An epiperimetric inequality approach to the regularity of the free boundary in the Signorini problem with variable coefficients

0.42 MB·English

by  Nicola Garofalo

AN EPIPERIMETRIC INEQUALITY APPROACH TO THE REGULARITY OF
THE FREE BOUNDARY IN THE SIGNORINI PROBLEM WITH VARIABLE
COEFFICIENTS
5
NICOLAGAROFALO,ARSHAKPETROSYAN,AND MARIANASMIT VEGA GARCIA
1
0
2 Abstract. In thispaperwe establish theC1,β regularity of theregular part of thefree boundary

n in the Signorini problem for elliptic operators with variable Lipschitz coefficients. This work is
a a continuation of the recent paper [GSVG14], where two of us established the interior optimal
J
regularity of the solution. Two of the central results of the present work are a new monotonicity

6 formula and a new epiperimetric inequality.
2

]
P
A Contents
.
h 1. Introduction 1
t
a 1.1. Statement of the problem and main assumptions 1
m
1.2. Main result 4
[ 1.3. Structure of the paper 6

1 2. Notation and preliminaries 6
v 2.1. Basic notation 6
8
2.2. Summary of known results 7
9
3. Regular free boundary points 10
4
6 4. A Weiss type monotonicity formula 14
0 5. Homogeneous blowups 16
.
1 6. An epiperimetric inequality for the Signorini problem 18
0 7. C1,β regularity of the regular part of the free boundary 28
5
References 39
1
:
v
i
X
1. Introduction
r
a
1.1. Statement of the problem and main assumptions. The purpose of the present paper is
to establish the C1,β regularity of the free boundary near so-called regular points in the Signorini
problem for elliptic operators with variable Lipschitz coefficients. Although this work represents
a continuation of the recent paper [GSVG14], where two of us established the interior optimal
regularity of the solution, proving the regularity of the free boundary has posed some major new
2010 Mathematics Subject Classification. 35R35.
Keywords andphrases. Thinobstacleproblem,Signoriniproblem,Lipschitzcoefficients,regularityoffreebound-
ary,Weiss-type monotonicity formula, Almgren’s frequency formula, epiperimetric inequality.
First authorsupported in part bya grant “Progetti d’Ateneo,2013,” University of Padova.
Second author supported in part bythe NSFGrant DMS-1101139.
This project was completed while the third author was visiting the Institute Mittag-Leffler for the program
“HomogenizationandRandomPhenomenon.” Shethanksthisinstitutionforthegracioushospitalityandtheexcellent
work environment.
1
2 NICOLAGAROFALO,ARSHAKPETROSYAN,ANDMARIANASMITVEGAGARCIA
challenges. Twoofthecentralresultsofthepresentworkareanewmonotonicity formula(Theorem
4.3) and a new epiperimetric inequality (Theorem 6.3). Both of these results have been inspired by
those originally obtained by Weiss in [Wei99] for the classical obstacle problem, but the adaptation
to the Signorini problem has required a substantial amount of new ideas.
The lower-dimensional (orthin)obstacle problem consistsofminimizingthe(generalized)Dirich-
let energy
(1.1) min A(x) u, u dx,
u∈KZΩh ∇ ∇ i
where u ranges in the closed convex set
K = K = u W1,2(Ω) u= g on ∂Ω, u ϕ on M Ω .
g,ϕ
{ ∈ | ≥ ∩ }
Here, Ω Rn is agiven boundedopenset, M is a codimension onemanifold which separates Ω into
two part⊂s, g is a boundarydatum and the function ϕ : M R represents the lower-dimensional, or
→
thin, obstacle. The functions g and ϕ are required to satisfy the standard compatibility condition
g ϕ on ∂Ω M. This problem is known also as (scalar) Signorini problem, as the minimizers
sa≥tisfy Signori∩ni conditions on M (see (1.7)–(1.9) below in the case of flat M).
Our assumptions on the matrix-valued function x A(x) = [a (x)] in (1.1) are that A(x) is
ij
7→
symmetric, uniformly elliptic, and Lipschitz continuous (in short A C0,1). Namely:
∈
(1.2) a (x) = a (x) for i,j = 1,...,n, and every x Ω;
ij ji
∈
there exists λ > 0 such that for every x Ω and ξ Rn, one has
∈ ∈
(1.3) λ ξ 2 A(x)ξ,ξ λ−1 ξ 2;
| | ≤ h i ≤ | |
there exists Q 0 such that
≥
(1.4) a (x) a (y) Q x y , x,y Ω.
ij ij
| − | ≤ | − | ∈
By standard methods in the calculus of variations it is known that, under appropriate assumptions
on the data, the minimization problem (1.1) admits a unique solution u K, see e.g. [Fri88], or
∈
also [Tro87]. The set
Λϕ(u) = x M Ω u(x) = ϕ(x)
{ ∈ ∩ | }
is known as the coincidence set, and its boundary (in the relative topology of M)
Γϕ(u) = ∂MΛϕ(u)
is known as the free boundary. In this paper we are interested in the local regularity properties of
Γϕ(u). When ϕ= 0, we will write Λ(u) and Γ(u), instead of Λ0(u) and Γ0(u).
We also note that since we work with Lipschitz coefficients, it is not restrictive to consider the
situation in which the thin manifold is flat, which we take to be M = x = 0 . We thus consider
n
{ }
the Signorini problem (1.1) when the thin obstacle ϕ is defined on
B′ = M B = (x′,0) B x′ < 1 ,
1 ∩ 1 { ∈ 1 || | }
which we call the thin ball in B . In this case we will also impose the following conditions on the
1
coefficients
(1.5) a (x′,0) = 0 in B′, for i < n,
in 1
which essentially means that the conormal directions A(x′,0)ν are the same as normal directions
±
ν = e . We stress here that the condition (1.5) is not restrictive as it can be satisfied by means
± n
∓
of a C1,1 transformation of variables, as proved in Appendix B of [GSVG14].
AN EPIPERIMETRIC INEQUALITY APPROACH, ETC. 3
We assume that ϕ C1,1(B′) and denote by u the unique solution to the minimization problem
∈ 1
(1.1). (Notice that, by letting ϕ˜(x′,x ) = ϕ(x′), we can think of ϕ C1,1(B ), although we will
n 1
∈
not make such distinction explicitly.) Such u satisfies
(1.6) Lu = div(A u) = 0 in B+ B−,
∇ 1 ∪ 1
(1.7) u ϕ 0 in B′,
− ≥ 1
(1.8) A u,ν + A u,ν 0 in B′,
h ∇ +i h ∇ −i≥ 1
(1.9) (u ϕ)( A u,ν + A u,ν ) = 0 in B′.
− h ∇ +i h ∇ −i 1
The conditions (1.7)–(1.9) are known as Signorini or complementarity conditions. It has been
recently shown in [GSVG14] that, under the assumptions above, the unique solution of (1.6)–(1.9)
is in C1,1/2(B± B′). This regularity is optimal since the function u(x) = (x +ix )3/2 solves
1 ∪ 1 ℜ 1 | n|
the Signorini problem for the Laplacian and with thin obstacle ϕ 0.
≡
Henceforth, we assume that 0 is a free boundary point, i.e., 0 Γϕ(u), and we suppose without
∈
restriction that A(0) = I. Under these hypothesis, we consider the following normalization of u
(1.10) v(x) = u(x) ϕ(x′)+bx , b= ∂ u(0).
− n ν+
Note that ∂ u(0)+∂ u(0) = 0, which follows from (1.9), by taking a limit from inside the set
ν+ ν−
(x′,0) B′ u(x′,0) > ϕ(x′) , and from the hypothesis A(0) = I. This implies that
{ ∈ 1 | }
(1.11) v(0) = v(0) =0.
|∇ |
Next, note that, in view of (1.4) above and of the assumption ϕ C1,1(B ), we have
1
∈
L(ϕ(x′) bx ) d=ef f L∞(B ).
n 1
− − ∈
Hence, we can rewrite (1.6)–(1.9) in terms of v as follows:
(1.12) Lv = div(A v) = f in B+ B−, f L∞(B ),
∇ 1 ∪ 1 ∈ 1
(1.13) v 0 in B′,
≥ 1
(1.14) A v,ν + A v,ν 0 in B′,
h ∇ +i h ∇ −i≥ 1
(1.15) v( A v,ν + A v,ν ) = 0 in B′.
h ∇ +i h ∇ −i 1
Note that from (1.12) we have
(1.16) ( A v, η +fη)= ( A v,ν + A v,ν )η, η C∞(B ).
h ∇ ∇ i h ∇ +i h ∇ −i ∈ 0 1
ZB1 ZB1′
and thus the conditions (1.13)–(1.15) imply that v satisfies the variational inequality
( A v, (w v) +f(w v)) 0, for any w K ,
v,0
h ∇ ∇ − i − ≥ ∈
ZB1
where K = w W1,2(B ) w = v on ∂B ,w 0 on B′ . Since K is convex, and so is the
v,0 { ∈ 1 | 1 ≥ 1} v,0
energy functional
(1.17) ( A w, w +2fw),
h ∇ ∇ i
ZB1
this is equivalent to saying that v minimizes (1.17) among all functions w K . Notice as well
v,0
∈
that
Λϕ(u) = u(,0) = ϕ = v(,0) = 0 = Λ0(v) = Λ(v).
{ · } { · }
We will write Γϕ(u) = Γ0(v) = Γ(v), and thus 0 Γ(v) now and (1.11) holds.
∈
4 NICOLAGAROFALO,ARSHAKPETROSYAN,ANDMARIANASMITVEGAGARCIA
1.2. Main result. To state the main result of this paper, we need to further classify the free
boundary points. This is achieved by means of the truncated frequency function
σ(r) 1−δ d 1
N(r)= N (v,r) d=ef eK′r 2 logmax v2µ,r3+δ .
L 2 dr σ(r)rn−2
(cid:26) ZSr (cid:27)
Here, µ(x) = A(x)x,x /x 2 is a conformal factor, σ(r) is an auxiliary function with the property
h i | |
that σ(r)/r α > 0 as r 0, 0 < δ < 1, and K′ is a universal constant (see Section 2 for exact
→ →
definitions and properties). This function was introduced in [GSVG14], and represents a version of
Almgren’s celebrated frequency function (see [Alm79]), adjusted for the solutions of (1.12)–(1.15).
By Theorem 2.4 in [GSVG14], N(r) is monotone increasing and hence the limit
r
N˜(0+) = limN˜(r), where N˜(r) = N(r)
r→0 σ(r)
exists. The remarkable fact is that either N˜(0+) = 3/2, or N˜(0+) (3+δ)/2, see Lemma 2.5
≥
below. This leads to the following definition.
Definition 1.1. We say that 0 Γ(v) is a regular point iff N˜(0+) = 3/2. Shifting the origin to
∈
x Γ(v), and denoting the corresponding frequency function by N˜ , we define
0 ∈ x0
Γ (v) = x Γ(v) N˜ (0+) = 3/2 ,
3/2 { 0 ∈ | x0 }
the set of all regular free boundary points, also known as the regular set.
The remaining part of the free boundary is divided into the sets Γ (v), according to the cor-
κ
responding value of N˜(0+) d=ef κ. We note that the range of possible values for κ can be further
refined, provided more regularity is known for the coefficients A(x). This can be achieved by re-
placing the truncation function r3+δ in the formula for N(r) with higher powers of r, similarly to
what was done in [GP09] in the case of the Laplacian. This will provide more information on the
set of possible values of N˜(0+) which will serve as a classification parameter.
The following theorem is the central result of this paper.
Theorem 1.2. Let v be a solution of (1.12)–(1.15) with x Γ (v). Then, there exists η > 0,
0 3/2 0
∈
depending on x , such that, after a possible rotation of coordinate axes in Rn−1, one has B′ (x )
0 η0 0 ∩
Γ(v) Γ (v), and
3/2
⊂
B′ Λ(v) = B′ x g(x ,...,x )
η0 ∩ η0 ∩{ n−1 ≤ 1 n−2 }
for g C1,β(Rn−2) with a universal exponent β (0,1).
∈ ∈
Thisresultisknownandwell-understoodinthecaseL =∆,see[ACS08]orChapter9in[PSU12].
However, the existing proofs are based on differentiating the equation for v in tangential directions
e Rn−1 and establishing the nonnegativity of ∂ v in a cone of directions, near regular free
e
∈
boundary points (directional monotonicity). This implies the Lipschitz regularity of Γ (v), which
3/2
can bepushed to C1,β with the help of the boundaryHarnack principle. Theidea of the directional
monotonicity goesback atleasttothepaper[Alt77],whiletheapplication oftheboundaryHarnack
principle originated in [AC85]; see also [Caf98] and the book [PSU12]. In the case L = ∆, we
also want to mention two recent papers that prove the smoothness of the regular set: Koch, the
second author, and Shi [KPS14] establish the real analyticity of Γ by using hodograph-type
3/2
transformation and subelliptic estimates, and De Silva and Savin [DSS14b] prove C∞ regularity of
Γ by higher-order boundary Harnack principle in slit domains.
3/2
Takingdirectional derivatives, however, does notwork wellfor theproblemstudiedinthis paper,
particularly so since we are working with solutions of the non-homogeneous equation (1.12), which
AN EPIPERIMETRIC INEQUALITY APPROACH, ETC. 5
corresponds to nonzero thin obstacle ϕ. In contrast, the methods in this paper are purely energy
based, and they are new even in the case of the thin obstacle problem for the Laplacian. They
are inspired by the homogeneity improvement approach of Weiss [Wei99] in the classical obstacle
problem. The latter consists of a combination of a monotonicity formula and an epiperimetric
inequality. In this connection we mention that recently in [FGS13], Focardi, Gelli, and Spadaro
extended Weiss’ method to the classical obstacle problem for operators with Lipschitz coefficients.
We also mention a recent preprintby Koch, Ru¨land, and Shi[KRS15a] in which they use Carleman
estimatestoestablishthealmostoptimalinteriorregularityofthesolutioninthevariablecoefficient
Signoriniproblem,whenthecoefficientmatrixisW1,p,withp > n+1. Inapersonalcommunication
[KRS15b] these authors have informed us of work in progress on the optimal interior regularity as
well as the C1,β regularity of the regular set for W1,p coefficients with p > 2(n+1). Their preprint
was not available to us when the present paper was completed.
We next describe our proof of Theorem 1.2 above. The first main ingredient consists of the
“almost monotonicity” of the Weiss-type functional
1 3/2
W (v,r) = [ A(x) v, v +vf] v2µ,
L rn+1 h ∇ ∇ i − rn+2
ZBr ZSr
for solutions of (1.12)–(1.15). In Theorem 4.3 below we prove that W (v,r) + Cr1/2 is nonde-
L
creasing for a universal constant C. Here, we were inspired by [Wei99] and [Wei98], where Weiss
introduced related monotonicity formulas in the classical obstacle problem. In [GP09] two of us
also proved a similar monotonicity formula in the Signorini setting, in the case of the Laplacian.
In the present paper we use the machinery established in [GSVG14] to treat the case of variable
Lipschitz coefficients. We mention that the geometric meaning of the functional W above is that
L
it measures the closeness of the solution v to the prototypical homogeneous solutions of degree 3/2,
i.e., the functions
a ( x′,ν +ix )3/2, a 0, ν S .
n 1
ℜ h i | | ≥ ∈
The second central ingredient in the proof is the epiperimetric inequality for the functional
3
W(v) = W (v,1) = v 2 v2,
∆
|∇ | − 2
ZB1 ZS1
which states that if a (3/2)-homogeneous function w, nonnegative on B′, is close to the solution
1
h(x) = (x +ix )3/2 in W1,2(B )-norm, then there exists ζ in B with ζ = w on ∂B such that
1 n 1 1 1
ℜ | |
W(ζ) (1 κ)W(w),
≤ −
for a universal 0 < κ < 1, see Theorem 6.3 below.
The combination of Theorems 4.3 and Theorem 6.3 provides us with a powerful tool for esta-
blishing the following geometric rate of decay for the Weiss functional:
W (v,r) Crγ,
L
≤
for a universal γ > 0. In turn, this ultimately implies that
v v C x¯ y¯β
x¯,0 y¯,0
| − | ≤ | − |
ZS1′
for properly defined homogeneous blowups v and v at x¯,y¯ Γ (v). This finally implies the
x¯,0 y¯,0 3/2
∈
C1,β regularity of Γ (v) in a more or less standard fashion.
3/2
6 NICOLAGAROFALO,ARSHAKPETROSYAN,ANDMARIANASMITVEGAGARCIA
1.3. Structure of the paper. The paper is organized as follows.
In Section 2 we recall those definitions and results from [GSVG14] which constitute the
•
background results of this paper.
In Section 3 we give a more in depth look at regular free boundary points and prove some
•
preliminarybutimportantpropertiessuchastherelativeopennessoftheregularsetΓ (v)
3/2
in Γ(v) and the local uniformconvergence of the truncated frequency function N˜ (r) 3/2
x¯
→
on Γ (v). We also introduce Almgren type scalings for the solutions, see (3.2), which play
3/2
an important role in Section 7.
In Section 4 we establish the first main technical tool of this paper, the Weiss-type mono-
•
tonicity formula discussed above (Theorem 4.3). This resultis instrumentalto studyingthe
homogeneous blowups of our function, which we do in Section 5.
Section 6 is devoted to proving the second main technical result of this paper, the epiperi-
•
metric inequality (Theorem 6.3) which we have discussed above.
Finally, in Section 7 we combine the monotonicity formula and the epiperimetric inequa-
•
lity to prove the main result of this paper, the C1,β-regularity of the regular set Γ (v)
3/2
(Theorem 1.2).
2. Notation and preliminaries
2.1. Basic notation. Throughoutthe paper we use following notation. We work in the Euclidean
space Rn, n 2. We write the points of Rn as x = (x′,x ), where x′ = (x ,...,x ) Rn−1.
n 1 n−1
Very often, w≥e identify the points (x′,0) with x′, thus identifying the “thin” space Rn−1 ∈0 with
Rn−1. ×{ }
For x Rn, x′ Rn−1 and r > 0, we define the “solid” and “thin” balls
∈ ∈
B (x) = y Rn x y < r , B′(x′) = y′ Rn−1 x′ y′ < r
r { ∈ || − | } r { ∈ || − | }
as well as the corresponding spheres
S (x) = y Rn x y = r , S′(x′) = y′ Rn−1 x′ y′ = r .
r { ∈ || − | } r { ∈ || − | }
We typically do not indicate the center, if it is the origin. Thus, B = B (0), S = S (0), etc. We
r r r r
also denote
B±(x′,0) = B (x′,0) x > 0 , Rn = Rn x > 0 .
r r ∩{± n } ± ∩{± n }
For a given direction e, we denote the corresponding directional derivative by
∂ u= u,e ,
e
h∇ i
whenever it makes sense. For the standard coordinate directions e = e , i = 1,...,n, we also
i
abbreviate ∂ u= ∂ u.
i ei
In the situation when a domain Ω Rn is divided by a manifold M into two subdomains Ω
+
and Ω , ν and ν stand for the exter⊂ior unit normal for Ω and Ω on M. Moreover, we always
− + − + −
understand ∂ u (∂ u) on M as the limit from within Ω (Ω ). Thus, when Ω = B±, we have
ν+ ν− + − ± r
ν = e , ∂ u(x′,0) = lim ∂ u d=ef ∂+u(x′,0), ∂ u(x′,0) = lim ∂ ud=ef ∂−u(x′,0)
± ∓ n − ν+ y→(x′,0) n n ν− y→(x′,0) n n
y∈Br+ y∈Br−
In integrals, we often do not indicate the measure of integration if it is the Lebesgue measure on
subdomains of Rn, or the Hausdorff Hk measure on manifolds of dimension k.
AN EPIPERIMETRIC INEQUALITY APPROACH, ETC. 7
Hereafter, whenwesay thataconstant is universal, wemeanthatitdependsexclusively on n,on
the ellipticity bound λ on A(x), and on the Lipschitz bound Q on the coefficients a (x). Likewise,
ij
wewillsay thatO(1), O(r), etc, areuniversalif O(1) C, O(r) Cr,etc, with C 0universal.
| | ≤ | | ≤ ≥
2.2. Summary of known results. For the convenience of the reader, in this section we briefly
recall the definitions and results proved in [GSVG14] which will be used in this paper.
As stated in Section 1, we work under the nonrestrictive situation in which the thin manifold
M is flat. More specifically, we consider the Signorini problem (1.1) when the thin obstacle ϕ is
defined on
B′ = M B = (x′,0) B x′ < 1 .
1 ∩ 1 { ∈ 1 || | }
We assume that ϕ C1,1(B′) and denote by u the unique solution to the minimization problem
∈ 1
(1.1), which then satisfies (1.6)–(1.9). We assume that 0 is a free boundary point, i.e., 0 Γϕ(u),
∈
and that A(0) = I, and we consider the normalization of u as in (1.10), i.e.,
v(x) = u(x) ϕ(x′)+bx , with b = ∂ u(0).
− n ν+
As remarked on Section 1, v satisfies (1.12)–(1.15) with f d=ef L(ϕ(x′) bx ) L∞(B ), and has
n 1
− − ∈
the additional property that v(0) = v(0) = 0, see (1.11) above.
|∇ |
We then recall the following definitions from [GSVG14]. The Dirichlet integral of v in B is
r
defined by
D(r)= D (v,r) = A(x) v, v ,
L
h ∇ ∇ i
ZBr
and the height function of v in S is given by
r
(2.1) H(r) = H (v,r) = v2µ,
L
ZSr
where µ is the conformal factor
A(x)x,x
µ(x)= µ (x) = h i.
L x 2
| |
We notice that, when A(x) I, then µ 1. We also define the generalized energy of v in B
r
≡ ≡
(2.2) I(r)= I (v,r) = v A v,ν = D (v,r)+ vf
L L
h ∇ i
ZSr ZBr
where ν indicates the outer unit normal to S . The following result is Lemma 4.4 in [GSVG14].
r
Lemma 2.1. The function H(r) is absolutely continuous, and for a.e. r (0,1) one has
∈
(2.3) H′(r)= 2I(r)+ v2L x .
| |
ZSr
As it was explained in [GSVG14], the second term in the right-hand side of (2.3) above repre-
sents a serious difficulty to overcome if one wants to establish the monotonicity of the generalized
frequency. To bypass this obstacle, one of the main ideas in [GSVG14] was the introduction of the
following auxiliary functions, defined for v satisfying (1.12)–(1.15) and 0 < r < 1:
R v2L|x|
(2.4) ψ(r) = eR0rG(s)ds, σ(r) = rψn(−r2), where G(r) = nSR−rS1r,v2µ , iiff HH((rr)) =6= 00.,
 r
When L = ∆, it is easy to see that ψ(r) = rn−1 and that σ(r) = r. We have the following simple

and useful lemma which summarizes the most relevant properties of ψ(r) and σ(r).
8 NICOLAGAROFALO,ARSHAKPETROSYAN,ANDMARIANASMITVEGAGARCIA
Lemma 2.2. There exists a universal constant β 0 such that
≥
n 1 d n 1
(2.5) − β logψ(r) − +β, 0 <r < 1,
r − ≤ dr ≤ r
and one has
(2.6) e−β(1−r)rn−1 ψ(r) eβ(1−r)rn−1, 0< r < 1.
≤ ≤
This implies, in particular, ψ(0+) = 0. In terms of the function σ(r) = ψ(r)/rn−2 we have
d σ(r)
(2.7) log β, 0< r < 1
dr r ≤
(cid:12) (cid:12)
(cid:12) (cid:12)
and (cid:12) (cid:12)
(cid:12) (cid:12)
σ(r)
(2.8) e−β(1−r) eβ(1−r), 0 < r < 1.
≤ r ≤
In particular, σ(0+) = 0.
The next result is essentially Lemma 5.6 from [GSVG14].
Lemma 2.3. There exist α > 0 such that
σ(r)
(2.9) α βeβr, r (0,1),
r − ≤ ∈
(cid:12) (cid:12)
(cid:12) (cid:12)
for β as in Lemma 2.2. In particu(cid:12)lar, (cid:12)
(cid:12) (cid:12)
σ(r)
(2.10) α = lim .
r→0+ r
Moreover, we also have that e−β α eβ.
≤ ≤
With ψ as in (2.4), we now define
1 1
(2.11) M (v,r) = H (v,r), J (v,r) = I (v,r).
L L L L
ψ(r) ψ(r)
The next relevant formulas are those of J′(r) = d J (v,r) and M′(r)= d M (v,r). Using formula
dr L dr L
(5.28) in [GSVG14], we have
ψ′(r) n 2 1 A v,ν 2
(2.12) J′(r) = + − +O(1) J(r)+ 2 h ∇ i
−ψ(r) r ψ(r) µ
(cid:18) (cid:19) (cid:26) ZSr
2 n 2
Z, v f − +O(1) vf + vf ,
− r h ∇ i − r
ZBr (cid:18) (cid:19)ZBr ZSr (cid:27)
where the vector field Z is given by
rA r A(x)x
(2.13) Z = ∇ = .
µ µ
We also recall that (5.26) in [GSVG14] gives
(2.14) M′(r)= 2J(r).
The central result in [GSVG14] is the following monotonicity formula.
AN EPIPERIMETRIC INEQUALITY APPROACH, ETC. 9
Theorem 2.4 (Monotonicity of the truncated frequency). Let v satisfy (1.11)–(1.15) with f
∈
L∞(B ). Given δ (0,1) there exist universal numbers r ,K′ > 0, depending also on δ and
1 0
∈
f , such that the function
L∞
k k
(2.15) N(r)= N (v,r) d=ef σ(r)eK′r1−2δ d logmax M (v,r),r3+δ .
L L
2 dr
n o
is monotone non-decreasing on (0,r ).
0
We call N(r) the truncated frequency function, by analogy with Almgren’s frequency function
[Alm79] (see [GSVG14,GP09,CSS08] for more insights on this kind of formulas).
We then define a modification of N as follows:
r r 1−δ d
N˜(r)= N˜ (v,r) d=ef N (v,r) = eK′r 2 logmax M (v,r),r3+δ .
L L L
σ(r) 2 dr { }
We notice that by Theorem 2.4 the limit N(0+) exists. Combining that with Lemma 2.3 above,
which states that lim σ(r) = α > 0, we see that N˜(0+) also exists.
r
r→0+
The following lemma provides a summary of estimates which are crucial for our further study.
The lower bound on N˜(0+) is proved in Lemma 6.3 in [GSVG14] (whose proof contains also that
of the gap on the possible values of N˜(0+)), the bound on v(x) is Lemma 6.6 and the bound on
| |
v(x) is proved in Theorem 6.7 there.
|∇ |
Lemma 2.5. Let v satisfy (1.11)–(1.15) with f L∞(B ), and let r (0,1/2] be as in Theo-
1 0
∈ ∈
rem 2.4. Then, N˜(0+) 3, and actually N˜(0+) = 3 or N˜(0+) 3+δ.
≥ 2 2 ≥ 2
Moreover, there exists a universal C depending also on δ, H(r ) and f such that
0 k kL∞(B1)
(2.16) v(x) C x 3/2, v(x) C x 1/2, x r .
0
| | ≤ | | |∇ | ≤ | | | | ≤
Corollary 2.6. With r as in Theorem 2.4, one has
0
(2.17) H(r) Crn+2, I(r) Crn+1, r r .
0
≤ | | ≤ ≤
Proof. It is enough to use (2.16) in definitions (2.1) and (2.2) above. (cid:3)
The results of this section have been stated when the free boundary point in question is the
origin. However, given any x Γ(v), we can move x to the origin by letting
0 0
∈
v (x) = v(x +A1/2(x )x) b x , where b = A1/2(x ) v(x ),e ,
x0 0 0 − x0 n x0 h 0 ∇ 0 ni
A (x) = A−1/2(x )A(x +A1/2(x )x)A−1/2(x ),
x0 0 0 0 0
µ (x) = A (x)ν(x),ν(x) ,
x0 h x0 i
L = div(A ).
x0 x0∇·
(Note that, by the C1,21-regularity of v established in [GSVG14], the mapping x0 7→ bx0 is C12 on
Γ(v).) Then, by construction we have the normalizations A (0) = I , µ (0) = 1. We also know
x0 n x0
that 0 Γ(v ), and that
∈ x0
v (0) = v(x )= 0, v (0) = 0.
x0 0 |∇ x0 |
Besides, v satisfies (1.12)–(1.15) for the operator L . Thus, all results stated above for v are
x0 x0
also applicable to v .
x0
We thus also have the versions of the quantities defined in this sections, such as M , N , etc,
L L
centered at x (if we replace L with L ). But instead of using the overly bulky notations M ,
0 x0 Lx0
N , etc, we will use M , N , etc.
Lx0 x0 x0
10 NICOLAGAROFALO,ARSHAKPETROSYAN,ANDMARIANASMITVEGAGARCIA
3. Regular free boundary points
Using Theorem 2.4, in this section we explore in more detail the notion of regular free boundary
points and establish some preliminary properties of the regular set. We begin by recalling the
following definition from Section 1.
Definition 3.1. We say that x Γ(v) is regular iff N˜ (v ,0+) = 3 and let Γ (v) be the set
0 ∈ x0 x0 2 3/2
of all regular free boundary points. Γ (v) is also called the regular set.
3/2
In Lemma 3.3 below we prove that Γ (v) is a relatively open subset of the free boundary Γ(v).
3/2
To accomplish this we prove that N˜ (0+) = 3/2 for x¯ in a small neighborhood of x Γ (v).
x¯ 0 3/2
∈
Since the definition of N˜(r) involves a truncation of M(r), we first need to establish the following
auxiliary result.
Lemma 3.2. Let v satisfy (1.12)–(1.15) with 0 Γ (v). Then
3/2
∈
rM′(r) 3
as r 0+.
2 M(r) → 2 →
In particular, for every ε > 0 there exists r > 0 and C > 0 such that
ε ε
r M′(r) 3+ε
, M(r) C r3+ε, for 0< r r .
ε ε
2 M(r) ≤ 2 ≥ ≤
Proof. We first claim that since 0 Γ (v), then M(r) r3+δ for r > 0 small. Indeed, if there
3/2
∈ ≥
was a sequence s 0 such that M(s ) < s3+δ, then
j → j j
3+δ 1−δ 3+δ 3
N˜(sj) = eK′sj2 = = N˜(0+),
2 → 2 6 2
which is a contradiction. Hence, for r small we have
r 1−δ M′(r)
N˜(r) = eK′r 2 .
2 M(r)
1−δ
Since eK′r 2 1 as r 0 and N˜(0+) = 3/2, we obtain the first part of the lemma. Hence, for
→ →
every ε > 0 there exists a small r > 0 such that
ε
M′(r)
r 3+ε, r < r .
ε
M(r) ≤
Integrating from r to r , this gives
ε
M(r ) r 3+ε
ε ε
,
M(r) ≤ r
(cid:16) (cid:17)
from which we conclude, with C = M(r )/r3+ε, that M(r) C r3+ε. (cid:3)
ε ε ε ≥ ε
Lemma 3.3. Let v satisfy (1.12)–(1.15) with x Γ (v). Then, there exists η = η (x ) > 0
0 3/2 0 0 0
∈
such that Γ(v) B′ (x ) Γ (v) and, moreover, the convergence
∩ η0 0 ⊂ 3/2
N˜ (r) 3/2 as r 0+
x¯
→ →
is uniform for x¯ Γ(v) B′ (x ).
∈ ∩ η0/2 0