Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces

34 Pages·2003·0.283 MB·English

by  Maz'ya V.

ContemporaryMathematics
Lectures on Isoperimetric and Isocapacitary Inequalities in
the Theory of Sobolev Spaces
Vladimir Maz’ya
Abstract. Oldandnewauthor’sresultsonequivalenceofvariousisoperimet-
ricandisocapacitaryinequalities,ononehand,andSobolev’stypeimbedding
andcompactnesstheorems,ontheotherhand,aredescribed. Itisprovedthat
someimbeddingsintofractionalBesovandRieszpotential spaces areequiva-
lenttoisoperimetricinequalities ofanewtype. ItisshownalsothatSobolev
type inequalities follow from weighted one-dimensional inequalities with the
weight expressed in terms of a capacity minimizing function. The proof ap-
plies to functions on Riemannian manifolds, so that, for such functions, this
result provides a substitute for the rearrangement techniques used to obtain
sharpconstants inSobolevinequalitiesinRn.
Contents
Preface
1. Introduction
2. Classicalisoperimetricinequalityanditsapplicationstointegralinequalities
3. Sobolev type inequality for functions with unrestricted boundary values
4. Compactness criterion
5. The case p=1, q <1 in the Sobolev type inequality (3.1)
6. Imbeddings into fractional Sobolev spaces
7. Imbedding into a Riesz potential space
8. Capacity minimizing functions and their applications to Sobolev type in-
equalities
9. An application of p-capacity to Poincar´e’s inequality
References
Preface
This text is a somewhat extended version of my lectures in the Institut Henri
Poincar´e (June 2002) and the University of Helsinki (December 2002), where I
surveyed my results on applications of isoperimetric and isocapacitary inequalities
to the theory of Sobolev spaces.
2000 Mathematics Subject Classification. Primary54C40,14E20;Secondary 46E25,20C20.
Keywordsandphrases. Sobolevspaces,isoperimetricinequalities,isocapacitaryinequalities.
(cid:1)c0000 (copyright holder)
1
2 VLADIMIR MAZ’YA
I began to work on this topic many years ago, when as a fourth year un-
dergraduate student I discovered that Sobolev type inequalities are equivalent to
isoperimetric and isocapacitary inequalities [3], [4]. It turned out that classes of
domains and measures involved in imbedding and compactness theorems could be
completely described in terms of length, area and capacity minimizing functions.
Moreover,without change of proofs, the same remains true for spaces of functions
definedonRiemannianmanifolds[8],[1]. Nowadays,itisavastdomainofresearch
withapplicationstononlinearpartialdifferentialequations,geometry,spectralthe-
ory, Markov processes, and potential theory.
Most results presented in this paper can be found in the books [8] and [10],
where a lot more related information is contained. Sections 6 and 7 and Theorem
8.9 are new.
IproveinSections6and7thatsomeimbeddingsintofractionalBesovandRiesz
potential spaces are equivalent to isoperimetric inequalities of a new type. Theo-
rem8.9 showsthat Sobolevtype inequalitiesfollow fromweightedone-dimensional
inequalities with the weight expressed in terms of a capacity minimizing function.
The proof applies also to functions on Riemannian manifolds, so that, for such
functions, this result provides a substitute for the rearrangement techniques used
to obtain sharp constants in Sobolev inequalities in Rn.
I preserve a colloquial style of exposition and I make no attempts to achieve
generality. Sometechnicalproofsarereplacedbyplausibleargumentssupplemented
with references to corresponding places in [8].
ItgivesmeagreatpleasuretothanktheorganizersofthespecialprogramHeat
Kernels, Random Walks, Analysis on Manifolds and Graphs at the Institut Henri
Poincar´efor inviting me to give these lectures.
My heartfelt gratitude goes to Matti Vuorinen for the opportunity to give the
courseClassical Inequalities in Bad Domains atthe DepartmentofMathematicsof
the University of Helsinki.
I express my appreciation to Antti Rasila who took notes during the course
andpreparedthe final text in LATEX, and also for editing the text and correcting
a number of slips. In particular, all the figures below were made by him.
1. Introduction
Definition 1.1. LetΩ be anopensetinRn. FollowingSobolev,letus denote
by Wl(Ω) the space of functions in L (Ω), p ≥ 1, whose distributional derivatives
p p
belong to L (Ω). This space is supplied with the norm
p
(cid:2)u(cid:2)Wpl(Ω) =(cid:2)∇lu(cid:2)Lp(Ω)+(cid:2)u(cid:2)Lp(Ω),
where ∇u={∂|α|/∂xα1...∂xαn}, |α|=l.
l 1 n
Definition 1.2. A set Ω⊂Rn has the cone property if each point of Ω is the
vertex of a cone contained in Ω along with its closure, the cone being represented
bythe inequalities x2+...+x2 <ax2, 0<x <b insomeCartesiancoordinate
1 n−1 n n
system, where a,b are constants.
The following two theorems are known since Sobolev’s work of 1936 except for
thecasep=1inTheorem2whichisduetoGagliardo(1958)andNirenberg(1958).
LECTURES ON ISOPERIMETRIC AND ISOCAPACITARY INEQUALITIES 3
Figure 1. Sobolev’s cone property. (Cones have fixed size.)
Ω
Figure 2. This domain does not satisfy the cone property, and
thus it is bad for Sobolev theory whereasthe complementis good.
Theorem 1.3. (First Sobolev’s theorem) Let Ω be a domain in Rn satisfying
the cone property and let (cid:1)
p>1, pl >n,
p=1, l ≥n.
Then Wpl(Ω)⊂C(Ω)∩L∞(Ω). The inequality
(1.1) (cid:2)u(cid:2)L∞(Ω) ≤C(cid:2)u(cid:2)Wpl(Ω)
holds, where C depends only on Ω, l, p, and n but not on u.
Remark 1.4. W(cid:2) e cann(cid:2)ot put pl =n for p>1 in this theorem. Counterexam-
ple: Let u(x)=log(cid:2)log|x|(cid:2), where |x|<1/2. By the asymptotic relation
1
|∇lu(x)|≈ (cid:2)(cid:2)log|x|(cid:2)(cid:2)·|x|l for|x|<1/2
it follows that
(cid:3) (cid:3)
dx 1/2 rn−1dr
(cid:2) (cid:2) =C (cid:2) (cid:2) <∞.
|x|<1/2 |x|pl(cid:2)log|x|(cid:2)p 0 rn(cid:2)logr(cid:2)p
Hence u ∈ Wl(|x| < 1/2) while u(x) → ∞ as x → 0. If one wants a bounded
p
but no(cid:2)t contin(cid:2)uous function in Wpl(Ω), pl=n, one can choose, for example u(x)=
sinlog(cid:2)log|x|(cid:2).
Theorem 1.5. (Second Sobolev’s theorem) Let Ω be a domain in Rn satisfying
the cone property. Let pl ≤ n. Then Wl(Ω) ⊂ L (Ω) for p ≥ 1, q ≤ pn/(n−pl),
p q
n>pl and for all p≤q <∞, if n=pl. The inequality
(1.2) (cid:2)u(cid:2)Lq(Ω) ≤C(cid:2)u(cid:2)Wpl(Ω)
4 VLADIMIR MAZ’YA
holds.
x
2
Ω
0 x
1
Figure 3. The simplest domain without cone property is
{(x1,x2) : 0 < x1 < 1, 0 < x2 < xα1}, where α > 1. The do-
main has a cusp at 0 where the cone property fails.
Remark 1.6. Let Ω be as in Fig.3 and let u(x1,x2)=x−1γ, γ >0. When does
this function belong to the Sobolev space W1(Ω)? Clearly,
p
|∇u(x)|p =cx−(γ+1)p,
1
(cid:3) (cid:3) (cid:3) (cid:3) (cid:3)
|∇u|pdx1dx2 =c 1 xα1 dx2·x−1(γ+1)pdx1 =c 1xα1−(γ+1)pdx1.
Ω 0 0 0
Since the last integralconvergesfor α−p(γ+1)>−1 it follows that u∈W1(Ω) if
p
andonly if this inequality holds. Let us show that there is no Sobolev’s imbedding
Wp1(Ω)⊂L∞(Ω) for some p>2. We must find p satisfying
α+1
2<p< .
γ+1
We need to check that 2γ+2 < α+1, which is the same as α > 2γ+1. Clearly
we can choose γ small enough to satisfy the last inequality. Since the function u is
not bounded, the result follows.
Remark 1.7. Sobolev’s assertion concerning pl = n, p > 1 in Theorem 1.5
has been strengthened by Yudovich (1961) and Trudinger (1967) who showed that
u∈Wl(Ω) implies
p (cid:3)
(cid:4) (cid:5)
exp c|u|p−p1 dx<∞.
For u∈Wl(Ω) we know that
p
• Allassumptionsconcerningp,l,q andninSobolev’stheoremsareessential.
• The cone property cannot be omitted.
Butthe followingremarkshowsthatthe cone propertyis notnecessaryforthe
Sobolev’s imbedding theorem to hold.
Remark 1.8. The domain in Figure 4 does not satisfy the cone property be-
cause there is not enough space for a cone of any fixed size with the vertex on the
dashedline to be placed inside. On the other hand, this is a good domain for both
Sobolev’s assertions concerning W1(Ω).
p
LECTURES ON ISOPERIMETRIC AND ISOCAPACITARY INEQUALITIES 5
Ω
a
0
Figure 4. A non-cuspidal domain without cone property.
~
Ω
0
Figure 5. A domain with the cone property.
In fact, let u ∈ W1(Ω) and let u˜ be its mirror extension onto Ω˜ (see Fig.5).
p
Then (1.1) and (1.2) hold with l=1 for u˜ and Ω˜, which implies (1.1) and (1.2) for
u and Ω.
Definition 1.9. Let Ll(Ω) = {u ∈ D(cid:2) : Dαu ∈ L (Ω), ∀|α| = l}. By using
p p
Sobolev’sinequalitiesonecanshowthateachubelongstoL (Ω)whenu∈Ll(Ω)
q,loc p
with 
 q = pn , n>pl,
n−pl
∀q, n=pl,

q =∞, n<pl.
There are no requirements on Ω. However, there is a price to be paid and this is
the ’loc’ in the result; the functions may be very bad at the boundary.
Clearly,
Wl(Ω)=Ll(Ω)∩L (Ω).
p p p
Let us supply Ll(Ω) with the norm
p
(cid:2)u(cid:2)Llp(Ω) =(cid:2)∇lu(cid:2)Lp(Ω)+(cid:2)u(cid:2)Lp(ω),
where ω is an arbitrary open, nonempty set with ω ⊂Ω.
6 VLADIMIR MAZ’YA
Let us consider another space:
Vl(Ω)=∩l Lk(Ω),
p k=0 p
endowed with the norm
(cid:9)l
(cid:2)u(cid:2)Vp(Ω) = (cid:2)∇ku(cid:2)Lp(Ω).
k=0
It turns out that Ll(Ω), Wl(Ω), and Vl(Ω) might be nonisomorphic.
p p p
Example 1.10. In 1933, Nikodym constructed the domain depicted in Figure
6 with a certain strange property.
−m
2
3
A
m
2
B
m
1 ε
m
C
Figure 6. Domain of O. Nikodym (1933).
Let us introduce a continuous function u on the domain Ω of Fig.6 by setting:

 0 on C,
u(x)= α on A ,
 m m
linear on B .
m
Let the width of the rectangle A be equal to 2−m and the width of the thin
m
passage B be denoted by (cid:5) <<2−m. Consider the Dirichlet integral
m (cid:3) m
(cid:9) (cid:9)
|∇u|2dx= mes2Bmα2m = (cid:5)mα2m.
Ω
m m
On the other hand,
(cid:3)
(cid:9) (cid:9)
|u|2dx≥ α2mmes2Am = α2m2−m.
Ω
m m
Clearly, α can be chosen so that
m
(cid:9) (cid:9)
(cid:5) α2 <∞ and α2 2−m =∞.
m m m
m m
We see that the gradient is in L2(Ω) but the function itself is not. Thus:
L12(Ω)(cid:12)⊂L2(Ω).
LECTURES ON ISOPERIMETRIC AND ISOCAPACITARY INEQUALITIES 7
2−m
y
Pm −m/22
m
Sm −2
1
P
0 2 x
Figure 7. The domain Ω showing that derivatives of the first
order can be worse than derivatives of the second order.
Example 1.11. ([8], Sect. 1.1.4) Let us check that the spaces W2(Ω) and
2
V2(Ω) do not coincide for the domain Ω shown in Figure 7.
2
Let the width of S be equal to 2−4m and let
m
 0 on P,
u= 4m(y−1)2 on S (m=1,2,...),
 m
2m+1(y−1)−1 on P (m=1,2,...).
m
We can easily check that
(cid:3)
|∇2u|2dxdy ∼ 2−m,
Sm(cid:3)
u2dxdy ∼ 2−5m,
(cid:3)Sm
u2dxdy ∼ 2−m/2,
(cid:3) Pm
|∇u|2dxdy ∼ 2−3m,
(cid:3)Sm
|∇u|2dxdy ∼ 2m/2.
Pm
Therefore, (cid:2)∇u(cid:2)L2(Ω) =∞ and (cid:2)u(cid:2)W22(Ω) <∞.
2. Classical isoperimetric inequality and its applications to integral
inequalities
Consider the problem of maximizing the area a of a plane domain Ω with
rectifiable boundary of a fixed length l.
The maximizing property of the disk can be written as the isoperimetric in-
equality
(2.1) 4πa≤l2.
Thefollowingsimpleproofof(2.1)isborrowedfromthebookInequalitiesbyHardy,
Littlewood and Polya.
8 VLADIMIR MAZ’YA
Figure 8. The disc gives the best result.
Let x = x(ϕ), y = y(ϕ) be parametric equations of ∂Ω, where ϕ = 2πλ/l and
λ is the arc length on ∂Ω. For simplicity, we assume that the derivatives x(cid:2) and y(cid:2)
are continuous. We have
(cid:10) (cid:11) (cid:10) (cid:11) (cid:12)(cid:10) (cid:11) (cid:10) (cid:11) (cid:13)
dx 2 dy 2 dx 2 dy 2 l2 l2
+ = + = .
dϕ dϕ dλ dλ 4π2 4π2
(Here we have usedthe fact thatthe differential ofthe arc dr has the unit length.)
dλ
Now,
l2 (cid:3) 2π(cid:12)(cid:10)dx(cid:11)2 (cid:10)dy(cid:11)2(cid:13) (cid:3) 2π dx
−2a= + dϕ+2 y dϕ≥0.
2π dϕ dϕ dϕ
0 0
(cid:14) (cid:14)
Indeed, let x(ϕ)= x einϕ, y(ϕ)= y einϕ. We have
n n n n
(cid:9) (cid:4) (cid:5) (cid:15) (cid:9) (cid:16)
n2 |x |2+|y |2 +2(cid:14) i y x¯ ·n
n n n n
(cid:9) (cid:4) (cid:5) (cid:9)
≥ n2 |x |2+|y |2 −2 |x ||y |n≥0.
n n n n
which gives (2.1).
The n-dimensional generalization of (2.1) is
(2.2) (mesng)n−n1 ≤cnHn−1(∂g),
where g is a domain with smooth boundary ∂g and compact closure, and Hn−1 is
the (n−1)-dimensional area. The constant c is such that (2.2) becomes equality
n
for any ball, that is c = n−1v−1/n with v standing for the volume of the unit
n n n
ball. Inequality (2.2) holds for arbitrary measurable sets with Hn−1 replaced by
the so called perimeter in the sense of De Giorgi (1954-1955).
HowdoesthisgeometricfactconcernSobolevimbeddingtheorems? Theanswer
is given by the following result [2], [3].
Theorem 2.1. Let u∈C∞(Rn). There holds the inequality:
0
(cid:10)(cid:3) (cid:11) (cid:3)
n−1
(2.3) |u|n−n1dx n ≤C(n) |∇u|dx,
Rn Rn
where the best constant is the same as in the isoperimetric inequality (2.2).
Proof. First we provethe lowerestimate for C(n). Figure 9 shows the graph
ofthe functionu (|x|) to be insertedin(2.3). (The function is notsmoothbutit is
ε
Lipschitz and can be approximated by smooth functions in the norm (cid:2)∇u(cid:2)L(Rn)).
LECTURES ON ISOPERIMETRIC AND ISOCAPACITARY INEQUALITIES 9
1
ε
Figure 9. The function u .
ε
We have
(cid:10)(cid:3) (cid:11) (cid:3)
n−1
vnn−n1 ≤ |uε|n−n1dx n ≤C(n) |∇uε|dx
Rn Rn
(cid:3) (cid:2) (cid:2)
1+ε(cid:2)du (cid:2) (cid:4) (cid:5)
= C(n)nv (cid:2) ε(cid:2)rn−1dr = 1+O(ε) nv C(n).
n n
dr
1
It follows that
(cid:4) (cid:5)
n−1v−1/n ≤ 1+O(ε) C(n).
n
and finally
(cid:4) (cid:5)
C(n)≥ nv1/n −1 =c .
n n
In order to prove (2.3) we need the coarea formula:
(cid:3) (cid:3)
∞
(2.4) |∇u|dx= Hn−1(Et)dt,
Rn 0
where E ={x:|u(x)|=t}. It is known by Sard’s lemma that almost all level sets
t
are C∞ manifolds. (Note that by Whitney’s extension theorem, any multidimen-
sionalcompact set can be a levelset of a C∞ function and thus the words “almost
all” cannot be omitted.)
Next we give a plausible argumentin favour of the coareaformula. This is not
a rigorous proof, as we assume all level sets to be good.
|u|=t+dt
|u|=t
Figure 10. Level surfaces and lines of quickest descent, i.e. the
lines orthogonalto the level surfaces.
10 VLADIMIR MAZ’YA
We write dx= dHn−1dν and |∇u(x)| =dt/dν, where dν is the element of the
trajectory orthogonalto E . We obtain
t
(cid:3) (cid:3) (cid:3)
∞
dt
|∇u(x)|dx= dHn−1dν
dν
Rn 0 Et
(cid:3) (cid:3) (cid:3)
∞ ∞
= dt dHn−1 = Hn−1(Et)dt.
0 Et 0
A rigorous proof of the coarea formula for smooth functions can be found in [8],
Sect. 1.2.4. This formula was provedfor the so called asymptotically differentiable
functions of two variables, by A. S. Kronrod (1950). H. Federer obtained a more
general result for Lipschitz mappings Rn → Rm (1959). The result was extended
to the functions of bounded variation by Fleming and Rishel (1980).
Let us prove (2.3). By the coarea formula and by the isoperimetric inequality
(2.2),
(cid:3) (cid:3) (cid:3)
∞ ∞(cid:4) (cid:5)
|∇u|dx= Hn−1(Et)dx≥nvn1/n mesnNt nn−1dt,
Rn 0 0
where N ={x:|u(x)|≥t}. It follows from the definition of the Lebesgue integral
t
that
(cid:10)(cid:3) (cid:11)n−1 (cid:10)(cid:3) ∞ (cid:4) (cid:5)(cid:11)n−1
|u|n−n1dx n = mesnNtd tn−n1 n
Rn 0
(2.5) =(cid:10)n−n 1(cid:3) ∞(cid:4)mesnNt(cid:5)n−n1(cid:4)mesnNt(cid:5)n1tn−11dt(cid:11)nn−1.
0
By using the obvious inequality
(cid:3)
(cid:4) (cid:5)n−1 t(cid:4) (cid:5)n−1
t mes N n ≤ mes N n dτ.
n t n τ
0
we conclude that the right-hand side of (2.5) does not exceed
(cid:17) n (cid:3) ∞(cid:4)mes N (cid:5)n−n1(cid:10)(cid:3) t(cid:4)mes N (cid:5)n−n1dτ(cid:11)n−11dt(cid:18)nn−1
n−1 n t n τ
0 0
(cid:3)
∞(cid:4) (cid:5)
n−1
= mes N n dt
n t
0
Thuswehaveobtained(2.3)withthebestpossibleconstant. TheproofofTheorem
is complete.
Now, we consider a more general inequality:
(cid:10)(cid:3) (cid:11) (cid:3)
1/q
(2.6) |u|qdµ ≤C |∇u|dx,
Ω Ω
whereq ≥1, Ω is an open subset ofRn, µis anarbitrarymeasureandu∈C∞(Ω).
0
Theorem 2.2. Inequality (2.6) with q ≥1 holds if and only if
(2.7) µ(g)1/q ≤CHn−1(∂g)
for every bounded open set g with smooth boundary, g ⊂Ω.