The Variable Coefficient Thin Obstacle Problem: Carleman Inequalities

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by  Herbert Koch

THE VARIABLE COEFFICIENT THIN OBSTACLE PROBLEM:
CARLEMAN INEQUALITIES
5
1
0 HERBERTKOCH,ANGKANARU¨LAND,ANDWENHUISHI
2

y
Abstract. In this article we present a new strategy of addressing the (vari-
a
M able coefficient) thinobstacle problem. Ourapproach is based ona (variable
coefficient) Carleman estimate. This yields semi-continuity of the vanishing
order,loweranduniformuppergrowthboundsofsolutionsandsufficientcom-
9
pactness propertiesinordertocarryoutablow-upprocedure. Moreover,the
2
Carlemanestimateimpliestheexistenceofhomogeneousblow-uplimitsalong

certain sequences and ultimately leads to analmost optimal regularity state-
]
P ment. As itis a very robust tool, itallows us to consider the problem in the
settingofSobolevmetrics,i.e. thecoefficientsareonlyW1,p regularforsome
A
p>n+1.
h. These results provide the basis for our further analysis of the free boundary,
t the optimal (C1,1/2-) regularity of solutions and a first order asymptotic ex-
a pansion of solutions at the regular free boundary which is carried out in a
m follow-up article, [KRS15], in the framework of W1,p, p > 2(n+1), regular
[ coefficients.

2
v
6
9 Contents
4
1. Introduction 2
4
0 1.1. Main results 2
. 1.2. Context and literature 4
1
0 1.3. Difficulties and strategy 4
5 1.4. Organization of the paper 5
1 2. Preliminaries 5
:
v 2.1. Auxiliary results and assumptions 6
i 2.2. Notation 7
X
3. A Carleman Estimate 7
r
a 3.1. A Carleman estimate and variations of it 7
3.2. Consequences of the Carleman estimate (7) 18
3.3. Proof of Lemma 3.1 20
4. Consequences of the Carleman Estimate: Compactness, Homogeneity
and an Almost Optimal Regularity Result 24
4.1. The order of vanishing and growth estimates 24
4.2. Doubling and the blow-up procedure 29
2010 Mathematics Subject Classification. Primary35R35.
Keywordsandphrases. VariablecoefficientSignoriniproblem,variablecoefficientthinobstacle
problem,thinfreeboundary,Carlemanestimates.
A.R.acknowledgesthattheresearchleadingtotheseresultshasreceivedfundingfromtheEu-
ropeanResearchCouncilundertheEuropeanUnion’sSeventhFrameworkProgramme(FP7/2007-
2013)/ERCgrantagreementno291053andaJuniorResearchFellowshipatChristChurch. W.S.
issupportedbytheHausdorffCenterofMathematics.
1
2 HERBERTKOCH,ANGKANARU¨LAND,ANDWENHUISHI
4.3. Homogeneous blow-ups 30
4.4. Almost optimal regularity 35
References 37
1. Introduction
In this article we present a new, very robust strategy of analyzing solutions
of the (variable coefficient) thin obstacle or Signorini problem. Let aij : B+
1 →
R(n+1)×(n+1)beasymmetric,uniformlyelliptictensorfieldwhichisW1,p,p>n+1,
sym
regularandletB+ := x B Rn+1 x 0 denotetheupperhalf-ball. Then
1 { ∈ 1 ⊂ | n+1 ≥ }
consider local minimizers of the constrained Dirichlet energy:
J(w)= aij(∂ w)(∂ w)dx,
ˆ i j
B+
1
where we use the Einstein summation convention and assume that
w := v H1(B+) v 0 on B′ :=B+ x =0 .
∈K { ∈ 1 | ≥ 1 1 ∩{ n+1 }}
Thus, local minimizers of this variational problem solve a uniformly elliptic diver-
genceformequationintheinterioroftheupperhalf-ball,onthissettheyare“free”.
However,onthecodimensiononesurfaceB′ theyobeytheconvexconstraintw 0.
1 ≥
In this sense the obstacle is “thin”.
DuetoresultsofCaffarelli[Caf79],Kinderlehrer[Kin81](whoworkinthesetting
with C1,γ coefficients) andUraltseva [Ura87], local minimizers are C1,α regularfor
some α (0,1/2] and satisfy a second order elliptic equation with Signorini (or
∈
complementary) boundary conditions:
∂ aij∂ w =0 in B+,
i j 1
(1)
w 0, an+1,j∂ w 0 and w(an+1,j∂ w)=0 on B′.
≥ − j ≥ j 1
In the sequel we study (1) with the aim of obtaining optimal regularity estimates
for its solutions as well as a better understanding of the free boundary, i.e. the set
Γw = ∂B1′{x ∈ B1′| w(x) > 0}, which separates the coincidence set, Λw := {x ∈
B′ w(x) = 0 , in which the solution coincides with the obstacle w = 0, from the
1| }
positivity set, Ω := x B′ w(x)>0 .
w { ∈ 1| }
1.1. Main results. In studying the variable coefficient thin obstacle problem (1)
in a low regularity framework (we only assume that the coefficients aij are in an
appropriateSobolevclass),weintroduceCarleman estimates asakeyingredientof
obtaining first information on the solutions. Here the Carleman estimate replaces
a variable coefficient frequency function approach.
Carleman estimates have the advantage of being very flexible with respect to
perturbations: After deriving a constant coefficient Carleman inequality, it is of-
ten possible to deduce a variable coefficient analogue by perturbative techniques.
Comparing the Carleman estimate with monotonicity arguments, we note that
monotonicity of the frequency function (for harmonic functions) is equivalent to
THE VARIABLE COEFFICIENT THIN OBSTACLE PROBLEM 3
logarithmic convexity of the L2 norm. It depends on rigid calculations. In con-
trast, Carleman inequalities provide relaxed convexity statements (c.f. Corollary
3.1). The conjugated equation is central (c.f. (18)), and one stays in the more
flexible context of PDEs.
Making use of the Carleman estimate, we obtain the upper semi-continuity of
the vanishing order(c.f. Proposition4.2), lowerand uniformupper growthbounds
for solutions (c.f. Corollary 4.1 and Lemma 4.1), compactness of L2 normalized
blow-up sequences (c.f. Proposition 4.3). Moreover, we obtain that along certain
sequences, the blow-up limits are homogeneous (c.f. Proposition 4.5). This in
particular allows us to classify the blow-up limits when the vanishing order is less
than 2. Combining this information then allows us to prove the following almost
optimal regularity result:
Proposition 1.1 (Almost optimal regularity). Let aij : B+ R(n+1)×(n+1) be a
1 → sym
uniformly elliptic, symmetric W1,p tensor field with p (n+1, ]. Assume that
∈ ∞
w is a solution of the variable coefficient thin obstacle problem (1). Then, with
γ =1 n+1,
− p
w(x) w(y)
|∇ −∇ |
C(γ) w x y γ for p (n+1,2(n+1)),
k kL2(B1+)| − | ∈
≤( CkwkL2(B1+)|x−y|1/2ln(|x−y|)2 for p∈[2(n+1),∞],
for all x,y B+. Apart from the dependence on γ for the first case, the constants
∈ 1
2
C are also functions of aij ,p,n.
W1,p(B+)
1
Inparticular,thisimp(cid:13)rove(cid:13)stheregularityestimatesofCaffarelli[Caf79],Kinder-
(cid:13) (cid:13)
lehrer [Kin81] and Uraltseva [Ura87], by coming logarithmically close to the ex-
pected optimal threshold of C1,1/2 regularityif p 2(n+1) (or even attaining the
≥
optimal C1,γ regularity if p (n+1,2(n+1))).
∈
The Carleman estimate however permits us to obtain more information: The
existenceofhomogeneousblow-upsolutionsdirectlyallowsustoclassifythe lowest
possible vanishing rate at free boundary points without invoking the Friedland-
Hayman inequality [FH76] (c.f. the proof of Proposition 4.6). Since blow-up solu-
tionssatisfy the constantcoefficientthin obstacleproblem,itis not surprisingthat
this lowestblow-uphomogeneity coincides with the lowestpossible homogeneity of
solutions of the constant coefficient thin obstacle problem, κ=3/2. As in the case
oftheconstantcoefficientproblem,weshowthatthereisagaptothenextpossible
blow-up homogeneity which is, κ=2.
In our follow-up article [KRS15], we use this and the upper semi-continuity of
the mapping Γ x κ to separate the free boundary into
w x
∋ 7→
Γ =Γ (w) Γ (w),
w 3/2 κ
∪
κ≥2
[
whereΓ (w)isarelativelyopensetofthefreeboundary,theso-calledregular free
3/2
boundary and Γ (w) consists of all free boundary points which have a higher
κ
κ≥2
orderofvanishinSg. WorkingintheframeworkofW1,pmetricswithp (2(n+1), ],
∈ ∞
4 HERBERTKOCH,ANGKANARU¨LAND,ANDWENHUISHI
we prove that the regular free boundary is locally a C1,α graph for some α (0,1)
∈
in [KRS15]. This then allows us to improve the almost optimal regularity result
from Proposition 1.1 to an optimal regularity result. Moreover, we identify the
leading term in the asymptotic expansion of solutions of (1) at the regular free
boundary and provide explicit error bounds.
1.2. Context and literature. Let us comment on the context of our problem:
Apart from the (constant coefficient) two-dimensional problem, which was com-
pletelysolvedby[Lew72]and[Ric78],anddespite variousimpressiveresultsonthe
higher dimensional problem [Caf79], [Fre77], [Kin81], [Ura87], the optimal regular-
ityofthethinobstacleproblemwasonlyresolvedrelativelyrecentlybyCaffarelliet
al. [AC06], [ACS08]. In these seminal papers a frequency function was introduced
as the key tool in studying solutions of the thin obstacle problem.
Followingthistherehasbeengreatprogressinvariousdirectionsfortheconstant
coefficient problem: Relying on frequency function methods, the ground breaking
papers of Caffarelli et al. [AC06], [ACS08] establish the optimal regularity of so-
lutions, as well as the C1,α regularity of the regular free boundary. This has been
furtherextendedtotherelatedobstacleproblemforthefractionalLaplacian[Sil07],
[CSS08]andparabolicanalogues[DGPT13]. Relyingneitheronfrequencyfunctions
nor on comparison arguments, Andersson [And13] has shown that similar results
also hold for the full Lam´e system. Recently, Koch, Petrosyanand Shi [KPS14] as
well as De Silva and Savin [DSS14] have proved smoothness ([KPS14] proved an-
alyticity) of the regular free boundary. Moreover, Garofalo and Petrosyan [GP09]
give a structure theorem for the singular set of the thin obstacle problem.
The variable coefficient thin obstacle problems is much less understood. Here
the best regularity result in the literature in the setting of Sobolev regularity is
given by Uraltseva’s C1,α regularity result: In [Ura87] she proves that for the
thin obstacle problem with a W1,p, p > n+1 metric, aij, there exists a Ho¨lder
coefficientα (0,1/2],dependingonlyontheSobolevexponentpandtheellipticity
∈
constantsof aij, suchthat the correspondingsolutionis in C1,α (c.f. also [Ura89]).
Recentlytherehasbeenimportantprogressinderivinganimprovedunderstanding
in the low regularity setting: In [Gui09] Guillen derives optimal regularity results
for solutions of the variable coefficient thin obstacle problem. He works in the
setting of C1,γ, γ > 0 coefficients. This was generalized by Garofalo and Smit
Vega Garcia [GSVG14] to Lipschitz continuous coefficients by using the frequency
function approach. In a recent work in progress Garofalo, Petrosyan and Smit
VegaGarcia[GPSVG]furtherextendthisresultandproveHo¨ldercontinuityofthe
regular free boundary.
1.3. Difficulties and strategy. Let us elaborate a bit further on the main diffi-
culties in investigating the variable coefficient thin obstacle problem: The central
problemthathastobeovercomeandthatisreflectedinallstagesofourarguments
is the low regularity of the metric. This requires robust tools.
Inthisfirstpartofourdiscussionofthevariablecoefficientthinobstacleproblem,
Carlemanestimates,whicharewell-knownfromuniquecontinuationandthestudy
of inverse problems [Car39], [JK85], [KT01], [Isa04], [Ru¨l14b], are used to handle
THE VARIABLE COEFFICIENT THIN OBSTACLE PROBLEM 5
this low regularity setting: Although they are usually employed in the setting of
Lipschitz metrics (as for instance the unique continuation principle in general fails
for less regular coefficients), it is possible to extend these to the low regularity
framework we are interested in.
This allows us to carry out a blow-up argument and exploit information from
the constant coefficient setting. For this we argue in two steps:
A Carleman estimate for solutions of the variable coefficient thin obstacle
•
problem (4). Thisyieldssufficientlystrongcompactnesspropertiesinorder
tocarryoutablow-upprocedureatthefreeboundarypoints. Inparticular,
doubling inequalities (c.f. Proposition 4.3) are immediate consequences of
the Carleman estimate.
A blow-up procedure. The good compactness properties deduced in the
•
previous step permit to carry out a blow-up procedure with a non-trivial
limitsatisfyingaconstant coefficientthinobstacleproblem(c.f. Proposition
4.4). Thus, (for an appropriately normalized metric aij) the blown-up
solution is of the form:
∆w =0 in B+,
1
(2)
w 0, ∂ w 0 and w∂ w =0 on B′.
≥ − n+1 ≥ n+1 1
However, the Carleman estimate yields further information: It is possible
to show that the there are blow-up sequences such that the blow-up limits
are homogeneous solutions of (2). Moreover, the homogeneity is given by
the order of vanishing (c.f. Proposition 4.5). For a solution of the variable
coefficientthin obstacle problemthis then allowsto exploitthe existing in-
formationonsolutionsoftheconstantcoefficientthinobstacleproblemand
thus, for instance, obtainan almostoptimalregularityresult(c.f. Proposi-
tion 4.8).
1.4. Organization of the paper. Let us finally commenton the structure of the
remainder of the article: In the next section, we first briefly recall auxiliary results
(c.f. Proposition2.1)andthenintroduceournotationalconventionsinSection2.2.
Followingthis,Section3isdedicatedtoourCarlemanestimate,Proposition3.1. In
particular,wederiveanimportantcorollary,Corollary3.1,fromit. InSection4we
thendeducecrucialconsequencesoftheCarlemanestimate: Wederivecompactness
properties (c.f. Proposition 4.3), carry out a blow-up procedure (c.f. Proposition
4.4) and prove the existence of homogeneousof blow-up solutions (c.f. Proposition
4.5). This is then applied in proving the first (almost optimal) regularity result,
Proposition 4.8.
In our second article dealing with the thin obstacle problem we then use these
results to further analyze the regular free boundary, to derive optimal regularity
estimates and to obtain a first order expansion of solutions at the regular free
boundary.
2. Preliminaries
6 HERBERTKOCH,ANGKANARU¨LAND,ANDWENHUISHI
2.1. Auxiliary results and assumptions. In the sequel we recall certain auxil-
iaryresultswhichallowustoreformulateandsimplifytheproblem(1)inaparticu-
larlyusefulway. Moreover,wecollectallourassumptionsontheinvolvedquantities.
We startbyrecallinga(slightmodificationofa)resultduetoUraltseva[Ura89],
p.1183 which allows to simplify the complementary boundary conditions:
Proposition 2.1. Let aij : B+ R(n+1)×(n+1) be a uniformly elliptic W1,p,
1 → sym
p> n+1, tensor field and w H1(B+) be a solution to (1). Then for each point
∈ 1
x B′ there exist a neighborhood, U(x), and a W2,p diffeomorphism
∈ 1
2
T :U(x) B+ V B+, x T(x)=:y,
∩ 1 → ∩ 1 7→
such that in the new coordinates w˜(y) := w(T−1y) (weakly) solves a new thin
obstacle problem with
∂ bkℓ(y)∂ w˜ =0 in V B+,
(3) k ℓ ∩ 1
∂ w˜ 0, w˜ 0, w˜(∂ w˜)=0 on V B′,
n+1 ≤ ≥ n+1 ∩ 1
where B(y)=(bkℓ(y))= det(DT(x))−1(DT(x))tA(x)DT(x) satisfies
| | x=T−1(y)
bn+1,ℓ =0 on B′ for all ℓ 1,...,n , (cid:12)
1 ∈{ } (cid:12)
and remains a uniformly elliptic W1,p tensor field.
Here the boundary data in (1) and (3) are interpreted in a distributional sense:
Itisknownthatonecanseefromtheequationbylocalargumentsthat,ifw H1 is
∈
a weak solutionin the interior,then an+1,j∂ w H−1/2. This remains true
j |xn+1=0 ∈
inthenewcoordinates: bn+1,n+1∂ w˜ H−1/2. Sincebn+1,n+1isbounded
n+1 |xn+1=0 ∈
from below and in W1,p, the multiplication by its inverse defines a bounded op-
erator on H1/2 and H−1/2. Thus, w˜∂ w˜ defines a distribution on the
n+1 |xn+1=0
boundary which we require to be zero. Vice versa: If w˜ H1, ∂ bkl∂w˜ = 0 in
k l
V B+ and ∂ w˜ 0, w˜ 0 and w˜∂ w˜ = 0, then w∈˜ is a local solution to
∩ 1 n+1 ≤ ≥ n+1
the thin obstacle problem. In any of the two formulations the regularity theory of
Uraltseva [Ura87] applies.
By a further (affine) change of coordinates we may, without loss of generality,
assume that
bkℓ(x) δkℓ C xγ,
| − |≤ | |
1 for k =ℓ,
for γ =1 n+1 and δkℓ := denotes the Kronecker delta. In the
− p 0 else,
(cid:26)
sequel, for convenience, we will always assume that we are in a sufficiently small
coordinate patch such that our thin obstacle problem is formulated in this way.
This allows us to exploit the boundary conditions very efficiently. So, without loss
of generality, we pass from (1) to considering
∂ aij∂ w=0 in B+,
i j 1
(4)
w 0 ∂ w 0 and w(∂ w)=0 on B′,
≥ − n+1 ≥ n+1 1
with
(A0) w =1,
k kL2(B1+(0))
THE VARIABLE COEFFICIENT THIN OBSTACLE PROBLEM 7
(A1) ai,n+1(x′,0)=0 on Rn 0 for i=1,...,n,
×{ }
(A2) aij is symmetric and uniformly elliptic with eigenvalues in the interval
[1/2,2],
(A3) aij W1,p(B+(0)) for some p (n+1, ],
∈ 1 ∈ ∞
(A4) aij(0)=δij.
Moreover,throughout the paper we assume that the solution, w, of (4) satisfies
(W) there exists α (0,1] such that w C1,α.
∈ 2 ∈
Assumption (A0) is a normalization, which is no restriction of generality, since
positivemultiplesofsolutionsaresolutions. Property(A1)maybeassumedbecause
of Proposition 2.1. By Morrey’s inequality and assumption (A3)
aij(x) δij C aij xγ,
| − |≤ ∇ Lp(B1+(0))| |
where γ = 1 n+1 and hence the cond(cid:13)(cid:13)ition(cid:13)(cid:13)(A2) on the eigenvalues holds after
− p
rescalingifnecessary. Theregularityassumption(W)doesnotposeanyrestrictions
on the class of solutions, as Uraltseva [Ura87] proves that this property is true for
any H1 solution of (4). We however stress that most of the paper is independent
of this regularity assumption in the sense that assumption (W) is only invoked
in Proposition 4.6. Apart from this, all the other necessary regularity results are
proved “by hand” in the paper.
2.2. Notation. In this subsection we briefly introduce some notation. We set:
Rn+1 := x Rn+1 x 0 , Rn+1 := x Rn+1 x 0 .
• + { ∈ | n+1 ≥ } − { ∈ | n+1 ≤ }
Let x = (x′,0) Rn+1. For an upper half-ball of radius r > 0 around
• 0 0 ∈ +
x we write B+(x ) := x Rn+1 x x < r ; the projection onto the
0 r 0 { ∈ + | | − 0| }
boundary of Rn+1 is correspondingly denoted by B′(x ):= x Rn x
+ r 0 { ∈ | | −
x < r . If x = (0,0) we also write B+ and B′. Analogous conventions
0| } 0 r r
are used for balls in the lower half sphere: B−(x ). Moreover, we use the
r 0
notation B′′(x )=B′(x ) x =0 .
r 0 r 0 ∩{ n }
Annuli around a point x = (x′,0) in the upper half-space with radii 0 <
• r < R < as well as t0heir p0rojections onto the boundary of Rn+1 are
∞ +
denoted by by A+ (x ) := B+(x ) B+(x ) and A′ (x ) := B′ (x )
r,R 0 R 0 \ r 0 r,R 0 R 0 \
B′(x ) respectively. For annuli around x =(0,0) we also omit the center
r 0 0
point. Furthermore, we set A (x ):=A+ (x ) A− (x ).
r,R 0 r,R 0 ∪ r,R 0
Ω := x Rn 0 w(x)>0 denotes the positivity set.
w
• { ∈ ×{ }| }
Γw :=∂B′Ωw is the free boundary.
• 1
Λ :=B′ Ω is the coincidence set which we also denote by Λ .
• w 1\ w w
We use the symbol A . B to denote that there exists an only dimension
•
dependentconstant, C =C(n), suchthatA C(n)B. Similarconventions
≤
are used for the symbol &.
3. A Carleman Estimate
3.1. A Carleman estimate and variations of it. Inthissectionweintroducea
central tool of our argument: The Carleman estimate from Proposition 3.1 allows
us toobtaincompactnesspropertiesfor blow-upsolutions(c.f. Proposition4.4), to
deduce the existence of homogeneous blow-up solutions (c.f. Proposition 4.5) and
to derive the openness of the regular part of the free boundary (c.f. Proposition
8 HERBERTKOCH,ANGKANARU¨LAND,ANDWENHUISHI
4.2). It is a very flexible and robust tool replacing a variable coefficient frequency
function argument.
Proposition 3.1 (VariablecoefficientCarlemanestimate). Let n 2 and 0<ρ<
r<1. Let aij :A+ Rn×n be a tensor field which satisfies ≥
ρ,r → sym
(i) aij W1,n+1(A+ ),
∈ ρ,r
(ii) the off-diagonal assumption (A1),
(iii) the uniform ellipticity assumption (A2),
(iv) the following smallness condition: There exists δ =δ(n)>0 such that
(5) sup aij δ.
∇ Ln+1(A+ ) ≤
ρ≤r˜≤r r˜,2r˜
(cid:13) (cid:13)
Assume that w H1(B+) with su(cid:13)pp(w)(cid:13) A+ is a weak solution of the divergence
∈ 1 ⊂ ρ,r
form equation
∂ aij∂ w =f in A+ ,
i j ρ,r
(6)
w 0, ∂ w 0, w∂ w =0 on A′ ,
≥ n+1 ≤ n+1 ρ,r
where f : A+ R is an in A+ compactly supported L2(A+ ) function. Let φ be
ρ,r → ρ,r ρ,r
the following radial weight function:
1
φ(x)=φ˜(ln x) with φ˜:R R, φ˜(t)= t+c tarctant ln(1+t2) ,
0
| | → − − 2
(cid:18) (cid:19)
where 0<c 1 is an arbitrarily small but fixed constant. Then for any γ (0,1)
0
≪ ∈
and any τ >1 we have
(7)
τ23 eτφ x−1(1+ln(x)2)−21w +τ12 eτφ(1+ln(x)2)−12 w
| | | | L2(A+ρ,r) | | ∇ L2(A+ρ,r)
(cid:13) (cid:13) (cid:13) (cid:13)
≤c(cid:13)(cid:13)−01C(n) τ2C(aij) eτφ|x|γ(cid:13)(cid:13)−1w L2(A+ρ,r)+(cid:13)(cid:13) eτφ|x|f L2(A+ρ,r) , (cid:13)(cid:13)
where (cid:16) (cid:13) (cid:13) (cid:13) (cid:13) (cid:17)
(cid:13) (cid:13) (cid:13) (cid:13)
aij(x) δij
(8) C(aij)= sup − + sup x−γ aij .
xγ | | ∇ Ln+1(A+ )
A+ρ,r(cid:12) | | (cid:12) ρ≤r˜≤r/2 r˜,2r˜
(cid:12) (cid:12) (cid:13) (cid:13)
Remark 1 (Restrictions(cid:12)on τ). In ord(cid:12)er to deriv(cid:13)e the Carlem(cid:13)an estimate, it would
(cid:12) (cid:12)
not have been necessary to assume that τ > 1. In fact the estimate is valid for
arbitrary τ >0 if we replace the right hand side of (7) by
τ32 eτφ x−1(1+ln(x)2)−21w +τ12 eτφ(1+ln(x)2)−12 w
| | | | L2(A+ρ,r) | | ∇ L2(A+ρ,r)
(cid:13) (cid:13) (cid:13) (cid:13)
≤c(cid:13)(cid:13)−01C(n) max{1,τ2}C(aij)(cid:13)(cid:13)eτφ|x|γ−1w L(cid:13)(cid:13)2(A+ρ,r)+ eτφ|x|f L2(A+ρ(cid:13)(cid:13),r)
(cid:16) (cid:13) (cid:13) (cid:13) (cid:13)
+τ12 eτφ x−1(1+ln(x)4)−(cid:13)21w (cid:13) . (cid:13) (cid:13)
(cid:13) | | | | (cid:13)L2(A+ρ,r)(cid:19)
Choosing 0(cid:13)< r < 1 depending on τ(cid:13)sufficiently small, then allows to absorb the
(cid:13) (cid:13)
error terms in this case as well (c.f. Remark 2).
Remark 2 (Applications). In the sequel we will apply the Carleman estimate (7)
witha metric aij suchthataij(0)=δij andaij W1,p(B+), p (n+1, ]. Thus,
∈ 1 ∈ ∞
by Morrey’s and Ho¨lder’s inequalities, for γ = 1 n+1, C(aij) is bounded by a
− p
constant which depends only on n and aij .
k∇ kLp(B1+)
THE VARIABLE COEFFICIENT THIN OBSTACLE PROBLEM 9
In our applications, we will always consider r R = R (τ,n,p, aij )
≤ 0 0 k∇ kLp(B1+)
sufficiently small, such that the smallness condition (5) is satisfied and moreover,
the first term on the right hand side of (7) can be absorbed by the left hand side.
More precisely, we consider R such that
0
C(aij)R0γ ≤δ, τ12c−01C(n)C(aij)R0γ|lnR0|≤1/4.
Then the Carleman inequality (7) can be rewritten as
(9)
τ32 eτφ x−1(1+ln(x)2)−21w +τ12 eτφ(1+ln(x)2)−12 w
| | | | L2(A+ρ,r) | | ∇ L2(A+ρ,r)
(cid:13) (cid:13) (cid:13) (cid:13)
c(cid:13)−1C(n) eτφ xf (cid:13)for 0<ρ<r(cid:13) R and τ 1. (cid:13)
≤(cid:13)0 | | L2(A+ρ,r) (cid:13) (cid:13)≤ 0 ≥ (cid:13)
Remark 3 (Ca(cid:13)rleman e(cid:13)stimate for n=1). In one dimension it is possible to prove
(cid:13) (cid:13)
a similarCarlemanestimate asabove. However,slightmodifications arenecessary.
Instead of working with the W˙ 1,2 semi-norm (which would be the scale invariant
normin (1+1)-dimensions)of the metric and the smallness condition (5), we con-
siderLp norms, aij withp>2. Moreover,allSobolevembeddingarguments
∇ Lp
are then replaced by interpolation inequalities of the form
(cid:13) (cid:13)
(cid:13) (cid:13) 2 1−2
(10) w c(p,n) w p w p .
k kLp2−p2 ≤ k∇ kL2k kL2
This then yields the following Carleman estimate
(11)
τ32 eτφ x−1(1+ln(x)2)−21w +τ12 eτφ(1+ln(x)2)−12 w
| | | | L2(A+ρ,r) | | ∇ L2(A+ρ,r)
(cid:13) (cid:13) (cid:13) (cid:13)
≤c(cid:13)(cid:13)−01C(n,p) τ2 ∇aij Lp e(cid:13)(cid:13)τφ|x|γ−1w L2(A(cid:13)(cid:13)+ρ,r)+ eτφ|x|f L2(A+ρ,r)(cid:13)(cid:13).
Remark 4 (Gene(cid:16)raliz(cid:13)ations(cid:13)). (cid:13) With(cid:13)slight mod(cid:13)ifications(cid:13), the Ca(cid:17)rleman es-
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
•
timate is valid for more general (radial) pseudoconvex weight functions,
φ(x) = φ˜(ln(x)). In this more general setting (but with φ˜(t) t and
| | ∼ −
φ˜′(t) 1 asymptotically as t ) the Carleman estimate reads
| |∼ →−∞
τ32 eτφ x−1φ˜′(φ˜′′)21w +τ21 eτφ(φ˜′′)21 w
(12) | | L2(A+ρ,r) ∇ L2(A+ρ,r)
(cid:13) (cid:13) (cid:13) (cid:13)
≤c(cid:13)(cid:13)−01C(n) τ2C(aij)(cid:13)(cid:13)eτφ|x|γ−1w L(cid:13)(cid:13)2(A+ρ,r)+ eτφ(cid:13)(cid:13)|x|f L2(A+ρ,r) ,
where C(aij)(cid:16)is the con(cid:13)stant from ((cid:13)8). (cid:13) (cid:13) (cid:17)
(cid:13) (cid:13) (cid:13) (cid:13)
As we aim at absorbing the error terms on the right hand side of (7) or
•
(12) into the left hand side of the inequality if τ >1 is uniformly bounded
by some τ < , it is in general not possible to work with “linear” weight
0
∞
functions in (12). As a consequence, in the sequel we will always consider
strictly pseudoconvex weight functions with the exception of Lemma 3.2.
In contrast to the setting of unique continuation, where the metrics under
•
considerationareLipschitz, ourmetricisonly W1,p, p>n+1, regular. On
the one hand, this requires additional care in the commutator estimates of
theCarlemaninequality. Ontheotherhand,thelowregularityofthemetric
also implies that we cannot hope for the unique continuation principle to
hold in the setting of (4). Hence, the Carleman estimate can, for instance,
not exclude free boundary points with infinite order of vanishing. On a
10 HERBERTKOCH,ANGKANARU¨LAND,ANDWENHUISHI
technical level these observations are manifested in the error contributions
on the right hand side of the inequality (7) which carry higher powers of τ
than the analogous contributions on the left hand side.
It is possible to derive an analogous Carleman inequality for the setting of
•
the interior thin obstacle problem and the case in which inhomogeneities
are present in the thin obstacle problem. For a detailed discussion of this
we refer to Section 5 in [KRS15].
Before proving the Carlemanestimate, we state H2 estimates for the associated
conjugatedoperatorindyadicannuliwhichwillbe provedinSection3.3attheend
of this Section.
Lemma 3.1. Let n 2. Then there exists δ =δ(n)>0 such that the following is
≥
true: Let aij : B+ R(n+1)×(n+1) be a uniformly elliptic, symmetric tensor field
which satisfies (51).→Furtshyemr, assume that v : B+ R with v H1(B+) is a weak
1 → ∈ 1
solution of
∂ aij∂ v+τ2(∂ φ)aij(∂ φ)v τaij(∂ φ)∂ v
i j i j i j
−
τ(∂ φ)∂ (aijv) τaij(∂2φ)v =f in B+,
(13) − j i − ij 1
∂ v 0, v 0, v∂ v =0 on B′,
n+1 ≤ ≥ n+1 1
v =0 on ∂B+,
1
where φ(x) = φ˜(ln(x)) is defined as in Proposition 3.1 and f : B+ R is an
| | 1 →
L2(B+) function. Then on each dyadic (half-)annulus A+ := B+ B+ ,
m N1, m 1, and for each τ >1 m 2−m \ 2−m−1
∈ ≥
(14) ∇2v L2(A+m) . τ2 |x|−2v L2(A+m−1∪A+m∪A+m+1)+kfkL2(A+m−1∪A+m∪A+m+1).
(cid:13) (cid:13) (cid:13) (cid:13)
Remark(cid:13)5 (M(cid:13)odificationsf(cid:13)or n=(cid:13)1). Analogousto the Carlemanestimate, Lemma
3.1hasasimilarextensionton=1. Inthatcase,wereplacethesmallnesscondition
(5) and all Sobolev embedding arguments by interpolation arguments as in (10).
Proof of Proposition 3.1. WeinterprettheequationasaperturbationoftheLapla-
cian
(15) ∆w =∂ (aij δij)∂ w+f,
i j
−
and argue in two steps: First, we derive a Carleman inequality for the Laplacian
with Signorini boundary conditions. In a second step, we explain how to treat the
righthandsideof(15). However,beforecomingtothis,werecallthattheboundary
contributions in (6) are well-defined as distributions.
Step 1: Regularity of w. In order to make sense of the boundary contributions,
we refer to the discussion following (3). Moreover, using the H2 regularity of w
(c.f. Lemma 3.1), we may also deal with second derivatives of w in L2(B+) as well
1
asL2 gradientcontributionsontheboundary. Analternativemethodtodealwith,
for example, boundary contributions would have been a regularization mechanism
(c.f. Remark 6)