We combine a Korn type inequality with Widman’s hole filling technique to prove the interior regularity of minimizers for energies occurring in General Relativity. In addition we provide a new variant of this Korn type inequality valid for the nonquadratic case

Fuchs, Martin

Schirra, Oliver

Acronym

Universita¨t des SaarlandesUN IV E R SIT A SSAR A V I E NS ISFachrichtung 6.1 – MathematikPreprint Nr. 238An Application Of A Generalized Korn Inequality ToEnergies Studied In General Relativity Giving TheSmoothness Of MinimizersMartin Fuchs and Oliver SchirraSaarbru¨cken 2009Fachrichtung 6.1 – Mathematik Preprint No. 238Universita¨t des Saarlandes submitted: 10th of August, 2009An Application Of A Generalized Korn Inequality ToEnergies Studied In General Relativity Giving TheSmoothness Of MinimizersMartin FuchsSaarland UniversityDep. of MathematicsP.O. Box 15 11 50D-66041 Saarbru¨ckenGermanyfuchs@math.uni-sb.deOliver SchirraSaarland UniversityDep. of MathematicsP.O. Box 15 11 50D-66041 Saarbru¨ckenGermanyolli@math.uni-sb.deEdited byFR 6.1 – MathematikUniversita¨t des SaarlandesPostfach 15 11 5066041 Saarbru¨ckenGermanyFax: + 49 681 302 4443e-Mail: preprint@math.uni-sb.deWWW: http://www.math.uni-sb.de/AMS Classification: 49 N 60, 83 C.Keywords: generalized Korn’s inequality, regularity of minimizers.AbstractWe combine a Korn type inequality with Widman’s hole filling technique to provethe interior regularity of minimizers for energies occurring in General Relativity.In addition we provide a new variant of this Korn type inequality valid for thenonquadratic case.In a recent paper Dain [Da] proves Korn type inequalities in which the symmetric gradientε(u) =12(∂iuj + ∂jui)1≤i,j≤nof vector fields u : Ω → Rn defined on a bounded Lipschitz domain Ω ⊂ Rn is replacedby its trace free partεD(u) = ε(u)−1n(div u) 1,where 1 denotes the unit-matrix. More precisely, it is shown that in case n ≥ 3 it holds(1) ‖u‖2W 12(Ω) ≤ C∫Ω|u|2 dx+∫Ω∣∣εD(u)∣∣2 dxfor all functions u from the Sobolev space W 12 (Ω;Rn) (see [Ad] for a definition), whereasfor n = 2 we have(2)∫Ω|∇ϕ|2 dx ≤ 2∫Ω∣∣εD(ϕ)∣∣2 dxvalid now for fields ϕ ∈ C∞0 (Ω;R2). By approximation (2) extends to the space◦W12(Ω;R2)of Sobolev functions with zero trace, and another application of (2) yields(3) X :={u ∈ L2loc(Ω;R2) : εD(u) ∈ L2loc(Ω; S2)}=W 12,loc(Ω;R2).(Here S2 denotes the space of symmetric (2 × 2)-matrices.) In fact, if w ∈ X, then weconsider η w(ρν), where η is a localization function and w(ρν) is a sequence of mollifications.Since (2) is true for η w(ρν), we easily obtain that w belongs to W 12,loc(Ω;R2). The needfor inequalities of the form (1) or (2) origins from the question, if functionals of the form(4) E(u,Ω) :=∫Ω(∣∣εD(u)∣∣2 − g · u) dxstudied in General Relativity (see [BI]) are coercive under suitable boundary conditions.Due to (1) and (2) the answer is positive, and the goal of our paper is the investigation1of the smoothness properties of minimizers of functionals being more general than Eintroduced in (4) at least for the 2D–case. To be precise, consider an energy densityf : S2 → [0,∞) of class C2 such that with constants λ,Λ > 0 it holds(5) λ |σ|2 ≤ D2f(τ)(σ, σ) ≤ Λ |σ|2for all σ, τ ∈ S2. Suppose further that Ω is a domain in R2 and that we are given functions(6) u0 ∈W12 (Ω;R2), g ∈ L∞(Ω;R2) .We then define the energy(7) I[u,Ω] :=∫Ω[f(εD(u))− g · u]dxon the class(8) K := u0+◦W12(Ω;R2)and getTHEOREM 1. Let (5) - (8) hold. Then we have:a) The problem I[·,Ω]→ min on K admits a unique solution u ∈ K.b) This solution belongs to the space C1,α(Ω;R2) for some α ∈ (0, 1).From the proof we will deduceCorollary 1. Let (5) hold and consider u ∈ X, which locally minimizes the functional Ifrom (7) with g ∈ L∞loc(Ω;R2), i.e. I[u,Ω′] <∞ and I[u,Ω′] ≤ I[v,Ω′] for all v ∈ X suchthat spt(u− v) ⋐ Ω′, where Ω′ is any subdomain of Ω such that Ω′ is a compact subset ofΩ. Then the first derivatives of u are locally Ho¨lder continuous in Ω.Proof of Theorem 1Step 1: Existence and uniqueness of the minimizer2For any function v ∈ K it holds (on account of (5))∫Ωf(εD(v))dx ≥ c∫Ω∣∣εD(v)∣∣2 dx− |Ω|≥ c∫Ω∣∣εD(v − u0)∣∣2 dx−∫Ω|∇u0|2 dx− |Ω|(2)≥ c∫Ω|∇(v − u0)|2 dx−∫Ω|∇u0|2 dx− |Ω|≥ c∫Ω|∇v|2 dx−∫Ω|∇u0|2 dx− |Ω| ,where here and in what follows c represents a positive constant being independent of v,whose value may change from line to line. With Poincare´’s and Young’s inequality weobtain for any δ > 0∣∣∣∣∣∣∫Ωg · v dx∣∣∣∣∣∣ ≤ δ∫Ω|v − u0|2 dx+ c(δ)∫Ω|g|2 dx+ c∫Ω|g||u0| dx≤ δc∫Ω|∇(v − u0)|2 dx+ c(δ)∫Ω|g|2 dx+ c∫Ω|u0||g| dx ,so that appropriate choice of δ implies(9) I[v,Ω] ≥ c∫Ω|∇v|2 dx− c(|Ω|, λ,Λ, ‖g‖L2, ‖u0‖W 12),and alternatively we can replace∫Ω|∇v|2 dx on the r.h.s. by∫Ω|∇v −∇u0|2 dx. If {un}denotes on I-minimizing sequence in K, then (9) clearly impliessupn‖un‖W 12(Ω) <∞ ,so that after passing to a subsequence we can assume un ⇁: u in W12 (Ω;R2), un → uin L2(Ω;R2) for a function u ∈ K. The lower semicontinuity of I[·,Ω] yields that u isa solution of the minimization problem. Suppose that u˜ ∈ K is a second solution. IfεD(u) 6= εD(u˜) holds on a set of positive measure, then the strict inequalityf(εD(u+ u˜2))<12f(εD(u))+12f(εD(u˜))3is true on this set which leads to the contradictionI[u+ u˜2,Ω]< infKI .We therefore have εD(u − u˜) = 0 a.e. on Ω and (2) shows u = u˜. Thus part a) ofthe Theorem is proved. We note that for this step it is enough to have g ∈ L2(Ω;R2)(compare (6)) which is also sufficient to carry out the next step.Step 2: Higher weak differentiability of the minimizer uWe claim that(10) u ∈W 22,loc(Ω;R2)is true, which by Sobolev’s theorem implies that u is locally Ho¨lder continuous on Ω forany exponent α < 1. For proving (10), we fix a coordinate direction i, i = 1, 2, and let∆h denote the difference quotient of functions in this direction. From the minimality ofu we deduce(11)∫Ω(Df(εD(u)): εD(v)− g · v)dx = 0for all v ∈◦W 12(Ω;R2). We let v := ∆−h(η2∆hu) with η ∈ C∞0 (Ω), 0 ≤ η ≤ 1, and deducefrom (11)(12)∫Ω∆h(Df(εD(u))): εD(η2∆hu) dx = −∫Ωg ·∆−h(η2∆hu) dx .AbbreviatingBx :=1∫ΩD2f(εD(u)(x) + th εD(∆hu)(x))dtwe get∆h(Df(εD(u)))= Bx(εD(∆hu) , ·),4hence(Zh := εD(η2∆hu)− η2εD(∆hu))l.h.s. of (12) =∫ΩBx(εD(∆hu), εD(η2∆hu))dx=∫ΩBx(εD(∆hu), εD(∆hu))η2 dx+∫ΩBx(εD(∆hu), Zh)dx(5)≥ λ∫Ωη2∣∣εD(∆hu)∣∣2 dx− Λ∫Ω|εD(∆hu)|2η|∇η||∆hu| dx≥λ2∫Ωη2∣∣εD(∆hu)∣∣2 dx− c∫Ω|∇η|2 |∆hu|2 dx .Therefore (12) implies(13)λ2∫Ωη2∣∣εD(∆hu)∣∣2 dx ≤∫Ω|g||∆−h(η2∆hu)| dx+ c∫Ω|∇η|2 |∆hu|2 dx .If |h| < dist (spt η, ∂Ω), then we quote Lemma 7.23 of [GT] and get(14)∫Ω|∇η|2 |∆hu|2 dx ≤ ‖∇η‖2L∞(Ω)∫Ω|∇u|2 dx .At the same time we have with Young’s inequality∫Ω|g|∣∣∆−h(η2∆hu)∣∣2 dx ≤ δ∫Ω∣∣∆−h(η2∆hu)∣∣2 dx+ c(δ)∫Ω|g|2 dx≤ δ∫Ω∣∣∇ (η2∆hu)∣∣2 dx+ c(δ)∫Ω|g|2 dx(2)≤ 2δ∫Ω∣∣εD(η2∆hu)∣∣2 dx+ c(δ)∫Ω|g|2 dx≤ 2δ∫Ωη2∣∣εD(∆hu)∣∣2 dx+c(δ)∫Ω|g|2 dx+ ‖∇η‖2L∞(Ω)∫Ω|∇u|2 dx ,where we also used Lemma 7.23 of [GT] again. Combining (13) with (14) and the aboveestimate, we deduce after appropriate choice of δ:(15)∫Ω∣∣∆hεD(u)∣∣2 η2 dx ≤ c∫Ω|g|2 dx+ ‖∇η‖2L∞(Ω)∫Ω|∇u|2 dx .5Since (15) is valid for all |h| ≪ 1, we see that εD(u) is of class W 12,loc(Ω; S2). ThereforeεD(∂iu) = ∂iεD(u) is in L2loc(Ω; S2) and we infer from (3) that ∂iu ∈ W12,loc(Ω;R2) is truefor i = 1, 2. This proves our claim (10).Step 3: C1,α–regularity of the minimizer uHere we are going to prove part b) of the Theorem applying the hole filling techniquegoing back to Widman’s work [Wi]. Recall that according to our assumption (6) g is abounded function but we remark that the following arguments can be carried out underthe weaker hypothesis g ∈ L2(Ω;R2) together with g ∈ L2,µloc (Ω;R2) for some µ > 0, whereL2,µloc (. . .) is the local Morrey space (see, e.g. [Gi], Chapter III). From (10) and (11) we get(from now on summation w.r.t. k = 1, 2)(16)∫Ω∂k(Df(εD(u))): εD(η2∂k[u− Px])dx = −∫Ωg · ∂k(η2∂k[u− Px])dxvalid for η ∈ C∞0 (Ω), 0 ≤ η ≤ 1, and any (2× 2)–matrix P .Letting Wk := εD(η2∂k[u − Px])− η2εD(∂ku) and observing |Wk| ≤ cη|∇η||∇u− P |, wededuce from (16) (recalling also (5))∫Ωη2∣∣∇εD(u)∣∣2 dx≤ c∫Ωη|∇εD(u)||∇η||∇u− P | dx+∫Ω|g||∇η|η|∇u− P | dx+∫Ω|g|η2|∇2u| dx ,and if we apply Young’s inequality we find(17)∫Ωη2∣∣∇εD(u)∣∣2 dx ≤ c∫Ω|∇η|2 |∇u− P |2 dx+∫Ω|g|2η2 dx+∫Ω|g|η2|∇2u| dx .Let us fix a coordinate direction k. We have∫Ωη2 |∇∂ku|2 dx =∫Ωη2 |∇(∂ku− Pk)|2 dx≤ c∫Ω∣∣∇ (η2[∂ku− Pk])∣∣2 dx+∫Ω|∇η|2 |∇u− P |2 dx(2)≤ c∫Ω∣∣εD (η2[∂ku− Pk])∣∣2 dx+∫Ω|∇η|2 |∇u− P |2 dx6≤ c∫Ωη2∣∣εD(∂ku)∣∣2 dx+∫Ω|∇η|2 |∇u− P |2 dx≤ c∫Ωη2∣∣∇εD(u)∣∣2 dx+∫Ω|∇η|2 |∇u− P |2 dx , i.e.(18)∫Ωη2∣∣∇2u∣∣2 dx ≤ c∫Ωη2∣∣∇εD(u)∣∣2 dx+∫Ω|∇η|2 |∇u− P |2 dx ,and we may insert (17) into the r.h.s. of (18) with the result(19)∫Ωη2∣∣∇2u∣∣2 dx ≤ c∫Ω|∇η|2 |∇u− P |2 dx+∫Ω|g|2η2 dx+∫Ω|g|η2|∇2u| dx .If we use Young’s inequality for the last integral in (19), it is shown that(20)∫Ωη2∣∣∇2u∣∣2 dx ≤ c∫Ωη2 |g|2 dx+∫Ω|∇η|2 |∇u− P |2 dx .In a final step we fix a disc B2R(x0) ⋐ Ω and choose η such that η = 1 on BR(x0),spt η ⊂ B2R(x0) and |∇η| ≤ c/R. Inequality (20) implies(21)∫BR(x0)∣∣∇2u∣∣2 dx ≤ c[R2 +R−2∫TR(x0)|∇u− P |2 dx],where TR(x0) := B2R(x0) − BR(x0). Choosing P as the mean value of ∇u over the ringTR(x0) and applying Poincare´’s inequality, we deduce from (21)∫BR(x0)∣∣∇2u∣∣2 dx ≤ c[R2 +∫TR(x0)∣∣∇2u∣∣2 dx].Adding c∫BR(x0)|∇2u|2dx on both sides we find∫BR(x0)∣∣∇2u∣∣2 dx ≤ cc+ 1R2 +cc + 1∫B2R(x0)∣∣∇2u∣∣2 dx ,and a standard iteration argument shows that∫Br(x0)|∇2u|2dx grows at most like rκ forsome κ > 0 which proves the claim of part b) of the Theorem.7REMARK 1. 1.) Having established the continuity of ∇u it is possible to prove theinterior Ho¨lder continuity of ∇u for any exponent α ∈ (0, 1) by combining the freez-ing technique with Campanato type estimates valid for constant coefficient systemsinvolving εD which have been established in [Sc].2.) Using these Campanato estimates one can also prove partial C1–regularity for I–minimizers now on domains Ω ⊂ Rn, n ≥ 3, and with iterative applicationof the difference quotient technique one obtains C∞–regularity for minimizers of∫Ω[A(εD(u), εD(u)) − g · u] dx, Ω ⊂ Rn, n ≥ 2, provided A is a positive definite,symmetric bilinear form on Sn (:= space of symmetric (n × n)–matrices) havingconstant coefficients and g is a smooth function. For details we again refer to [Sc].Let us look at the minimization problem(22) J [u,Ω] :=∫ΩF (εD(u)) dx→ min in u0+◦W1p(Ω;R2)with u0 from the space W1p (Ω;R2) and with integrand F : S2 → R being strictly convexsatisfying in addition the growth condition(23) a|σ|p − b ≤ F (σ) ≤ A|σ|p +B, σ ∈ S2 ,with constants a, A > 0, b, B ≥ 0. The following Korn type inequality provides theexistence of a unique solution to (22) under the hypothesis (23).Proposition 1. Let Ω ⊂ R2 denote a bounded Lipschitz domain and let p ∈ (1,∞).Then there is a constant Cp(Ω) such that(24) ‖∇v‖Lp(Ω) ≤ Cp(Ω)‖εD(v)‖Lp(Ω)holds for any v ∈◦W1p(Ω;R2).Proof: It is sufficient to consider the case v ∈ C∞0 (Ω;R2). From (26) in [Da] we deduce∂iεDij(v) =12∆vj, j = 1, 2 ,i.e. h := vj ∈◦W1p(Ω) solves(25) −∆h = divH on Ωwith H := (−2εDij(v))1≤i≤2 of course from the space Lp(Ω;R2). Applying standard Lp–arguments to equation (25) (see, e.g. [Me]) we end up with‖∇h‖Lp(Ω) ≤ Cp(Ω)‖H‖Lp(Ω)8and our claim (24) follows.If (23) is replaced by the stronger requirement that F is a smooth p–elliptic integrand,then we believe that the result of the Theorem extends to the p– case by combining theKorn type inequality (24) with suitable regularity techniques as applied for example in[BFZ].References[Ad] Adams, R. A., Sobolev spaces. Academic Press, New York-San Francisco-London(1975).[BI] Bartnik, R., Isenberg, J., The constraint equation. In: Chrus´ciel, P. T., Friedrich,H., (eds.), The Einstein equations and large scale behaviour of gravitational fields,pp.1–38. Birkha¨user Verlag, Basel-Boston-Berlin (2004).[BFZ] Bildhauer, M., Fuchs, M., Zhong, X., A lemma on the higher integrability of func-tions with applications to the regularity theory of two dimensional generalizedNewtonian fluids. Manus. Math. 116 (2), 135–156 (2005).[Da] Dain, S., Generalized Korn’s inequality and conformal Killing vectors.Calc. Var. 25 (4), 535–540 (2006).[Gi] Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear el-liptic systems. Princeton University Press (1983).[GT] Gilbarg, D., Trudinger, N. S., Elliptic partial differential equations of secondorder. Springer-Verlag, Berlin-Heidelberg-New York (1998).[Me] Meyers, N. G., An Lp–estimate for the gradient of solutions of second orderelliptic divergence equations. Ann. SNS Pisa III (17), 189–206 (1963).[Sc] Schirra, O., PhD-thesis.[Wi] Widman, K. O., Ho¨lder continuity of solutions of elliptic systems.Manus. Math. 5, 299–308 (1971).9

An application of a generalized Korn inequality to energies studied in general relativity giving the smoothness of minimizers

Scientific documents from the Saarland University,

Universita¨t des Saarlandes
U
N I
V E R SIT A S
S
A
R A V I E N
S I
S
Fachrichtung 6.1 – Mathematik
Preprint Nr. 238
An Application Of A Generalized Korn Inequality To
Energies Studied In General Relativity Giving The
Smoothness Of Minimizers
Martin Fuchs and Oliver Schirra
Saarbru¨cken 2009
Fachrichtung 6.1 – Mathematik Preprint No. 238
Universita¨t des Saarlandes submitted: 10th of August, 2009
An Application Of A Generalized Korn Inequality To
Energies Studied In General Relativity Giving The
Smoothness Of Minimizers
Martin Fuchs
Saarland University
Dep. of Mathematics
P.O. Box 15 11 50
D-66041 Saarbru¨cken
Germany
fuchs@math.uni-sb.de
Oliver Schirra
Saarland University
Dep. of Mathematics
P.O. Box 15 11 50
D-66041 Saarbru¨cken
Germany
olli@math.uni-sb.de
Edited by
FR 6.1 – Mathematik
Universita¨t des Saarlandes
Postfach 15 11 50
66041 Saarbru¨cken
Germany
Fax: + 49 681 302 4443
e-Mail: preprint@math.uni-sb.de
WWW: http://www.math.uni-sb.de/
AMS Classification: 49 N 60, 83 C.
Keywords: generalized Korn’s inequality, regularity of minimizers.
Abstract
We combine a Korn type inequality with Widman’s hole filling technique to prove
the interior regularity of minimizers for energies occurring in General Relativity.
In addition we provide a new variant of this Korn type inequality valid for the
nonquadratic case.
In a recent paper Dain [Da] proves Korn type inequalities in which the symmetric gradient
ε(u) =
1
2
(
∂iu
j + ∂ju
i
)
1≤i,j≤n
of vector fields u : Ω → Rn defined on a bounded Lipschitz domain Ω ⊂ Rn is replaced
by its trace free part
εD(u) = ε(u)−
1
n
(div u) 1,
where 1 denotes the unit-matrix. More precisely, it is shown that in case n ≥ 3 it holds
(1) ‖u‖2W 1
2
(Ω) ≤ C

∫
Ω
|u|2 dx+
∫
Ω
∣∣εD(u)∣∣2 dx


for all functions u from the Sobolev space W 12 (Ω;R
n) (see [Ad] for a definition), whereas
for n = 2 we have
(2)
∫
Ω
|∇ϕ|2 dx ≤ 2
∫
Ω
∣∣εD(ϕ)∣∣2 dx
valid now for fields ϕ ∈ C∞0 (Ω;R
2). By approximation (2) extends to the space
◦
W12(Ω;R
2)
of Sobolev functions with zero trace, and another application of (2) yields
(3) X :=
{
u ∈ L2loc(Ω;R
2) : εD(u) ∈ L2loc(Ω; S
2)
}
=W 12,loc(Ω;R
2).
(Here S2 denotes the space of symmetric (2 × 2)-matrices.) In fact, if w ∈ X, then we
consider η w(ρν), where η is a localization function and w(ρν) is a sequence of mollifications.
Since (2) is true for η w(ρν), we easily obtain that w belongs to W 12,loc(Ω;R
2). The need
for inequalities of the form (1) or (2) origins from the question, if functionals of the form
(4) E(u,Ω) :=
∫
Ω
(∣∣εD(u)∣∣2 − g · u) dx
studied in General Relativity (see [BI]) are coercive under suitable boundary conditions.
Due to (1) and (2) the answer is positive, and the goal of our paper is the investigation
1
of the smoothness properties of minimizers of functionals being more general than E
introduced in (4) at least for the 2D–case. To be precise, consider an energy density
f : S2 → [0,∞) of class C2 such that with constants λ,Λ > 0 it holds
(5) λ |σ|2 ≤ D2f(τ)(σ, σ) ≤ Λ |σ|2
for all σ, τ ∈ S2. Suppose further that Ω is a domain in R2 and that we are given functions
(6) u0 ∈W
1
2 (Ω;R
2), g ∈ L∞(Ω;R2) .
We then define the energy
(7) I[u,Ω] :=
∫
Ω
[
f(εD(u))− g · u
]
dx
on the class
(8) K := u0+
◦
W
1
2(Ω;R
2)
and get
THEOREM 1. Let (5) - (8) hold. Then we have:
a) The problem I[·,Ω]→ min on K admits a unique solution u ∈ K.
b) This solution belongs to the space C1,α(Ω;R2) for some α ∈ (0, 1).
From the proof we will deduce
Corollary 1. Let (5) hold and consider u ∈ X, which locally minimizes the functional I
from (7) with g ∈ L∞loc(Ω;R
2), i.e. I[u,Ω′] <∞ and I[u,Ω′] ≤ I[v,Ω′] for all v ∈ X such
that spt(u− v) ⋐ Ω′, where Ω′ is any subdomain of Ω such that Ω′ is a compact subset of
Ω. Then the first derivatives of u are locally Ho¨lder continuous in Ω.
Proof of Theorem 1
Step 1: Existence and uniqueness of the minimizer
2
For any function v ∈ K it holds (on account of (5))
∫
Ω
f
(
εD(v)
)
dx ≥ c

∫
Ω
∣∣εD(v)∣∣2 dx− |Ω|


≥ c

∫
Ω
∣∣εD(v − u0)∣∣2 dx−
∫
Ω
|∇u0|
2 dx− |Ω|


(2)
≥ c

∫
Ω
|∇(v − u0)|
2 dx−
∫
Ω
|∇u0|
2 dx− |Ω|


≥ c

∫
Ω
|∇v|2 dx−
∫
Ω
|∇u0|
2 dx− |Ω|

 ,
where here and in what follows c represents a positive constant being independent of v,
whose value may change from line to line. With Poincare´’s and Young’s inequality we
obtain for any δ > 0
∣∣∣∣∣∣
∫
Ω
g · v dx
∣∣∣∣∣∣ ≤ δ
∫
Ω
|v − u0|
2 dx+ c(δ)
∫
Ω
|g|2 dx+ c
∫
Ω
|g||u0| dx
≤ δc
∫
Ω
|∇(v − u0)|
2 dx+ c(δ)
∫
Ω
|g|2 dx+ c
∫
Ω
|u0||g| dx ,
so that appropriate choice of δ implies
(9) I[v,Ω] ≥ c
∫
Ω
|∇v|2 dx− c
(
|Ω|, λ,Λ, ‖g‖L2, ‖u0‖W 1
2
)
,
and alternatively we can replace
∫
Ω
|∇v|2 dx on the r.h.s. by
∫
Ω
|∇v −∇u0|
2 dx. If {un}
denotes on I-minimizing sequence in K, then (9) clearly implies
sup
n
‖un‖W 1
2
(Ω) <∞ ,
so that after passing to a subsequence we can assume un ⇁: u in W
1
2 (Ω;R
2), un → u
in L2(Ω;R2) for a function u ∈ K. The lower semicontinuity of I[·,Ω] yields that u is
a solution of the minimization problem. Suppose that u˜ ∈ K is a second solution. If
εD(u) 6= εD(u˜) holds on a set of positive measure, then the strict inequality
f
(
εD
(
u+ u˜
2
))
<
1
2
f
(
εD(u)
)
+
1
2
f
(
εD(u˜)
)
3
is true on this set which leads to the contradiction
I
[
u+ u˜
2
,Ω
]
< inf
K
I .
We therefore have εD(u − u˜) = 0 a.e. on Ω and (2) shows u = u˜. Thus part a) of
the Theorem is proved. We note that for this step it is enough to have g ∈ L2(Ω;R2)
(compare (6)) which is also sufficient to carry out the next step.
Step 2: Higher weak differentiability of the minimizer u
We claim that
(10) u ∈W 22,loc(Ω;R
2)
is true, which by Sobolev’s theorem implies that u is locally Ho¨lder continuous on Ω for
any exponent α < 1. For proving (10), we fix a coordinate direction i, i = 1, 2, and let
∆h denote the difference quotient of functions in this direction. From the minimality of
u we deduce
(11)
∫
Ω
(
Df
(
εD(u)
)
: εD(v)− g · v
)
dx = 0
for all v ∈
◦
W 12(Ω;R
2). We let v := ∆−h(η
2∆hu) with η ∈ C
∞
0 (Ω), 0 ≤ η ≤ 1, and deduce
from (11)
(12)
∫
Ω
∆h
(
Df
(
εD(u)
))
: εD(η2∆hu) dx = −
∫
Ω
g ·∆−h(η
2∆hu) dx .
Abbreviating
Bx :=
1∫
Ω
D2f
(
εD(u)(x) + th εD(∆hu)(x)
)
dt
we get
∆h
(
Df
(
εD(u)
))
= Bx
(
εD(∆hu) , ·
)
,
4
hence
(
Zh := ε
D(η2∆hu)− η
2εD(∆hu)
)
l.h.s. of (12) =
∫
Ω
Bx
(
εD(∆hu), ε
D
(
η2∆hu
))
dx
=
∫
Ω
Bx
(
εD(∆hu), ε
D(∆hu)
)
η2 dx+
∫
Ω
Bx
(
εD(∆hu), Zh
)
dx
(5)
≥ λ
∫
Ω
η2
∣∣εD(∆hu)∣∣2 dx− Λ
∫
Ω
|εD(∆hu)|2η|∇η||∆hu| dx
≥
λ
2
∫
Ω
η2
∣∣εD(∆hu)∣∣2 dx− c
∫
Ω
|∇η|2 |∆hu|
2 dx .
Therefore (12) implies
(13)
λ
2
∫
Ω
η2
∣∣εD(∆hu)∣∣2 dx ≤
∫
Ω
|g||∆−h
(
η2∆hu
)
| dx+ c
∫
Ω
|∇η|2 |∆hu|
2 dx .
If |h| < dist (spt η, ∂Ω), then we quote Lemma 7.23 of [GT] and get
(14)
∫
Ω
|∇η|2 |∆hu|
2 dx ≤ ‖∇η‖2L∞(Ω)
∫
Ω
|∇u|2 dx .
At the same time we have with Young’s inequality∫
Ω
|g|
∣∣∆−h(η2∆hu)∣∣2 dx ≤ δ
∫
Ω
∣∣∆−h(η2∆hu)∣∣2 dx+ c(δ)
∫
Ω
|g|2 dx
≤ δ
∫
Ω
∣∣∇ (η2∆hu)∣∣2 dx+ c(δ)
∫
Ω
|g|2 dx
(2)
≤ 2δ
∫
Ω
∣∣εD(η2∆hu)∣∣2 dx+ c(δ)
∫
Ω
|g|2 dx
≤ 2δ
∫
Ω
η2
∣∣εD(∆hu)∣∣2 dx
+c(δ)

∫
Ω
|g|2 dx+ ‖∇η‖2L∞(Ω)
∫
Ω
|∇u|2 dx

 ,
where we also used Lemma 7.23 of [GT] again. Combining (13) with (14) and the above
estimate, we deduce after appropriate choice of δ:
(15)
∫
Ω
∣∣∆hεD(u)∣∣2 η2 dx ≤ c

∫
Ω
|g|2 dx+ ‖∇η‖2L∞(Ω)
∫
Ω
|∇u|2 dx

 .
5
Since (15) is valid for all |h| ≪ 1, we see that εD(u) is of class W 12,loc(Ω; S
2). Therefore
εD(∂iu) = ∂iε
D(u) is in L2loc(Ω; S
2) and we infer from (3) that ∂iu ∈ W
1
2,loc(Ω;R
2) is true
for i = 1, 2. This proves our claim (10).
Step 3: C1,α–regularity of the minimizer u
Here we are going to prove part b) of the Theorem applying the hole filling technique
going back to Widman’s work [Wi]. Recall that according to our assumption (6) g is a
bounded function but we remark that the following arguments can be carried out under
the weaker hypothesis g ∈ L2(Ω;R2) together with g ∈ L2,µloc (Ω;R
2) for some µ > 0, where
L2,µloc (. . .) is the local Morrey space (see, e.g. [Gi], Chapter III). From (10) and (11) we get
(from now on summation w.r.t. k = 1, 2)
(16)
∫
Ω
∂k
(
Df
(
εD(u)
))
: εD
(
η2∂k[u− Px]
)
dx = −
∫
Ω
g · ∂k
(
η2∂k[u− Px]
)
dx
valid for η ∈ C∞0 (Ω), 0 ≤ η ≤ 1, and any (2× 2)–matrix P .
Letting Wk := ε
D(η2∂k[u − Px])− η
2εD(∂ku) and observing |Wk| ≤ cη|∇η||∇u− P |, we
deduce from (16) (recalling also (5))
∫
Ω
η2
∣∣∇εD(u)∣∣2 dx
≤ c

∫
Ω
η|∇εD(u)||∇η||∇u− P | dx+
∫
Ω
|g||∇η|η|∇u− P | dx+
∫
Ω
|g|η2|∇2u| dx

 ,
and if we apply Young’s inequality we find
(17)
∫
Ω
η2
∣∣∇εD(u)∣∣2 dx ≤ c

∫
Ω
|∇η|2 |∇u− P |2 dx+
∫
Ω
|g|2η2 dx+
∫
Ω
|g|η2|∇2u| dx

 .
Let us fix a coordinate direction k. We have∫
Ω
η2 |∇∂ku|
2 dx =
∫
Ω
η2 |∇(∂ku− Pk)|
2 dx
≤ c

∫
Ω
∣∣∇ (η2[∂ku− Pk])∣∣2 dx+
∫
Ω
|∇η|2 |∇u− P |2 dx


(2)
≤ c

∫
Ω
∣∣εD (η2[∂ku− Pk])∣∣2 dx+
∫
Ω
|∇η|2 |∇u− P |2 dx


6
≤ c

∫
Ω
η2
∣∣εD(∂ku)∣∣2 dx+
∫
Ω
|∇η|2 |∇u− P |2 dx


≤ c

∫
Ω
η2
∣∣∇εD(u)∣∣2 dx+
∫
Ω
|∇η|2 |∇u− P |2 dx

 , i.e.
(18)
∫
Ω
η2
∣∣∇2u∣∣2 dx ≤ c

∫
Ω
η2
∣∣∇εD(u)∣∣2 dx+
∫
Ω
|∇η|2 |∇u− P |2 dx

 ,
and we may insert (17) into the r.h.s. of (18) with the result
(19)
∫
Ω
η2
∣∣∇2u∣∣2 dx ≤ c

∫
Ω
|∇η|2 |∇u− P |2 dx+
∫
Ω
|g|2η2 dx+
∫
Ω
|g|η2|∇2u| dx

 .
If we use Young’s inequality for the last integral in (19), it is shown that
(20)
∫
Ω
η2
∣∣∇2u∣∣2 dx ≤ c

∫
Ω
η2 |g|2 dx+
∫
Ω
|∇η|2 |∇u− P |2 dx

 .
In a final step we fix a disc B2R(x0) ⋐ Ω and choose η such that η = 1 on BR(x0),
spt η ⊂ B2R(x0) and |∇η| ≤ c/R. Inequality (20) implies
(21)
∫
BR(x0)
∣∣∇2u∣∣2 dx ≤ c
[
R2 +R−2
∫
TR(x0)
|∇u− P |2 dx
]
,
where TR(x0) := B2R(x0) − BR(x0). Choosing P as the mean value of ∇u over the ring
TR(x0) and applying Poincare´’s inequality, we deduce from (21)
∫
BR(x0)
∣∣∇2u∣∣2 dx ≤ c
[
R2 +
∫
TR(x0)
∣∣∇2u∣∣2 dx
]
.
Adding c
∫
BR(x0)
|∇2u|
2
dx on both sides we find
∫
BR(x0)
∣∣∇2u∣∣2 dx ≤ c
c+ 1
R2 +
c
c + 1
∫
B2R(x0)
∣∣∇2u∣∣2 dx ,
and a standard iteration argument shows that
∫
Br(x0)
|∇2u|
2
dx grows at most like rκ for
some κ > 0 which proves the claim of part b) of the Theorem.

7
REMARK 1. 1.) Having established the continuity of ∇u it is possible to prove the
interior Ho¨lder continuity of ∇u for any exponent α ∈ (0, 1) by combining the freez-
ing technique with Campanato type estimates valid for constant coefficient systems
involving εD which have been established in [Sc].
2.) Using these Campanato estimates one can also prove partial C1–regularity for I–
minimizers now on domains Ω ⊂ Rn, n ≥ 3, and with iterative application
of the difference quotient technique one obtains C∞–regularity for minimizers of∫
Ω
[A(εD(u), εD(u)) − g · u] dx, Ω ⊂ Rn, n ≥ 2, provided A is a positive definite,
symmetric bilinear form on Sn (:= space of symmetric (n × n)–matrices) having
constant coefficients and g is a smooth function. For details we again refer to [Sc].
Let us look at the minimization problem
(22) J [u,Ω] :=
∫
Ω
F (εD(u)) dx→ min in u0+
◦
W
1
p(Ω;R
2)
with u0 from the space W
1
p (Ω;R
2) and with integrand F : S2 → R being strictly convex
satisfying in addition the growth condition
(23) a|σ|p − b ≤ F (σ) ≤ A|σ|p +B, σ ∈ S2 ,
with constants a, A > 0, b, B ≥ 0. The following Korn type inequality provides the
existence of a unique solution to (22) under the hypothesis (23).
Proposition 1. Let Ω ⊂ R2 denote a bounded Lipschitz domain and let p ∈ (1,∞).
Then there is a constant Cp(Ω) such that
(24) ‖∇v‖Lp(Ω) ≤ Cp(Ω)‖ε
D(v)‖Lp(Ω)
holds for any v ∈
◦
W1p(Ω;R
2).
Proof: It is sufficient to consider the case v ∈ C∞0 (Ω;R
2). From (26) in [Da] we deduce
∂iε
D
ij(v) =
1
2
∆vj, j = 1, 2 ,
i.e. h := vj ∈
◦
W1p(Ω) solves
(25) −∆h = divH on Ω
with H := (−2εDij(v))1≤i≤2 of course from the space L
p(Ω;R2). Applying standard Lp–
arguments to equation (25) (see, e.g. [Me]) we end up with
‖∇h‖Lp(Ω) ≤ Cp(Ω)‖H‖Lp(Ω)
8
and our claim (24) follows.

If (23) is replaced by the stronger requirement that F is a smooth p–elliptic integrand,
then we believe that the result of the Theorem extends to the p– case by combining the
Korn type inequality (24) with suitable regularity techniques as applied for example in
[BFZ].
References
[Ad] Adams, R. A., Sobolev spaces. Academic Press, New York-San Francisco-London
(1975).
[BI] Bartnik, R., Isenberg, J., The constraint equation. In: Chrus´ciel, P. T., Friedrich,
H., (eds.), The Einstein equations and large scale behaviour of gravitational fields,
pp.1–38. Birkha¨user Verlag, Basel-Boston-Berlin (2004).
[BFZ] Bildhauer, M., Fuchs, M., Zhong, X., A lemma on the higher integrability of func-
tions with applications to the regularity theory of two dimensional generalized
Newtonian fluids. Manus. Math. 116 (2), 135–156 (2005).
[Da] Dain, S., Generalized Korn’s inequality and conformal Killing vectors.
Calc. Var. 25 (4), 535–540 (2006).
[Gi] Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear el-
liptic systems. Princeton University Press (1983).
[GT] Gilbarg, D., Trudinger, N. S., Elliptic partial differential equations of second
order. Springer-Verlag, Berlin-Heidelberg-New York (1998).
[Me] Meyers, N. G., An Lp–estimate for the gradient of solutions of second order
elliptic divergence equations. Ann. SNS Pisa III (17), 189–206 (1963).
[Sc] Schirra, O., PhD-thesis.
[Wi] Widman, K. O., Ho¨lder continuity of solutions of elliptic systems.
Manus. Math. 5, 299–308 (1971).
9


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An application of a generalized Korn inequality to energies studied in general relativity giving the smoothness of minimizers

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