www.elsevier.com/locate/anihpc
A “quasi maximum principle” for I -surfaces
Ruben Jakob
1Mathematisches Institut der Universität Duisburg-Essen, Germany Received 14 December 2005; accepted 30 March 2006
Available online 25 September 2006
Abstract
The result of this paper yields amaximum principlefor the components of surfaces whose distortion by a certain GL3(R)matrix are minimizers of a dominance functionalIof a parametric functionalJ with dominant area term within boundary value classes Hϕ1,2(B,R3), termedI-surfaces. Finally we derive acompactnessresult for sequences ofI-surfaces inC0(B,¯ R3), which serves as a preparation for the forthcoming article [R. Jakob, Unstable extremal surfaces of the “Shiffman functional” spanning rectifiable boundary curves, Calc. Var., submitted for publication] whose aim is a proof of a sufficient condition for the existence of extremal surfaces ofJ which do not furnish global minima ofJ within the classC∗(Γ )ofH1,2-surfaces spanning an arbitrary closed rectifiable boundary curveΓ ⊂R3that merely has to satisfy a chord-arc condition.
©2006 Elsevier Masson SAS. All rights reserved.
1. Introduction and main result
Following Shiffman [12] we consider as in [6] and [8] the functional I(X):=
B
F (Xu∧Xv)+k
2|DX|2dudv=:F(X)+kD(X),
on surfacesX∈H1,2(B,R3)of the type of the open discB:=B12(0)⊂R2. The LagrangianF is assumed to satisfy the following list of requirements (A):
F ∈C0 R3
∩C2
R3\ {0}
, (1)
F (t z)=t F (z) ∀t0, ∀z∈R3, (2)
m1|z|F (z)m2|z| ∀z∈R3, 0< m1m2, (3)
F is convex onR3. (4)
Moreover we have to impose the following requirement onF:
1 The author was supported by the Deutsche Forschungsgemeinschaft and would also like to thank Prof. Dr. S. Hildebrandt, Prof. Dr. H. von der Mosel and especially Prof. Ph.D. R. Montgomery who kindly pointed out to the author the counterexample discussed in Section 2.
0294-1449/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpc.2006.03.006
(R∗) The restriction of the functiong(z):=F (z)+F (−z)to theS2shall have three linearly independent critical points, i.e. there have to be at least three linearly independent unit vectorsa1, a2, a3∈S2at which∇g(aj)=rjaj, for somerj∈R,j=1,2,3. Finally we assume that
k >max
S2 F =m2. (5)
ThusI is a controlled perturbation of the Dirichlet functional D, where F depends only on the normal Xu∧Xv, but not on the position vectorX itself. Now only imposing the requirements (A) it was proved in Lemma 2.2 and Theorem 4.3 of [6] that in every boundary value classHϕ1,2(B,R3)there exists a unique minimizer ofI, termed I-surface, which is additionally of the classC0(B,¯ R3)ifϕ∈C0(∂B,R3)∩H1/2,2(∂B,R3)by Theorem 5.2 in [6].
Shiffman claimed the results of this paper, Theorems 1.1 and 1.2, in Sections 6 and 7 of [12], but his proof of Theorem 1.1 is incomplete. We emphasize in particular that on p. 552 in [12] Shiffman asserts the incorrect statement that any integrandF meeting (A) satisfies the requirement (R∗) only by the fact that the function g, defined by g(z):=F (z)+F (−z), is even (see the wrong proof in footnote 7 on the mentioned page). In fact one can easily construct counterexamples, see Section 2. On the other hand we will see in Section 2 that any integrand F that satisfies the requirements
(A∗):= requirements (1)–(3) andF −λ| · |has to be convex onR3,
for someλ >0, can be “approximated” by a family of Lagrangians{F}>0meeting the conditions (A)+(R∗) for sufficiently small , which will be used in the forthcoming article [8]. Now combining property (R∗) of F with the method of “levelling” real valued functions onB, used by Shiffman in Section 6 of [12] and by McShane in¯ Theorem 3.1 in [10], the author was able to carry out a rigorous proof of the “quasi maximum principle” for I- surfaces, Theorem 1.1, which will imply a compactness result for sequences of those, Theorem 1.2. Firstly we need Definition 1.1.Letf∈C0(B)¯ andG⊆Bbe an open subset ofB. We set
mG(f ):=max maxG¯
f −max
∂G f,min
∂G f−min
¯ G
f
(6) and call md(f ):=supG⊆BmG(f )the monotonic diefficiency off, where the supremum is taken over all open subsets G⊆B.
Now letF be a fixed integrand satisfying (A)+(R∗) andg(z):=F (z)+F (−z). By the requirement (R∗) the functionggives rise to a matrixA:=(a1, a2, a3) ∈GL3(R), having chosen three linearly independent critical points a1, a2, a3ofg|S2 arbitrarily. The “quasi maximum principle” forI-surfaces reads (see also Theorem 6.1 on p. 554 in [12]):
Theorem 1.1.Let ϕ ∈C0(∂B,R3)∩H1/2,2(∂B,R3) be prescribed boundary values. Then the corresponding I- surfaceX∗∈Hϕ1,2(B,R3)∩C0(B,¯ R3), i.e. the unique minimizer ofIinHϕ1,2(B,R3), satisfiesmd((AX∗)i)=0for i=1,2,3.
Combining this result with Lemma 1 on p. 719 in [9] one easily obtains the following compactness result:
Theorem 1.2.Let {Xn} be a sequence ofI-surfaces with D(Xn)const,∀n∈N, and with equicontinuous and uniformly bounded boundary values. Then there exists a subsequence{Xnj}such that
Xnj−→ ¯X inC0B,¯ R3
and XnjX¯ inH1,2 B,R3
, (7)
for a surfaceX¯ ∈H1,2(B,R3)∩C0(B,¯ R3)withmd((AX)¯ i)=0,i=1,2,3.
2. Critical points of even functions onS2
This section is devoted to a discussion of the requirement (R∗) on the integrandF. Firstly we sketch a construc- tion of a counterexample of Shiffman’s assertion that any evenC1-function on theS2would possess three linearly independent critical points (see p. 552 in [12]).
To this end we consider a linear transformationA:R3−→R3which possesses exactly three linearly independent unit eigenvectorsa1,a2,a3, such thata3lies in a small neighborhood of the great circleGdetermined bya1anda2. Then we choose some pointb∈G\ {±a1,±a2}neara3and construct some smooth tangent vector fieldV on theS2 which vanishes outside a small neighborhoodUof the shortest arcγconnectinga3withband which induces a global smooth flowφ:R×S2−→S2, by Corollary 10.13 and Theorem 9.5 in [3], taking the pointa3viaγ ontobwithin a certain timet∗>0 and not effecting the points±a1and±a2in particular. Now we only consider the restriction of theC∞-diffeomorphismφ(t∗,·)to some appropriate (closed) hemisphereS, containingUin its interior, and extend it to an uneven diffeomorphismφ˜of theS2simply by reflection at the origin, i.e.
φ(z)˜ :=
φ(t∗, z) z∈S,
−φ(t∗,−z) z∈ −S,
which is well defined due to φ(t, z)≡z∀z∈∂S and∀t ∈R. Then the compositionq◦ ˜φ−1of the quadratic form q(z):= z, Azwithφ˜−1is indeed a smooth even function on theS2whose critical points are exactly the three linearly dependent unit vectorsa1,a2andb, which completes the construction of the asserted counterexample.
On the other hand there holds the following approximation result:
Proposition 2.1.LetF be an integrand satisfying the requirements(A∗), then there exists a family of approximations {F}>0meeting the requirements(A)and additionally(R∗)if <, for some sufficiently small¯ >¯ 0, and with
D2F−→D2F inC0
R3\Bρ(0)
, ∀ρ >0, (8)
∇F−→ ∇F inC0
R3\ {0}
, (9)
F−→F inC0 BR(0)
, ∀R >0, (10)
for0.
Proof. We setg(z):=F (z)+F (−z)and assume thatg|S2 has only critical points on some great circle which we suppose to be theS1without loss of generality, otherwise we were done. Now just arguing in the opposite way as in the above construction of the counterexample we claim the existence of some smooth tangent vector fieldV on theS2 which vanishes outside a small neighborhoodU of some chosen critical pointbofg|S2 and whose induced flowφ, which is globally defined and smooth onR×S2by Corollary 10.13 and Theorem 9.5 in [3], satisfies(φ(t, b))3>0,
∀t >0, and does not effect the antipodal pairs±a1and±a2of two further linearly independent critical pointsa1,a2
ofg|S2. As above we consider now the restriction ofφ(t,·)to some appropriate (closed) hemisphereS, containingU in its interior, and extend it to an uneven smooth flowφ˜on theS2by
φ(t, z)˜ :=
φ(t, z) z∈S,
−φ(t,−z) z∈ −S,
which is well defined due toφ(t, z)≡z∀z∈∂Sand∀t∈R. We extend this flow homogeneously of first degree onto R3, i.e. by setting
¯
φ(t, z):= |z| ˜φ
t, z
|z| forz∈R3\ {0} (11)
and φ(t,¯ 0)≡0, for any t∈R. As we know thatφ˜ is smooth on R×S2 we infer together withφ(t,¯ 0)≡0 that φ¯∈C∞(R×(R3\ {0}),R3)∩C0(R×R3,R3). Now since∂ijφ¯ is uniformly continuous on[−c, c] ×S2, for any c >0, andφ(0,¯ ·)=idR3one easily sees that
∂ijφ(t,¯ ·)|S2−→∂ijφ(0,¯ ·)|S2=0 inC0 S2
, fort→0 andi, j∈ {1,2,3}, where we denote∂i:=∂z∂
i. Now together with the homogeneity of∂ijφ(t,¯ ·)of degree
−1 by (11) one infers immediately:
∂ijφ(t,¯ ·)−→0 inC0
R3\Bρ(0)
, (12)
fort→0 and anyρ >0. Analogously one obtains Dφ(t,¯ ·)|S2−→Dφ(0,¯ ·)|S2 =13 inC0
S2 ,
and together with the homogeneity ofDφ(t,¯ ·)of degree 0 by (11):
Dφ(t,¯ ·)−→13 inC0
R3\ {0}
, (13)
fort→0. And finally one achieves similarly, using the homogeneity ofφ(t,¯ ·)of degree 1 andφ(t,¯ 0)≡0 for any t∈R:
φ(t,¯ ·)−→idB
R(0) inC0 BR(0)
, (14)
fort→0 and anyR >0. Now we setF:=F (φ(,¯ ·)−1)≡F (φ(¯ −,·))onR3, for >0. Then we can immediately infer from the homogeneity of degree 1 ofφ(¯ −,·)and its regularity thatF inherits the properties (1)–(3) fromF. Additionally we see by(φ(, b))¯ 3>0,∀ >0, and by the invariance ofa1, a2∈S1w. r. toφ¯thata1,a2andφ(, b)¯ are three linearly independent critical points of the restrictiong|S2ofg(z):=F(z)+F(−z)=g(φ(¯ −, z))on the S2, for any >0, where we used in the last equality thatφ(¯ −,·)is uneven onR3. Furthermore we calculate
∂iF(z)=
∇Fφ(¯ −, z)
, ∂iφ(¯ −, z) and
∂ijF(z)=
∇Fφ(¯ −, z)
, ∂ijφ(¯ −, z) +
D2Fφ(¯ −, z)
∂jφ(¯ −, z), ∂iφ(¯ −, z) ,
for z=0 and i, j∈ {1,2,3}. Hence, on account of (12), (13) and (14) together with the homogeneity ofD2F of degree−1 and of∇F of degree 0 onR3\ {0}and ofF of degree 1 onR3in combination with (11) and with the uniform continuity ofD2F,∇F andF onS2we obtain the asserted convergences (8), (9) and (10). Now by (8) we conclude that
ξ,
D2F(z)−D2F (z)
ξD2F−D2F
C0(S2)−→0 (15)
for 0, ∀z, ξ ∈S2. Moreover the required convexity of F −λ| · |, for some fixed λ >0, implies the positive semi-definiteness ofD2(F (z)−λ|z|)∀z∈R3\ {0}and thus by a short computation:
ξ, D2F (z)ξ
λ
ξ, D2
|z| ξ
= λ
|z|
|ξ|2−z, ξ2
|z|2 , (16)
∀z∈R3\ {0}and∀ξ ∈R3. Now we fix somez=0, consider the orthogonal decomposition Span(z)⊕Span(z)⊥of R3and note that |z,ξ|z|| is just the length of the orthogonal projectionξ:= |zz|, ξ|zz| ofξ onto Span(z). Now we conclude by the homogeneity of∇F (z)of order 0:
D2F (z) z=0· ∇F (z)=0 forz=0,
showing that Span(ξ)is contained in the kernel ofD2F (z)for anyξ∈R3and also ofD2F(z),∀ >0, by the same reasoning. Now we introduceξ⊥:=ξ−ξ, i.e. the orthogonal projection ofξ onto Span(z)⊥, and obtain together with the symmetry ofD2(F−F )(z):
ξ, D2(F−F )(z) ξ
=
ξ⊥, D2(F−F )(z)ξ⊥
(17)
∀z∈R3\ {0}and∀ξ∈R3. From (15) we infer in particular the existence of some >¯ 0 such that ζ, D2(F−F )(z)ζλ
2
for anyz∈S2andζ∈S2∩Span(z)⊥, if <, and thus together with (17):¯ ξ, D2(F−F )(z)ξ=ξ⊥, D2(F−F )(z)ξ⊥λ
2|ξ⊥|2 for anyz∈S2andξ∈R3, if <. Hence, recalling (16) we achieve¯
ξ, D2F(z)ξ
=
ξ, D2F (z)ξ +
ξ, D2(F−F )(z)ξ
λ−λ
2 |ξ⊥|2=λ 2
|ξ|2− z, ξ2 ,
for anyz∈S2andξ∈R3, if <. Thus we obtain together with the homogeneity of¯ D2F of degree−1:
ξ, D2F(z) ξ
λ
2|z|
|ξ|2−z, ξ2
|z|2 ∀ξ∈R3
and∀z∈R3\ {0}, which is equivalent to the positive semi-definiteness ofD2(F(z)−λ2|z|), for anyz=0, by the second equation in (16). Hence, we obtain the convexity ofF−λ2| · |onR3from the next lemma and thus especially the asserted convexity ofFfor <.¯ 2
Lemma 2.1.Letq∈C0(R3)∩C2(R3\ {0})have a positive semi-definite Hessian pointwise onR3\ {0}, thenq is convex onR3.
Proof. LetH be an arbitrary open halfspace in R3 whose boundary contains the origin. A well known argument yields the convexity ofqonHon account of the requirement of the lemma, i.e. there holds
q
t z1+(1−t )z2
t q(z1)+(1−t )q(z2) ∀t∈ [0,1], (18)
for any pairz1, z2∈H. Now letz∗1, z∗2∈∂H be arbitrarily given. Then we can choose two sequences{zi1},{zi2} ⊂H with zi1→z∗1 and zi2→z∗2, for i→ ∞, and infer from (18) applied to the pairs z1i, zi2 in combination with the continuity ofq onR3the convexity relation (18) in the limit also for the pairz∗1, z∗2. This proves the convexity ofq onR3. 2
3. Preparing propositions
Let F be a fixed integrand meeting (A) and (R∗),g(z):=F (z)+F (−z),a1, a2, a3 three linearly independent critical points ofg|S2andA:=(a1, a2, a3) ∈GL3(R). We choose two vectorsb1,c1, such thatO1:=(a1, b1, c1) ∈ SO(3)and setF:=F◦O1−1,g:=g◦O1−1. We prove
Lemma 3.1.There are real constantsk2andk3such that F
(z1, z2, z3)
−F
(z1,0,0)
k2z2+k3z3 (19)
∀z1, z2, z3∈R.
Proof. SinceO1−1·(1,0,0) =O1 ·(1,0,0) =a1and sincea1is a critical point ofg|S2 we calculate:
∇g
(1,0,0)
= ∇g(a1)·O1−1=r1a1 ·O1 =r1(O1·a1) =r1(1,0,0),
for somer1∈R. Hence,(1,0,0) is a critical point ofg|S2, implying in particular the equations:
0=gz
2
(1,0,0)
=Fz
2
(1,0,0)
−Fz
2
(−1,0,0) , 0=gz
3
(1,0,0)
=Fz
3
(1,0,0)
−Fz
3
(−1,0,0) ,
where we dropped the “ ”-sign. Now using that∇Fis homogeneous of degree 0 onR3\ {0}by (2) we obtain:
Fz
2≡const=:k2, Fz
3≡const=:k3
on thez1-axis except{0}. Furthermore we infer from the convexity ofF∈C2(R3\ {0})forz1=0:
F
(z1, z2, z3)
−F
(z1,0,0)
∇F
(z1,0,0)
, (z1, z2, z3)−(z1,0,0)
=Fz
2
(z1,0,0) z2+Fz
3
(z1,0,0)
z3=k2z2+k3z3, (20)
∀z2, z3∈R. Now lettingz1−→0 in (20) and usingF∈C0(R3)we achieve the assertion (19) also forz1=0. 2 If we choose vectorsb2,c2, andb3,c3, such thatO2:=(b2, a2, c2) ,O3:=(b3, c3, a3) ∈SO(3)and setF2:=
F ◦O2−1,F3:=F ◦O3−1, then we obtain analogously:
F2
(z1, z2, z3)
−F2
(0, z2,0)
constz1+constz3 (21)
and F3
(z1, z2, z3)
−F3
(0,0, z3)
constz1+constz2 (22)
∀z1, z2, z3∈R. Next we need
Definition 3.1.Letϕ∈C0(∂B,R3)∩H1/2,2(∂B,R3)be prescribed boundary values. Then we define M(ϕ):=
{Yn} ⊂C0B,¯ R3
∩H1,2 B,R3
|Yn|∂B−→ϕinC0
∂B,R3 and
m(ϕ):= inf
{Yn}∈M(ϕ)lim inf
n→∞ I Yn
. (23)
Clearly one hasm(ϕ)infH1,2
ϕ (B)∩C0(B)¯ Iand
Proposition 3.1.There exists a minimizing element{Xj}forIinM(ϕ), i.e.{Xj} ∈M(ϕ)satisfies
jlim→∞I Xj
=m(ϕ).
Proof. By the definition ofm(ϕ)we can choose a minimizing sequence{{Yn}j}j∈Nof sequences forIinM(ϕ), i.e.
we have{{Yn}j}j∈N⊂M(ϕ)such that
jlim→∞lim inf
n→∞ I Ynj
=m(ϕ).
We setmj:=lim infn→∞I({Yn}j). For eachj∈Nwe can choose an integern(j )such that I
Yn(j )j
−mj<1
j and Yn(j )j
∂B−ϕ
C0(∂B)<1 j. Now we chooseXj:= {Yn(j )}j ∀j∈Nand see that{Xj} ∈M(ϕ)satisfies
I Xj
−m(ϕ)I Xj
−mj+mj−m(ϕ)−→0. 2
Proposition 3.2.For anyX∈C0(B,¯ R3)∩H1,2(B,R3)there is a mollified family{X} ⊂Cc∞(B1+2δ(0),R3), for ∈(0, δ)and someδ >0, that satisfies:
X−→X inC0B,¯ R3
∩H1,2 B,R3
. (24)
Proof. Due to the continuation theorem for Sobolev functions there is a continuationXˆ ∈H1,2(B1+δ(0),R3)ofX, for someδ >0. An examination of this continuation, explicitly given in [2, p. 256], shows that we also haveXˆ ∈ C0(B1+δ
2(0),R3)on account ofX∈C0(B,¯ R3). Now we use a family{ϕ}of even Dirac kernels, with supp(ϕ)= B(0), to mollifyX:ˆ
X(·):=
B1+δ(0)
ϕ(· −w)X(w)ˆ dw∈Cc∞
B1+2δ(0),R3
for∈(0, δ). Due toXˆ∈H1,2(B1+δ(0),R3)we firstly obtain by [2, p. 108]:
X−XH1,2(B)= X− ˆX|BH1,2(B)−→0 for0.
Moreover, due to supp(ϕ)=B(0)and
B1+δ(0)ϕ(y−w)dw=1,∀y∈ ¯B,∀∈(0, δ), we gain:
X−XC0(B)¯ =max
y∈ ¯B
B1+δ(0)
ϕ(y−w)X(w)ˆ dw− ˆX(y)
=max
y∈ ¯B
B1+δ(0)
ϕ(y−w)X(w)ˆ − ˆX(y) dw
max
y∈ ¯B
B(y)
ϕ(y−w)X(w)ˆ − ˆX(y)dw
max
y∈ ¯B
max
w∈B(y)
X(w)ˆ − ˆX(y)−→0 for0, sinceXˆ is uniformly continuous onB1+δ
2(0), which completes the proof. 2
Next we state a proposition due to McShane in [9, p. 719] (see [11, p. 416], for a detailed proof):
Proposition 3.3.Letϕ∈C0(∂B)be prescribed boundary values and{fn}a sequence inC0(B)¯ ∩H1,2(B)with the following properties:
fn|∂B−→ϕ inC0(∂B), (25)
md fn
−→0 forn→ ∞, (26)
D fn
const ∀n∈N. (27)
Then there exists a subsequence{fnj}and a functionf∗∈C0(B)¯ ∩H1,2(B)such thatmd(f∗)=0and fnj−→f∗ inC0(B).¯
In [7, p. 7], the Lipschitz continuity of the integrandF onR3, with Lip.-const=m2, is derived from its required properties (A). Together with the Hölder inequality one can easily deduce (see [12, p. 548]):
Proposition 3.4.For anyX, X∈H1,2(B,R3)and any open subsetΩ⊆Bthere holds:
IΩ(X)−IΩ(X)(2m2+k)
DΩ(X)+
DΩ(X)
DΩ(X−X). (28)
4. Levelling ofCc∞(R2)-functions
In this section we discuss the process of “levelling” a functionf ∈C∞c (R2)on the unit discB¯ for a given fineness δ >0 (see also [12, p. 553], and [10, p. 558]). To this end let
Z: min
¯ B
f=l0< l1<· · ·< lN< lN+1=max
¯ B
f
be a partition of the interval[minB¯f,maxB¯f]such thatZ:=maxi=1,...,N+1{li−li−1}< δand such thatl1, . . . , lN
are regular values off, which is possible for any choice ofδby Sard’s theorem (see [4, p. 205]).
The levelling process starts on the level l1. Since l1 is a regular value of f ∈ Cc∞(R2) (especially l1 =0) f−1([l1,∞)) is a compact 2-dimensional C∞-manifold with boundary by the implicit function theorem (see [5, p. 303]). Hence,f−1([l1,∞))is locally connected, in particular, and has therefore only a finite number of connected components. Now we consider the (disjoint) unionU+l1 of those connected components off−1([l1,∞))that are con- tained inB, in particular we have¯
f (w) > l1 ∀w∈U˚+l1 and f (w)=l1 ∀w∈∂U+l1, (29) asl1is a regular value off and asf is continuous, and we set
f+l1(w):=
l1 w∈U+l1,
f (w) w∈R2\U+l1. ()
We go on by considering the compactC∞-manifoldf−1((−∞, l1])which again consists of only finitely many con- nected components, and termU−l1 the union of those connected components that are contained inB. By (29) we infer¯ U˚+l1∩U˚−l1= ∅ and therefore
f+l1(w) < l1 ∀w∈U˚−l1 and f+l1(w)=l1 ∀w∈∂U−l1,
again sincel1is a regular value off, by()and asf is continuous, and we set fl1(w):=
l1 w∈U−l1,
f+l1(w) w∈R2\U−l1. ()
Next we apply the same process tofl1 on the levell2and note that for connected componentsP1ofU±l1 andP2of U+l2 we haveP1∩P2= ∅and for connected componentsP1ofU±l1 andP2ofU−l2 we have eitherP1∩P2= ∅or P1P2. After that we apply the process to(fl1)l2on the levell3and so on, until we have performed the last levelling step on the levellN. Thus after 2×N steps we arrive at a finite collection of “level sets”U±lj,j =1, . . . , N, and at a functionfLonR2, that we term the ”levelled” function off, possessing the following properties:
Lemma 4.1.Letf∈Cc∞(R2)and a finenessδbe given arbitrarily. Firstly there holdsU±lj⊂ ¯Band U˚+lj∩U˚−lj= ∅for j=1, . . . , N. Secondly for connected componentsPj ofU±lj andPi ofU+li, withj < i, there holdsPj∩Pi= ∅and for connected componentsPj ofU±lj andPi ofU−li (j < i)there holds eitherPj∩Pi= ∅orPjPi. Furthermore U±lj are compact 2-dimensional C∞-manifolds with boundary and ∂U±lj are closed 1-dimensional C∞-manifolds.
In particular,U±lj consist of only a finite number of connected components and∂U±lj are Lebesgue-measurable with L2(∂U±lj)=0. MoreoverfLsatisfies:
fL∈C0(B)¯ ∩H1,2(B), fL|∂B=f|∂B, md fL|B¯
δ. (30)
Proof. The assertionsU±lj ⊂ ¯B and ˚U+lj ∩U˚−lj = ∅follow immediately from the definition ofU±lj and as thelj are regular values off forj =1, . . . , N. Next one obtains simultaneouslyfL∈C0(B)¯ and the relations between the connected componentsPj ofU±lj andPiofU+liresp.U−li, withj < i, by induction during the finite levelling process.
As the levelslj are regular values off∈Cc∞(R2)the implicit function theorem yields the assertions about the level sets U±lj and their boundaries∂U±lj at once. Furthermore one has to note that manifoldsM are locally connected, thus their connected components are open inMand compact manifolds can only consist of finitely many. Moreover L2(∂U±lj)=0 follows immediately from the implicit function theorem and Proposition 8 of Section 1.11 in [5, p. 101].
Furthermore by construction of the first levelling step we obtainf+l1 ∈H1,1(B)due to Lemma A 6.9 in [2, p. 254], where we have to use that∂U+l1 is a closedC∞-manifold, thus in particular a Lipschitz boundary. Moreover it is also clear that we have∇f+l1∈L2(B,R2)asf+l1 ≡f onR2\U+l1 and∇f+l1≡0 on ˚U+l1 and since∂U+l1especially satisfies L2(∂U+l1)=0. Hence, we havef+l1∈H1,2(B). Now, using that∂U−l1 is a closedC∞-manifold again, especially with L2(∂U−l1)=0 the same reasoning as above yields thatfl1∈H1,2(B)and again using that∂U±l2 is a C∞-manifold just the same reasoning as above yields that(fl1)l2∈H1,2(B). Hence, after 2×Nsteps we arrive atfL∈H1,2(B).
Next, ifU+l1∩∂B= ∅we havef+l1|∂B≡f|∂B, but ifU+l1∩∂B= ∅we obtain by the construction off+l1: f+l1≡l1≡f along∂U+l1∩∂B.
Since this argument holds true for each step of the levelling process we finally see thatfL|∂B≡f|∂B. If we suppose that there exists an open subsetGofB such that maxG¯fL−max∂GfL> δ, then due toZ< δthere would be some levellj∈Zsuch that max∂GfL< lj but maxG¯fL> lj. Hence, together with the continuity offLwe would have on a connected componentG(= ∅) ofG∩(fL)−1((lj,∞))G
fL(w) > lj ∀w∈G and fL(w)=lj ∀w∈∂G,
which implies thatfL≡f on G and G⊂U+lj. Therefore we must have fL≡li on G for some ij by the construction offLand the second part of the assertion of the lemma, which is a contradiction. Similarly one proves
that min∂GfL−minG¯fLδfor all open subsetsGofBagain by the construction offLand the second part of the assertion of the lemma, hence md(fL|B¯)δ. 2
5. Levelling of the components of distorted surfacesAπ
As in Section 3 we consider a fixed integrandF meeting (A) and (R∗), some smooth surfaceπ∈C∞c (R2,R3)and its distortionπ˜ :=Aπ, whereA:=(a1, a2, a3) ∈GL3(R)is defined at the beginning of Section 3. Its components satisfy
˜
πi= ai, π =(Oiπ )i=πii (31)
fori=1,2,3, where we termedπi :=Oiπ. We setm:=mini=1,2,3{minB¯π˜i}andM:=maxi=1,2,3{maxB¯π˜i}and choose a partition
Z:m=l0< l1<· · ·< lN< lN+1=M
of the interval [m, M]of finenessZ< δ, for an arbitrarily givenδ >0, such that the levelslj,j =1, . . . , N, are regular values of the three components π˜i simultaneously. At first we level the first component, i.e. π˜1→(π˜1)L, abbreviate(π1)L:=((π11)L, π21, π31)and prove (see also (6.6) in [12])
Lemma 5.1.For an arbitraryπ∈Cc∞(R2,R3)there holds:
F(π )F O1−1
π1L
. (32)
Proof. We abbreviate π:=π1=O1π. It will suffice to consider only the first step of the levelling process on the levell1 applied toπ1= ˜π1. LetD be the open kernel of a connected componentD¯ of the level set U+l1 which is a compact C∞-manifold with boundary by Lemma 4.1. Now we choose an atlasA:= {(Vj, ψj)j=0,...,k}of D¯ such that ∂D⊂k
j=1Vj and a subordinate partition of unity{ηj}j=0,...,k. Furthermore a careful examination of the implicit function theorem (see [5, p. 303]) shows that we may arrange the chartsψj:Br+
j(0)−→∼= Vj∩ ¯Dsuch thatγj:=ψj|[−rj,rj]:[−rj, rj]−→∼= Vj∩∂Dyields a parametrization ofVj∩∂Dwith respect to its arc length, for j =1, . . . , k, implying that((γj)2,−(γj)1)yields an outward pointing unit normal fieldνj alongVj∩∂D. Since we haveπ1≡l1along∂Dwe infer:
d dsπ1
γj(s)
≡0 ∀s∈ [−rj, rj], (33)
for j =1, . . . , k. Now we consider the vector fieldh(z1, z2, z3):=(−z2,0,0)onR3. Firstly we note that rot h≡ (0,0,1), thus settingN:=(N1, N2, N3):=πu ∧πv we haveN3= roth(π), πu∧πvonR2. Furthermore we set w:=(h(π), πv,−h(π), πu)∈Cc∞(R2,R2). Usingπuv =πvu due to Schwarz one easily calculates:
divw=
roth(π), πu ∧πv
onR2.
Now combining this with the divergence theorem for Lipschitz boundaries (see [2, p. 252]) and (33) we arrive at:
D
N3dudv=
D
divwdudv=
∂D
w, νds= k j=1
∂D∩Vj
ηjw, νjds
= k j=1
rj
−rj
(ηjw1) γj(s)
(γj)2−(ηjw2) γj(s)
(γj)1ds
= k j=1
rj
−rj
ηj
−π2(π1)v
γj(s)
(γj)2−
ηjπ2(π1)u
γj(s) (γj)1ds
