h|^p|S_F|^p_F+ | ∇h|^p_F dµ_F

¹

p

(12)

for allh∈C_c^∞(M).

Proof. From [14] we infer

M

|h|^nnp−pdµ ^n−p

np C(n, p)

M

|h|^p|H|^p+ |∇h|^p dµ

¹

p

,

so the obvious inequality|H|²n|S|²together with (6), (7) and (8) gives the desired result. 2

3. L^p-estimate

Let 2n5, let F be an elliptic integrand satisfying F −AC⁴ < δ(n), and let X:M→Rⁿ⁺¹ be an F-stable hypersurface. Suppose that B1(x0)M with µ(B1(x0))K. The main result of this section is the followingL^p-estimate:

Proposition 3.1. Letf:= |S_F|^p_F,p >1, and letη∈C^∞_c (M)be a nonnegative testfunction with supp(η)⊂Bτ(x₀), 0< τ <1. Define

q= n

n−2, if n3, 2, if n=2.

Then we have

M

(f η)^2qdµ_F 1/q

Cp^α

M

f²η²dµ_F+C

M

f²| ∇η|²_Fdµ_F, (13)

whereC=C(n, F, K, τ )andα=α(n, F ) >1.

Proof. Puth:=f ηin the Sobolev inequality (12). By approximation, this choice ofhis valid and ifn3, then we obtain

M

(f η)ⁿ²ⁿ−2 dµ_F ⁿ⁻²_n

C(n, F )

M

| ∇f|²Fη²+f²| ∇η|²F+f²η²|S_F|²F

dµ_F.

On the other hand, ifn=2, then we have

M

(f η)^22r−r dµ_F ^2−r

2r C(n, r, F )

M

∇(f η)^r

F +f^rη^r|S_F|^r_F dµ_F

¹

r

for allr∈(1,2), and by virtue of Hölder’s inequality together with supp(η)⊂B1(x₀)andµ_F(B1(x₀))C(F )K, see (8), we infer

M

∇(f η)^r

F+f^rη^r|S_F|^r_F

dµ_F C(r, F, K)

M

∇(f η)²

Fdµ_F ^r

2

+C(r, F, K)

M

f²η²|S_F|²_Fdµ_F ^r

2

,

hence

M

(f η)^22r−r dµ_F ^2−r

r C(n, r, F, K)

M

| ∇f|²_Fη²+f²| ∇η|²_F+f²η²|S_F|²_F dµ_F.

Chooser=⁴₃, so that ₂₋^r_r =2. Then we have shown

M

(f η)^2qdµ_F ¹

q C₁(n, F, K)

M

| ∇f|²_Fη²+f²| ∇η|²_F +f²η²|S_F|²_F

dµ_F (14)

with q=

_n

n−2, ifn3, 2, ifn=2.

We now use the generalized Simons inequality to estimate the first integral on the right hand side as follows:

Lemma 3.2. We have

M

| ∇f|²Fη²dµF C2(n, F )p

M

f²η²|SF|²FdµF +4

M

f²| ∇η|²FdµF. (15)

Proof of Lemma 3.2. We have 1

2F|SF|^2p_F =p(2p−1)|SF|^2p_F ⁻²| ∇|SF|F|²F+p|SF|^2p_F ⁻¹F|SF|F

=2p(p−1)|SF|^2p_F ⁻²| ∇|SF|F|²F+1

2p|SF|^2p_F ⁻²F|SF|²F

in a weak sense, and applying the generalized Simons inequality (11) we obtain 1

2F|SF|^2p_F p(p−1)|SF|^2p_F ⁻²| ∇|SF|F|²F +p

p−1+ 1−η

1+θ

1+2 n

|SF|^2p_F ⁻²| ∇|SF|F|²F

−p 1

λ(F )+C(η, θ )ε(F )

|SF|^2p_F ⁺² for allθ >0 andη∈(0,1]. Chooseη(n)andθ (n)so small that

1−η 1+θ

1+2

n

1.

Then we infer 1

2_F|S_F|^2p_F p²|S_F|^2p_F ⁻²| ∇|S_F|F|²F −p 1

λ(F )+C(η, θ )ε(F )

|S_F|^2p_F ⁺², whence

1

2_Ff²| ∇f|²_F−C₃(n, F )p|S_F|²_Ff².

We now multiply this inequality byη²and integrate by parts. This gives

M

| ∇f|²Fη²dµ_F −2

M

f η ∇f∇ηdµ_F +C₃p

M

f²η²|S_F|²Fdµ_F and in view of Young’s inequality

2f η| ∇f|F| ∇η|F 1

2| ∇f|²_Fη²+2f²| ∇η|²_F the desired estimate follows. 2

Combining (14) and (15) we obtain

M

(f η)^2qdµ_F ¹

q C₄(n, F, K)p

M

f²η²|S_F|²_Fdµ_F +C₅(n, F, K)

M

f²| ∇η|²_Fdµ_F. (16) We now employ the integral curvature estimate to estimate the first integral on the right-hand side.

Lemma 3.3. We have

M

f²η²|S_F|²Fdµ_FC₆γ

M

(f η)^2qdµ_F ¹

q +γ^−s−11

M

f²η²dµ_F (17)

for allγ >0, whereC₆=C₆(n, F, K, τ )ands=s(n, F ) >1.

Proof of Lemma 3.3. We need the following interpolation inequality from [10, p. 145]

abγ a^s+γ^−s−11 b^s−1s

fora, b0,γ >0 ands >1. Lettinga= |S_F|²_F andb=1, we obtain

M

f²η²|S_F|²Fdµ_Fγ

M

f²η²|S_F|^2sF dµ_F+γ^−s−¹1

M

f²η²dµ_F

for arbitraryγ >0 ands >1. Applying the Hölder inequality with¹_q+_q¹ =1 and noticing that supp(η)⊂Bτ(x₀), it follows that

M

f²η²|S_F|²Fdµ_Fγ

M

(f η)^2qdµ_F ¹_q

Bτ(x0)

|S_F|^2sq_F dµ_{F q}¹

+γ^−s−¹1

M

f²η²dµ_F. (18)

We now choosesin the following way: According to the definition ofδ(n)there exists a numbert=t (F )∈ (4,4+√

8/n)witht > n, such thatF −AC⁴< δ(n, t ). Choose s=

_t

n, ifn3,

t

4, ifn=2.

Then, clearly,s=s(n, F ) >1. Moreover, on account of q=

_n

2, ifn3, 2, ifn=2 we have

2sq=t. (19)

Now, letϕbe the cut-off function defined by ϕ(x):=Φ

r(x)−τ 1−τ

, x∈M,

whereΦ∈C¹(R)is a nonincreasing function withΦ(y)=1 fory0,Φ(y)=0 fory1 and|Φ(y)|2 for ally∈R. Thenϕ=1 inBτ(x₀),ϕ=0 inM\B1(x₀)and using (7) we see that

| ∇ϕ|F C(F )|∇ϕ|2C(F ) 1−τ

µ_F-a.e.. Also note that ϕ is compactly supported in M subject to B1(x₀)M. Hence, we can applyϕ in the integral curvature estimate (9) with exponenttand together with (19) andµ_F(B1(x₀))C(F )Kthis leads to

Bτ(x0)

|SF|^2sq_F dµF

M

|SF|^t_Fϕ^tdµF C(n, F, t )

B1(x0)

| ∇ϕ|^t_FdµF C(n, F, K)

2C(F ) 1−τ

t

.

Thus, we have

Bτ(x0)

|SF|^2sq_F dµF

_q¹

C6(n, F, K, τ )

and inserting this into (18) gives the desired result. 2 According to (16) and (17) we now have

M

(f η)^2qdµ_F ¹

q C₇(n, F, K, τ )pγ

M

(f η)^2qdµ_F ¹

q

+C₄pγ^−s−¹1

M

f²η²dµ_F+C₅

M

f²| ∇η|²_Fdµ_F

for allγ >0, and choosingγ:=_2C¹_7p we finally arrive at

M

(f η)^2qdµ_F ¹

q 2C₄p^1+s−¹1(2C₇)^s−¹1

M

f²η²dµ_F+2C₅

M

f²| ∇η|²_Fdµ_F

Cp^α

M

f²η²dµ_F +C

M

f²| ∇η|²Fdµ_F,

whereα=α(n, F ):=_s−^s₁>1 andC=C(n, F, K, τ ). This completes the proof of theL^p-estimate. 2

4. Proof of the curvature estimate

Before we turn to the proof of Theorem 1.1, let us make sure that it suffices to establish the curvature estimate (4) forR=1.

Indeed, suppose thatXsatisfies the assumptions of Theorem 1.1. We then consider the hypersurface X:M→Rⁿ⁺¹, X:= 1

RX.

On account of δF(X,ϕ)˜ = d

dεF(X+εϕ˜N )

ε=0= 1

RⁿδF(X, Rϕ)˜ =0 and

δ²F(X,ϕ)˜ = 1

Rⁿδ²F(X, Rϕ)˜ 0

for allϕ˜∈C_c^∞(M), we see thatXisF-stable as well. Moreover, ifrdenotes the abstract distance function ofX, i.e. the Lipschitz function withr(x₀)=0 and|∇r|1µ-a.e., then

˜

r:M→R, r˜:= 1 Rr

defines an abstract distance function forXand for the corresponding balls we have Bθ(x₀):=

x∈M: r(x) < θ˜ =Bθ R(x₀)

for allθ >0. In particular, we see thatB1(x₀)Mand since the measure scales with 1/Rⁿwe obtain µ˜ B1(x₀)

= 1 Rⁿµ

BR(x₀)

K.

Assume now, that the curvature estimate (4) holds forR=1. Then we can apply it toXand obtain

sup

Bθ(x₀)

|S|²_g˜C(n, F, K, θ )

for allθ∈(0,1). Moreover, since the principal curvatures scale withR, we have

|S|²_g˜=R²|S|²_g and therefore

sup

Bθ R(x0)

|S|²_g= sup

Bθ(x₀)

|S|²_g˜

R² C(n, F, K, θ ) R²

with the same constantC. Hence, the curvature estimate (4) holds for everyR >0.

Proof of Theorem 1.1. According to the discussion above we only have to consider the case R=1, so let us assume thatX:M→Rⁿ⁺¹is F-stable,B1(x₀)M andµ(B1(x₀))K. It is our goal to prove the following estimate

sup

Bθ(x₀)

|S_F|²_FC(n, F, K, θ ) (20)

for allθ∈(0,1)since this will imply the desired curvature estimate sup

Bθ(x0)

|S|²C(n, F, K, θ ) in view of (6).

To establish (20), letθ˜:=¹⁺₂^θ,τ:=^{1+ ˜}₂^θ and letρ,ρbe radii satisfying

θ < ρ< ρθ < τ <˜ 1 (21)

and

ρρ−1

2(ρ−θ ). (22)

Define a cut-off functionηby η(x):=Φ

r(x)−ρ ρ−ρ

, x∈M,

whereΦis to be chosen as in the proof of Lemma 3.3. Then we haveη=1 inBρ(x₀),η=0 inM\Bρ(x₀)and

| ∇η|F 2C(F ) ρ−ρ

µ_F-a.e.. Insertingηinto theL^p-estimate (13) we infer

Bρ(x₀)

f^2qdµ_F ¹

q C₁p^α

Bρ(x₀)

f²dµ_F +C₁

2C(F ) ρ−ρ

₂

Bρ(x₀)

f²dµ_F,

whereC₁=C₁(n, F, K, τ )andα=α(n, F ) >1. Sincep >1 and_2(ρ^θ˜−₋^θ_ρ)1 it follows that

Bρ(x₀)

f^2qdµ_F ¹

q C₂(n, F, K, θ ) (ρ−ρ)² p^α

Bρ(x₀)

f²dµ_F

withC₂=^C1(θ˜₄^{−θ )2} +4C₁C(F )². Now, let u:= |S_F|²_F. Then we have f²= |S_F|^2p_F =u^p andf^2q=u^qp, and therefore

Bρ(x₀)

u^qpdµ_F ¹

qp C

1 p

2p^αp(ρ−ρ)^−2p

Bρ(x₀)

u^pdµ_F ¹

p

.

Hence, abbreviating I (σ, t ):=

Bσ(x₀)

u^tdµ_F ¹

t

we obtain

I (ρ, qp)C

1 p

2p^αp(ρ−ρ)^−p2I (ρ, p) (23)

for allp >1 andρ,ρsatisfying (21) and (22).

From here we intend to employ a well-known iteration scheme originally due to Moser [15]. Let ρ_k:=θ+2^−k(θ˜−θ ),

ρ_k:=ρ_k+1=ρ_k−1

2(ρ_k−θ ) and

p_k:=q^kp₀ withp₀>1

fork=0,1,2, . . . .Then we infer from (23) I (ρk+1, pk+1)=I (ρ_k, qpk)C

1 pk

2 p

α pk

k (ρk−ρ_k)⁻

2

pkI (ρk, pk)

=C

1 pk

2 p

α pk

0 q

αk

pk4^k+1pk (θ˜−θ )⁻

2

pkI (ρk, pk)C

k+1pk

3 p

α pk

0 I (ρk, pk), whereC₃=C₃(n, F, K, θ ):=max(4C₂(θ˜−θ )⁻²,4q^α), and iterating this inequality yields

I (ρk+1, pk+1)C

k j=0

j+1 pj

3 p

k j=0 α

pj

0 I (ρ0, p0).

Now we have_∞

j=0 α

p_j =_p0(q^αq₋₁₎ and_∞

j=0 j+1

p_j =_p ^q2

0(q−1)².Consequently, I (ρ_k+1, p_k+1)C

q2 p0(q−1)2

4 p

αq p0(q−1)

0 I (ρ₀, p₀) (24)

fork=0,1,2, . . .withC₄=C₄(n, F, K, θ ):=max(C₃,1). Now we need the following simple lemma:

Lemma 4.1. We have sup

Bθ(x₀)

ulim inf

k→∞ I (ρ_k, p_k).

Proof of Lemma 4.1. Givenε >0, there exists a setA_ε⊂Bθ(x₀)withµ_F(A_ε) >0 and u sup

Bθ(x₀)

u−ε

inAε. Hence, I (ρ_k, p_k)=

Bρk(x₀)

u^pkdµ_F ¹

pk

Aε

u^pkdµ_F ¹

pk

sup

Bθ(x0)

u−ε

µ_F(A_ε)¹

pk,

and lettingk→ ∞we obtain lim inf

k→∞ I (ρk, pk) sup

B^θ(x₀)

u−ε.

The assertion now follows asε→0. 2 We now apply Lemma 4.1 to (24) and find

sup

Bθ(x₀)

uC

q2 p0(q−1)2

4 p

αq p0(q−1)

0 I (ρ₀, p₀)=C

q2 p0(q−1)2

4 p

αq p0(q−1)

0

Bθ˜(x₀)

u^p0dµ_F ¹

p0

.

Here,p0>1 can be arbitrarily chosen. In particular, lettingp0→1 we obtain sup

Bθ(x0)

|S_F|²_FC₅(n, F, K, θ )

Bθ˜(x₀)

|S_F|²_Fdµ_F,

whereC5:=C

q2 (q−1)2

4 .

Finally, using a suitable cut-off function in a similar fashion as we did before, we infer

Bθ˜(x₀)

|S_F|²_Fdµ_FC(F, K,θ )˜

from our stability inequality (10). Hence, we have

Bsupθ(x₀)

|S_F|²FC₆(n, F, K, θ )

which is the desired estimate (20), and this completes the proof of Theorem 1.1. 2

Acknowledgement

The author wishes to thank Prof. Dr. U. Dierkes for many interesting and helpful discussions on this work.

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