Sharp Poincar\'e-type inequality for the Gaussian measure on the boundary of convex sets

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by  Alexander V. Kolesnikov

Sharp Poincar´e-type inequality for the Gaussian measure
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on the boundary of convex sets
1
0
2
l Alexander V. Kolesnikov1 and Emanuel Milman2
u
J

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1 Abstract

] AsharpPoincar´e-typeinequalityisderivedfortherestrictionoftheGaussian
A
measure on the boundary of a convex set. In particular, it implies a Gaussian
F mean-curvature inequality and a Gaussian iso-second-variation inequality. The
h. new inequality is nothing but an infinitesimal equivalent form of Ehrhard’s in-
t equality for the Gaussian measure. While Ehrhard’s inequality does not extend
a
togeneralCD(1,∞)measures,weformulateasufficientconditionforthevalidity
m
of Ehrhard-type inequalities for general measures on Rn via a certain property
[
of an associated Neumann-to-Dirichlet operator.

2
v
5 1 Introduction
2
9
We consider Euclidean space (Rn,h·,·i) equipped with the standard Gaussian mea-
2
0 sure γ = Ψ dx, Ψ (x) = (2π) n/2exp(−|x|2/2). Let K ⊂ Rn denote a convex
γ γ −
1. domain with C2 smooth boundary and outer unit-normal field ν = ν∂K. The second
0 fundamental form II = II of ∂K at x ∈ ∂K is as usual (up to sign) defined by
∂K
6
II (X,Y) = h∇ ν,Yi, X,Y ∈ T ∂K. The quantities:
1 x X x
:
v
H(x) := tr(II ) , H (x) := H(x)−hx,ν(x)i ,
i x γ
X
r are called the mean-curvature and Gaussian mean-curvature of ∂K at x ∈ ∂K, re-
a
spectively. It is well-known that H governs the first variation of the (Lebesgue)
1Faculty of Mathematics, Higher School of Economics, Moscow, Russia. Supported by RFBR
project 14-01-00237 and the DFG project CRC 701. This study (research grant No 14-01-0056)
was supported by The National Research University – Higher School of Economics’ Academic Fund
Program in 2014/2015. Emails: [email protected], [email protected].
2Department of Mathematics, Technion - Israel Instituteof Technology, Haifa 32000, Israel. The
researchleadingtotheseresultsispartofaprojectthathasreceivedfundingfromtheEuropeanRe-
searchCouncil(ERC)undertheEuropeanUnion’sHorizon2020researchandinnovationprogramme
(grant agreement No 637851). In addition, E.M. was supported by BSF (grant no. 2010288) and
Marie-Curie Actions (grant no. PCIG10-GA-2011-304066). Email: [email protected].
1
boundary-measure Vol under the normal-map t 7→ exp(tν), and similarly H gov-
∂K γ
erns the first variation of the Gaussian boundary-measure γ := Ψ Vol , see e.g.
∂K γ ∂K
[15] or Section 2.
Recall that the Gaussian isoperimetric inequality of Borell [6] and Sudakov–
Tsirelson [20] asserts that if E is a half-plane with γ(E) = γ(K), then γ (∂K) ≥
∂K
γ (∂E) (in fact, this applies not just to convex sets but to all Borel sets, with an
∂E
appropriate interpretation of Gaussian boundary measure). In other words:
γ (∂K) ≥ I (γ(K))
∂K γ
with equality for half-planes, where I :[0,1] → R denotes the Gaussian isoperimet-
γ +
ricprofile,givenbyI := ϕ◦Φ 1 withϕ(t) = 1 exp(−t2/2)andΦ(t)= t ϕ(s)ds.
γ − √2π
Note that I is concave and symmetric around 1/2, hence it is increasingR−o∞n [0,1/2]
γ
and decreasing on [1/2,1].
Our main result is the following new Poincar´e-type inequality for the Gaussian
boundary-measure on ∂K:
Theorem 1.1. For all convex K and f ∈ C1(∂K) for which the following expressions
make sense, we have:
2
H f2dγ −(logI )(γ(K)) fdγ ≤ II 1∇ f,∇ f dγ .
Z γ ∂K γ ′ (cid:18)Z ∂K(cid:19) Z −∂K ∂K ∂K ∂K
∂K ∂K ∂K(cid:10) (cid:11)
(1.1)
Here∇ f denotesthe gradientoff on∂K withitsinducedmetric, and(logI )(v) =
∂K γ ′
−Φ 1(v)/I (v).
− γ
This inequality is already interesting for the constant function f ≡1:
Corollary 1.2 (Gaussian mean-curvature inequality).
H dγ ≤ (logI )(γ(K))γ (∂K)2. (1.2)
γ ∂K γ ′ ∂K
Z
∂K
In particular, if γ(K)≥ 1/2 then necessarily H dγ ≤ 0.
∂K γ ∂K
R
The latter inequality is sharp, yielding an equality when K is any half-plane E.
Indeed, since I (γ(E)) = γ (∂E), it is enough to note that E = (−∞,t]×Rn 1 has
γ ∂E −
constant Gaussian mean-curvature H = −t = (logϕ)(t)= I (γ(E)).
γ ′ γ′
More surprisingly, we will see in Section 4 that Corollary 1.2 in fact implies the
Gaussianisoperimetricinequality (albeitonlyforconvex sets). Furthermore,wehave:
2
Corollary 1.3 (Gaussian iso-curvature inequality). If E is a half-plane with γ(E) =
γ(K) ≥ 1/2, then the following iso-curvature inequality holds:
H dγ ≤ H dγ ( ≤ 0).
γ ∂K γ ∂E
Z Z
∂K ∂E
Proof. Thisisimmediatefrom(1.2),theGaussianisoperimetricinequalityγ (∂K) ≥
∂K
γ (∂E), the assumption that (logI )(γ(K)) ≤0, and the equality in (1.2) for half-
∂E γ ′
planes.
Clearly, by passing to complements, the latter corollary yields a reverse inequality
when applied to K, the complement to a convex set C satisfying γ(K) ≤ 1/2 (since
∂K = ∂C with reverse orientation and thus their generalized mean-curvature simply
changes sign). It is also easy to check that a reverse inequality holds when K is a
small Euclidean (convex) ball centered at the origin. It is probably unreasonable to
expect that a reverse inequality holds for all convex K with γ(K) ≤ 1/2, but we have
not seriously searched for a counterexample.
Weproceedtogive followinginterpretation ofthelatter twocorollaries. Denoting:
δ0(K) = γ(K) , δ1(K) = γ (∂K) , δ2(K) = H dγ ,
γ γ ∂K γ γ ∂K
Z
∂K
we note that δi(K) is precisely the i-th variation of the function t 7→ γ(K ), where
γ t
K := {x ∈ Rn ; d(x,K) ≤ t}andddenotesEuclideandistance. Consequently,Corol-
t
lary 1.3 may be rewritten as:
Corollary 1.4 (Gaussian iso-second-variation inequality). If E is a half-plane with
γ(E) = γ(K) ≥ 1/2, then the following iso-second-variation inequality holds:
δ2(K)≤ δ2(E) ( ≤ 0).
γ γ
It is interesting to note that we are not aware of an analogous statement on any
other metric-measure space, and in particular, for the Lebesguemeasure in Euclidean
space, as all known isoperimetric inequalities only pertain (by definition) to the first-
variation (and with reversed direction of the inequality). Furthermore, in contrast
to the isoperimetric inequality, it is easy to see that the second-variation inequality
above is false without the assumption that K is convex, as witnessed for instance by
taking the complement of any non-degenerate slab {x ∈ Rn ; a ≤ x ≤ b} of measure
1
1/2. As for Corollary 1.2, we see that it may be rewritten as:
Corollary 1.5 (Minkowski’s second inequality for Euclidean Gaussian extensions).
δ2(K)≤ (logI )(δ0(K))(δ1(K))2. (1.3)
γ γ ′ γ γ
3
This already hints at the proof of Theorem 1.1. To describe the proof, and put
the latter interpretation in the appropriate context, let us recall some classical facts
from the Brunn-Minkowski theory (for the Lebesgue measure).
1.1 Brunn–Minkowski inequality
The Brunn–Minkowski inequality [19, 13] asserts that:
Vol((1−t)K +tL)1/n ≥ (1−t)Vol(K)1/n +tVol(L)1/n , ∀t ∈ [0,1] , (1.4)
for all convex K,L ⊂ Rn; it was extended to arbitrary Borel sets by Lyusternik.
Here Vol denotes Lebesgue measure and A + B := {a+b ; a ∈ A,b ∈ B} denotes
Minkowskiaddition. WerefertotheexcellentsurveybyR.Gardner[13]foradditional
details and references.
For convex sets, (1.4) is equivalent to the concavity of the function t 7→ Vol(K +
tL)1/n. By Minkowski’s theorem, extending Steiner’s observation for the case that L
is the Euclidean ball, Vol(K +tL) is an n-degree polynomial n n W (K,L)ti,
i=0 i n i
−
whose coefficients P (cid:0) (cid:1)
i
(n−i)! d
W (K,L) := Vol(K +tL) , (1.5)
n i (cid:12)
− n! (cid:18)dt(cid:19) (cid:12)
(cid:12)t=0
(cid:12)
arecalledmixed-volumes. Theaboveconcavityth(cid:12)usamountstothefollowing“Minkowski’s
second inequality”, which is a particular case of the Alexandrov–Fenchel inequalities:
W (K,L)2 ≥ W (K,L)W (K,L) . (1.6)
n 1 n 2 n
− −
Specializing tothecase thatL istheEuclidean unit-ballD, notingthatK = K+tD,
t
and denoting by δi(K) the i-th variation of t 7→ Vol(K ), we have as before:
t
δ0(K)= Vol(K) , δ1(K) = Vol (∂K) , δ2(K)= HdVol .
∂K ∂K
Z
∂K
The corresponding distinguished mixed-volumes W (K) = W (K,D), which are
n i n i
called intrinsic-volumes or quermassintegrals, are rel−ated to δi(K−) via (1.5). Conse-
quently, when L = D, Minkowski’s second inequality amounts to the inequality:
n−1 1
δ2(K) ≤ (δ1(K))2.
n δ0(K)
Theanalogy with (1.3) becomes apparent, in view of the fact that (logI)(v) = n 11,
′ −n v
whereI(v) = cnvn−n1 isthestandardisoperimetricprofileofEuclideanspace(Rn,h·,·i)
endowed with the Lebesgue measure.
4
An important difference to note with respect to the classical theory, is that in the
Gaussian theory, δ2(K) may actually be negative, in contrast to the non-negativity
γ
of all mixed-volumes, and in particular of δ2(K). One reason for this is that the
GaussianmeasureisfinitewhereastheLebesguemeasureisnot,sothatI ismonotone
increasingwhereasI isnot. Thisfeatureseemstoalso beresponsibleforthepeculiar
γ
iso-second-variation corollary.
1.2 Ehrhard inequality
A remarkable extension of the Brunn-Minkowski inequality to the Gaussian setting
was obtained by Ehrhard [11], who showed that:
Φ 1(γ((1−t)K +tL))≥ (1−t)Φ 1(γ(K))+tΦ 1(γ(L)) ∀t∈ [0,1],
− − −
for all convex sets K,L ⊂ Rn, with equality when K and L are parallel half-planes
(pointing in the same direction). This was later extended by Lata la [16] to the case
that only one of the sets is assumed convex, and finally by Borell [4, 5] to arbitrary
Borel sets. As before, for K,L convex sets, Ehrhard’s inequality is equivalent to the
concavity of the function t 7→ F (t) := Φ 1(γ((1−t)K +tL)).
γ −
To prove Theorem 1.1, we repeat an idea of A. Colesanti. In [9] (see also [10]),
Colesanti showed that the Brunn-Minkowski concavity of t 7→ F(t) := Vol((1−t)K+
tL)1/n is equivalent to a certain Poincar´e-type inequality on ∂K, by parametrizing
K,L viatheirsupportfunctionsandcalculating thesecondvariation ofF(t). Repeat-
ing the calculation for F (t), Theorem 1.1 turns out to be an equivalent infinitesimal
γ
reformulation of Ehrhard’s inequality for convex sets.
1.3 Comparison with Previous Results
Going in the other direction, we have recently shown in our previous work [15] how
to directly derive a Poincar´e-type inequality on the boundary of a locally-convex
subset of a weighted Riemannian manifold, which may then be used to infer a Brunn-
Minkowski inequality in the weighted Riemannian setting via an appropriate geomet-
ric flow. In particular, in the Euclidean setting, our results apply to Borell’s class
of 1/N-concave measures [7] (1 ∈ [−∞, 1]), defined as those measures µ on Rn
N n
satisfying the following generalized Brunn-Minkowski inequality:
N
µ((1−t)A+tB)≥ (1−t)µ(A)1/N +tµ(B)1/N ,
(cid:16) (cid:17)
for all t ∈ [0,1] and Borel sets A,B ⊂ Rn with µ(A),µ(B) > 0. It was shown by
Brascamp–Lieb [8]andBorell [7]thattheabsolutely continuous membersof thisclass
5
areprecisely characterized byhavingdensity Ψsothat(N−n)Ψ1/(N n) is concave on
−
its convex support Ω (interpreted as logΨ being concave when N = ∞), amounting
to the Bakry–E´mery CD(0,N) condition [2, 1, 14]. Our results from [15] then imply
that for any (say) compact convex K in the interior of Ω with C2 boundary, and any
f ∈C1(∂K), one has:
2
N −1 1
H f2dµ − fdµ ≤ II 1 ∇ f,∇ f dµ ,
Z µ ∂K N µ(K)(cid:18)Z ∂K(cid:19) Z −∂K ∂K ∂K ∂K
∂K ∂K ∂K(cid:10) (cid:11)
(1.7)
with µ = ΨVol denoting the boundary measure and H = H +hlogΨ,νi the
∂K ∂K µ
µ-weighted mean-curvature. Note that the Gaussian measure γ satisfies CD(1,∞)
and in particular CD(0,∞), as logΨ is concave on Rn. Consequently, applying(1.7)
γ
with N = ∞, we have:
2
1
H f2dγ − fdγ ≤ II 1 ∇ f,∇ f dγ . (1.8)
Z µ ∂K γ(K) (cid:18)Z ∂K(cid:19) Z −∂K ∂K ∂K ∂K
∂K ∂K ∂K(cid:10) (cid:11)
It is easy to verify that (logI )(v) < 1 for all v ∈ (0,1), and hence Theorem 1.1
γ ′ v
constitutes an improvement over (1.8).
A very important point is that the latter improvement is strict only for test func-
tions f with non-zero mean, fdγ 6= 0. Put differently, the entire significance of
∂K ∂K
Theorem 1.1 lies in the coeffiRcient infront of the fdγ 2 term, since by (1.7),
∂K ∂K
for zero-mean test functions, the inequality asser(cid:0)teRd in Theo(cid:1)rem 1.1 holds not only
for the Gaussian measure, butin fact for Borell’s entire class of concave (or CD(0,0))
measures (using our convention from [14, 15] that N 1 = −∞ when N = 0 and that
N−
∞·0= 0).
Unfortunately, our method from [15], involving L2-duality and the Reilly formula
from Riemannian geometry, cannot be used in the Gaussian setting without some
additional ingredients, like information on an associated Neumann-to-Dirichlet op-
erator, see Section 3. In particular, we observe in Section 4 that Theorem 1.1 (or
equivalently, Ehrhard’s inequality for convex sets) and even Corollary 1.2, are simply
false for a general CD(1,∞) probability measure in Euclidean space, having density
Ψ = exp(−V) with ∇2V ≥Id.
2 Proof of Theorem 1.1
The general formulation of Theorem 1.1 is reduced to the case that K is compact
with strictly-convex C3 smooth boundary (II > 0) by a standard (Euclidean)
∂K
approximation argument - this class of convex sets is denoted by C3 (and analogously
+
6
we define the class C2). As explained in the Introduction, the proof of Theorem 1.1
+
boils down to a direct calculation of the second variation of the function:
t 7→ Φ 1(γ((1−t)K +tL))
−
for an appropriately chosen L. Ehrhard’s inequality ensures that this function is
concave when K,L are convex.
The second variation will be conveniently expressed using support functions. Re-
call that the support function of a convex body (convex compact set with non-empty
interior) C is defined as the following function on the Euclidean unit sphere Sn 1:
−
h (θ):= sup{hθ,xi;x ∈ C} , θ ∈ Sn 1.
C −
ItiseasytoseethatthecorrespondenceC 7→ h betweenconvexbodiesandfunctions
C
on Sn 1 is injective and positively linear: h = ah +bh for all a,b ≥ 0. As
− aC1+bC2 C1 C2
K ∈ C3 we know that h is C3 smooth [19, p. 106].
+ K
Now let f ∈ C2(∂K), and consider the function h := f ◦ ν 1 : Sn 1 → R.
∆ ∂−K −
Since K ∈ C3 this function is well-defined and C2 smooth. Moreover, it is not hard
+
to show (e.g. [19, pp. 38, 111],[9]) that for ǫ > 0 small enough, h + th is the
K ∆
support function of a convex body K ∈ C2 for all t ∈ [0,ǫ]. It follows by linearity of
t +
the support functions that K = (1−t)K +tK for all t ∈ [0,1], and so Ehrhard’s
ǫt ǫ
inequality implies that:
t 7→ F (t) := Φ 1(γ(K ))
γ − t
is concave on [0,ǫ].
The first and second variations of t 7→ µ(K ) were calculated by Colesanti in [9]
t
for the case that µ is the Lebesgue measure, and for general measures µ with positive
density Ψ by the authors in [15] (in fact in a general weighted Riemannian setting,
with an appropriate interpretation of K avoiding support functions):
t
δ0 := (d/dt)0| µ(K ) = µ(K) ,
t=0 t
δ1 := (d/dt)1| µ(K ) = fdµ ,
t=0 t ∂K
Z
∂K
δ2 := (d/dt)2| µ(K ) = H f2dµ − II 1∇ f,∇ f dµ .
t=0 t Z µ ∂K Z −∂K ∂K ∂K ∂K
∂K ∂K(cid:10) (cid:11)
Applying the above formulae for µ = γ, calculating:
0 ≥ F (0) = (Φ 1) (δ0)(δ1)2+(Φ 1)(δ0)δ2,
γ′′ − ′′ − ′
dividing by (Φ 1)(δ0)> 0 and using that:
− ′
(Φ 1) (v)
− ′′
= −(logI )(v),
(Φ 1)(v) γ ′
− ′
7
Theorem 1.1 readily follows for f ∈ C2(∂K). The general case for f ∈ C1(∂K) is
obtained by a standard approximation argument.
Going in the other direction, it should already be clear that Theorem 1.1 implies
back Ehrhard’s inequality. Indeed, given K,L ∈ C2, consider K = (1−t)K +tL
+ t
for t ∈ [0,1], and note that h = (1−t)h +th . Fixing t ∈ (0,1), it follows that
Kt K L 0
h = h +ǫ(h −h ). Inspecting the proof above, we see that the statement of
Kt0+ǫ Kt0 L K
Theorem 1.1 for K and f = (h −h )◦ν ∈ C1(∂K), is precisely equivalent to the
t0 L K
concavity of the function ǫ 7→ Φ 1(γ(K )) at ǫ = 0. Since the point t ∈ (0,1) was
− t0+ǫ 0
arbitrary, we see that Theorem 1.1 implies the concavity of [0,1] ∋ t 7→ Φ 1(γ((1−
−
t)K+tL)) for K,L ∈ C2. The case of general convex K,L follows by approximation.
+
3 Neumann-to-Dirichlet Operator
Inthissection,wementionhowacertainpropertyofaNeumann-to-Dirichletoperator
can be used to directly obtain an Ehrhard-type inequality for general measures µ =
exp(−V(x))dx on Rn (say with C2 positive density). Define the associated weighted
Laplacian L =L as:
µ
L = L := exp(V)∇·(exp(−V)∇) = ∆−h∇V,∇i .
µ
Given a compact set Ω ⊂ Rn with C1 smooth boundary, note that the usual integra-
tion by parts formula is satisfied for f,g ∈ C2(Ω):
L(f)gdµ = f gdµ − h∇f,∇gidµ = (f g−g f)dµ + L(g)fdµ ,
ν ∂M ν ν ∂Ω
Z Z Z Z Z
Ω ∂M Ω ∂Ω Ω
where u = ν ·u.
ν
Given a compact convex body K ∈ C2 and f ∈ C1,α(∂K), let us now solve the
+
following Neumann Laplace equation:
1
Lu ≡ fdµ on K , u = f on ∂K. (3.1)
∂K ν
µ(K) Z
∂K
Since the compatibility condition Ludµ = fdµ is satisfied, it is known (e.g.
K ∂K ∂K
[15]) that a solution u ∈ C2,α(K)Rexists (andRis unique up to an additive constant).
The operator mapping f 7→ u is called the Neumann-to-Dirichlet operator.
Theorem 3.1. Assume that there exists a function F :R → R so that for all K, f
+
and u as above:
F(µ(K))( fdµ )2 ≤ (h∇ f,∇ ui+u f)dµ . (3.2)
∂K ∂K ∂K ν,ν ∂K
Z Z
∂K ∂K
8
Denote G(v) := 1 − F(v) and Φ 1(v) := v exp(− t G(s)ds)dt. Then for all
v −µ 1/2 1/2
f ∈C1(∂K): R R
2
H f2dµ −G(µ(K)) fdµ ≤ II 1∇ f,∇ f dµ , (3.3)
Z µ ∂K (cid:18)Z ∂K(cid:19) Z −∂K ∂K ∂K ∂K
∂K ∂K ∂K(cid:10) (cid:11)
and for all convex K,L ⊂ Rn and t ∈ [0,1]:
Φ 1((1−t)K +tL) ≥ (1−t)Φ 1(K)+tΦ 1(L). (3.4)
−µ −µ −µ
For the proof, we require the following lemma. We denote by ∇2u the Hilbert-
Schmidt norm of the Hessian ∇2u. (cid:13) (cid:13)
(cid:13) (cid:13)
Lemma 3.2. With the above notation:
∇2V ∇u,∇u + ∇2u 2 dµ = (h∇ f,∇ ui+u f)dµ .
∂K ∂K ν,ν ∂K
ZK(cid:16)(cid:10) (cid:11) (cid:13) (cid:13) (cid:17) Z∂K
(cid:13) (cid:13)
Remark 3.3. The integrand on the left-hand-side above is the celebrated Bakry–
E´mery iterated carr´e-du-champ Γ (u), associated to (K,h·,·i,µ) [2].
2
Proof. Denoting ∇u= (u ,...,u ), we calculate:
1 n
n n n
∇2u 2dµ = |∇u |2dµ = u u dµ − L(u )u dµ.
i i,ν i ∂K i i
Z Z Z Z
K(cid:13) (cid:13) KXi=1 ∂KXi=1 KXi=1
(cid:13) (cid:13)
To handlethe L(u ) terms, we take the i-th partial derivative in the Laplace equation
i
(3.1), yielding:
0= (Lu) = L(u )−h∇u,∇V i.
i i i
Consequently, we have:
n
L(u )u = ∇2V ∇u,∇u ,
i i
Xi=1 (cid:10) (cid:11)
and therefore:
n
∇2V ∇u,∇u + ∇2u 2 dµ = u u dµ .
i,ν i ∂K
ZK(cid:16)(cid:10) (cid:11) (cid:13) (cid:13) (cid:17) Z∂KXi=1
(cid:13) (cid:13)
Recalling that f = u , the assertion follows.
ν
9
Proof of Theorem 3.1. As in [15], our starting point is the generalized Reilly formula
[14], which is an integrated form of Bochner’s formula in the presence of a boundary.
Inthe Euclidean setting, it states thatfor any u∈ C2(K)(see [14] for less restrictions
on u):
(Lu)2dµ = ∇2u 2dµ+ ∇2V ∇u,∇u dµ+
Z Z Z
K K(cid:13) (cid:13) K(cid:10) (cid:11)
(cid:13) (cid:13)
H (u )2dµ + hII ∇ u,∇ uidµ −2 h∇ u ,∇ uidµ .
µ ν ∂K ∂K ∂K ∂K ∂K ∂K ν ∂K ∂K
Z Z Z
∂K ∂K ∂K
(3.5)
As we assume that II > 0, we may apply the Cauchy–Schwarz inequality to the
∂K
last-term above:
2h∇ u ,∇ ui≤ hII ∇ u,∇ ui+ II 1∇ u ,∇ u , (3.6)
∂K ν ∂K ∂K ∂K ∂K −∂K ∂K ν ∂K ν
(cid:10) (cid:11)
yielding:
(Lu)2dµ ≥ ∇2u 2dµ+ ∇2V ∇u,∇u dµ
Z Z Z
K K(cid:13) (cid:13) K(cid:10) (cid:11)
(cid:13) (cid:13)
+ H (u )2dµ − II 1 ∇ u ,∇ u dµ .
Z µ ν ∂K Z −∂K ∂K ν ∂K ν ∂K
∂K ∂K(cid:10) (cid:11)
Given f ∈ C1,α(∂K), we now apply the above inequality to the solution u of the
Neumann Laplace equation (3.1). Together with Lemma 3.2, this yields:
2
1
H (u )2dµ − fdµ + (h∇ f,∇ ui+u f)dµ
µ ν ∂K ∂K ∂K ν,ν ∂K
Z µ(K)(cid:18)Z (cid:19) Z
∂K K ∂K
≤ II 1 ∇ f,∇ f dµ .
Z −∂K ∂K ∂K ∂K
∂K(cid:10) (cid:11)
Invoking ourassumption (3.2), theasserted inequality (3.3)follows for f ∈ C1,α(∂K).
The case of a general f ∈C1(∂K) follows by a standard approximation argument.
Lastly, (3.4)isanequivalentversionof(3.3). Indeed,theproofprovidedinSection
2 demonstrates how to pass from (3.4) to (3.3), with:
(Φ 1) (v)
G(v) = −µ ′′ = (log((Φ 1)))(v).
(Φ 1)(v) −µ ′ ′
−µ ′
To see the other direction, repeat the argument described in the previous section.
After establishing (3.4) for K,L ∈ C2, the general case follows by a standard approx-
+
imation argument.
10