Strongly local nonlinear Dirichlet functionals
We introduce a new notion of Markov functional and we prove that its properties allows to define a notion of capacity associated with the functional.
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irk-123456789-1246272018-07-17T22:08:59Z Strongly local nonlinear Dirichlet functionals Biroli, M. We introduce a new notion of Markov functional and we prove that its properties allows to define a notion of capacity associated with the functional. 2004 Article Strongly local nonlinear Dirichlet functionals / M. Biroli // Український математичний вісник. — 2004. — Т. 1, № 4. — С. 485-500. — Бібліогр.: 20 назв. — англ. 1810-3200 2000 MSC. 31C25, 35B65, 35J70 http://dspace.nbuv.gov.ua/handle/123456789/124627 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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We introduce a new notion of Markov functional and we prove that its properties allows to define a notion of capacity associated with the functional. |
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Biroli, M. Strongly local nonlinear Dirichlet functionals Український математичний вісник |
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Strongly local nonlinear Dirichlet functionals |
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Strongly local nonlinear Dirichlet functionals |
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Strongly local nonlinear Dirichlet functionals |
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Strongly local nonlinear Dirichlet functionals |
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Strongly local nonlinear Dirichlet functionals |
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strongly local nonlinear dirichlet functionals |
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Інститут прикладної математики і механіки НАН України |
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Strongly local nonlinear Dirichlet functionals / M. Biroli // Український математичний вісник. — 2004. — Т. 1, № 4. — С. 485-500. — Бібліогр.: 20 назв. — англ. |
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Український математичний вісник |
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AT birolim stronglylocalnonlineardirichletfunctionals |
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2025-07-09T01:45:00Z |
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2025-07-09T01:45:00Z |
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1837131893896642560 |
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Український математичний вiсник
Том 1 (2004), № 4, 485 – 500
Strongly Local Nonlinear Dirichlet Functionals
Marco Biroli
(Presented by I. V. Skrypnik)
Abstract. We introduce a new notion of Markov functional and we
prove that its properties allows to define a notion of capacity associated
with the functional.
2000 MSC. 31C25, 35B65, 35J70.
Key words and phrases. Potential Theory, Dirichlet forms, Elliptic
degenerate equations.
1. Introduction
Our goal in this paper is an extension of the results connected with
the capacity associated to a (linear) Dirichlet form notion to the case of
nonlinear Markov functionals.
For the notion of Dirichlet form we refer to the book of Fukushima-
Oshima-Takeda, [13]. In [13] a purely analytical proof of fundamental
properties of Dirichlet form is given, this type of proof firstly appeared
in [18]; we recall also the papers [4], [7], where an analytical investigation
of the properties of the harmonics relative to a a strongly local “Rieman-
nian” Dirichlet forms is carried on. From Beuerling-Deny representation
formula, [1], a Dirichlet form is represented as the sum of a strongly local
part, of a “killing” part and of a global part. The Beuerling-Deny rep-
resentation theorem is the fundamental tool allowing to prove that same
properties of Dirichlet forms (in particular the Markov property) hold
again for energy measures in the strong local (regular) case. Using the
above mentioned properties of energy measure it can be proved that for
the energy measure of a strongly local (regular) Dirichlet form a chain
rule and a Leibnitz rule hold; those properties are the starting point for
an investigation of local regularity of harmonics relative to a strongly lo-
cal (regular) Dirichlet form, see in particular [4], [7]. The Beuerling-Deny
Received 15.12.2003
The Author has been supported by the MURST Research Project 2002012589
ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України
486 Strongly Local Nonlinear Dirichlet Functionals
representation theorem is proved using Riesz theorem on representation
of measures, which is an essentially linear tool, then it seems that in the
proof of a nonlinear version of this result is difficult.
Previous work on a possible extension of the notion of Dirichlet form
to the nonlinear case has been given by Benilan-Picard, [1], and Cipriani-
Grillo, [10] [11]. In particular in [1] the relations between maximum
principle and Markov property are investigated generalizing to nonlinear
monotone case previous results obtained in [13] and [15] in the linear
case. In [11] a notion of nonlinear Dirichlet form is given and the relations
with a class of nonlinear semigroups (the order preserving contractions
semigroups with a cyclically monotone generator) are investigated. The
above papers deal with the general global case and are interested in the
properties of the corresponding nonlinear semigroup; then the existence
of an energy measure is not ensured and there is no proof of chain or
Leibnitz rule for the energy measure, when such a measure exists.The
first paper concerning local forms was [17], where a suitable chain rule
for the energy measure connected with the form is assumed and Sobolev-
Morrey inequalities are proved as a consequence of a Poincaré inequality.
In [8], [7], [10], [11] some nonlinear forms on fractals are explicitely given
and it is proved that the assumptions in [17] hold (see also the more
recent papers [20], [14] on the p-Laplacian on the Sierpiski gasket).
In the paper [5] we have introduced the notion of nonlinear strongly lo-
cal Dirichlet forms and we give our assumptions (in particular the Markov
property) directly on the energy measure of the form, whose existence is
assumed. We are able to prove in this framework (by purely analytical
methods in the line of [18]) suitable Leibnitz and chain rule, which are
the starting point for an investigation of local regularity of the harmonics
relative to the form and in particular for a proof (under suitable assump-
tions) of an Harnack type inequality for positive harmonics (we observe
that the chain rule proved here is the same assumed in [17] and that
an Harnack nequality for positive harmonics in the linear case has been
proved in [4], [7]). This last part will be developped in a forthcoming
paper.
Here the notion of capacity relative to Markov (global) functional is
introduced and we prove that a theory for this capacity can be developed
essentially in connection with global assumptions in analogy with the
linear case (see [13]). We finally observe that our framework contains the
case of the subelliptic p-Laplacian, p > 1, related to some vector fields
Xi, i = 1, . . . ,m, which satisfy an Hörmander condition, considered on
RN endowed with the Lebesgue measure as well as the p-Laplacians on
fractals considered in [5], [8], [10] (see also [19], [13], where the Authors
give a construction of a p-Laplacian on the Sierpiski gasket and investigate
M. Biroli 487
the Hölder continuity of harmonics) or the global forms which arise in the
theory of Sobolev spaces.
2. The capacity
We consider a locally compact separable Hausdorff space X with a
metrizable topology and a positive Radon measure m on X such that
supp[m]= X. Let Φ : Lp(X,m) → [0,+∞], 1 < p, be a l.s.c. convex
functional with domain D, i. e. D = {v; Φ(v) < +∞}, with Φ(0) = 0.
We assume that D is dense in Lp(X,m) and that the following conditions
hold:
(H1)D is a dense linear subspace of Lp(X,m), which can be endowed with
a norm ||.||D; moreover D has a structure of uniformly convex Banach
space with respect to the norm ||.||D and the following estimate holds:
there exists s ≥ 0 such that
c1||v||pD ≤ Φs(v) = Φ(v) + s
∫
X
|v|pdm ≤ c2||v||pD
for every v ∈ D, where c1, c2 are positive constants.
(H2) We denote by D0 the closure of D ∩ C0(X) in D (with respect to
the norm ||.||D) and we assume that D∩C0(X) is dense in C0(X) for the
uniform convergence on X, moreover we assume that that Φs is locally
uniformly convex on D0, i. e. if we have lim
n→0
Φs(
un+u
2 ) = Φs(u) and
lim
n→0
un = u weakly in D0 then lim
n→0
un = u in D0 (this last assumption is
not necessary in the present paper, but simplify some proofs and will be
used in forthcoming paper on the theory of capacity with respect to Φs).
Remark 2.1. We observe that, since Φ is convex, Φ is l.s.c. also with
respect to the weak toplogy of Lp(X,m). We remark that the assumption
(H1) substantially does not allow us to deal with the case p = 1 or with
sublinear functionals. Moreover from the assumption (H1) it follows that
Φ is continuous on D for the norm ||.||D, [19] Ch. 1 Sec. 2 pg. 20, then
from (H2) the restriction of Φ to D0 coincides with the relaxation of Φ
defined on D ∩ C0(X).
(H3) For every u, v ∈ D ∩ C0(X) we have u ∨ v ∈ D ∩ C0(X), u ∧ v ∈
D ∩ C0(X) and
Φ(u ∨ v) + Φ(u ∧ v) ≤ Φ(u) + Φ(v).
Moreover for every u ∈ D ∩ C0(X) we have that u ∧ 1 ∈ D ∩ C0(X)
and Φ(u ∧ 1) ≤ Φ(u). We observe that from (H3), from Remark 2.1 and
488 Strongly Local Nonlinear Dirichlet Functionals
from the l.s.c. of our functional on Lp(X,m) we have that the above
inequalities hold again for every u, v ∈ D0.
Remark 2.2. We observe, [12] pg. 15–19, that given an open set O
whose closure is contained in an open relatively compact open set Ω,
there exists a function ũ ∈ C0(X) such that ũ ≥ 1 + ǫ, ǫ > 0, on O
and ũ = 0 on Ωc, then from (H2) and (H3) there exists u ∈ D ∩ C0(X)
with u ≥ 1 on O. Moreover we observe that, since C0(X) are dense in
Lp(X,m), we have that D0 is dense in Lp(X,m).
Remark 2.3. We observe that the assumption (H3) is connected with
the assumptions in [11], moreover if Φ has a subdifferential ∂Φ on D0
with values in D′
0 (the dual space of D0), then the first inequality in (H3)
can be derived from the T -monotonicity of ∂Φ.
If the functional Φ satisfies the assumptions (H1)(H2)(H3) we call Φ
a (global) Markov functional.
The assumptions (H1)(H2) and (H3) allow us to define a capacity
relative to the functional Φ (and the measure space(X,m)). The capacity
of an open set O is defined as
capΦ,s(O) = capΦ(O) = inf{Φs(v); v ∈ D0, v ≥ 1 a. e. on O}
if the set {v ∈ D0 , v ≥ 1 a. e. on O} is not empty and
capΦ,s(O) = capΦ(O) = +∞
if the set {v ∈ D0 , v ≥ 1 a. e. on O} is empty (we drop out the index s
from the notation of capacity when it is considered as fixed). Let E be a
subset of X we define
capΦ(E) = inf{capΦ(O);O open set with E ⊂ O}.
We observe that from Remark 2.2 it follows that given an open set O
whose closure is contained in an open relatively compact open set Ω we
have capΦ(O) < +∞.
Proposition 2.1. Consider an open set O ⊂ X such that capΦ(O) <
+∞; there exists eO ≥ 0 in {v ∈ D0, v ≥ 1 a. e. on O}, such that
capΦ(O) = Φs(eO).
We say that eO ∈ D0 is a potential of O with respect to Ω. The potential
eO is unique up to sets of measure zero. Moreover if O1 ⊂ O2 are open
sets in X we have eO1 ≤ eO2 a. e.
M. Biroli 489
Proof. Let M = capΦ(O). Denote K = {v ∈ D0; v ≥ 1 a. e. on O}.
The set K is closed and convex in Lp(X,m), then K is weakly closed in
Lp(X,m). Since Φs is l.s.c. on Lp(X,m) for the strong and then for the
weak topology, there is a minimum point eO of Φs on K. Moreover we
have Φs(eO) = inf{Φs(v); v ∈ D0, v ≥ 1 a. e. on O} = capΦ(O). The
uniqueness of the potential in Lp(X,m) follows from the strong convexity
of Φs on Lp(X,m).
The positivity of eO follows from the inequality Φs(v ∨ 0) ≤ Φ1(v),
which is a consequence of (H3).
For the second and last part of the result we observe that from (H3)
eO1 ∧ eO2 and eO1 ∨ eO2 are in D0. Then again from (H3) we have
Φs(eO1 ∧ eO2) ≤ Φs(eO1) + Φs(eO2) − Φs(eO1 ∨ eO2)
= capΦ(O1) + capΦ(O2) − Φs(eO1 ∨ eO2).
Since eO1 ∨ eO2 ≥ 1 a. e. on O2, we have Φs(eO1 ∨ eO2) ≥ capΦ(O2); then
Φs(eO1 ∧ eO2) ≤ capΦ(O1).
Since eO1 ∧ eO2 ≥ 1 a. e. on O1, we have also
Φs(eO1 ∧ eO2) = capΦ(O1)
then eO1 ∧ eO2 = eO1 a. e., so eO1 ≤ eO2 a. e.
Remark 2.4. The assumption (H3) implies also that for an open set O
with finite capacity we have e0 = 1 a. e. (and then up to sets of zero
capacity, see Proposition 2.3) on O.
We prove that our notion of capacity has all the set theoretic proper-
ties of a Choquet capacity:
Proposition 2.2. The following properties hold:
(a) For every subset E of X we have s m(E) ≤ capΦ(E,Ω).
(b) Let E1 and E2 be subsets of X with E1 ⊂ E2 then capΦ(E1) ≤
capΦ(E2) (monotonicity property).
(c) Let E1 and E2 be subsets of X, then
capΦ(E1 ∪ E2) + capΦ(E1 ∩ E2) ≤ capΦ(E1) + capΦ(E2).
(d) Let En be an increasing sequence of subsets of X then
capΦ(∪+∞
n=1En) = lim
n→+∞
capΦ(En).
(e)x Let En be a sequence of subsets of X then
capΦ(∪+∞
n=1En) ≤
+∞∑
n=1
capΦ(En,Ω).
490 Strongly Local Nonlinear Dirichlet Functionals
Proof. The property (a) holds if capΦ(E) = +∞ and if capΦ(E,Ω) <
+∞ easily follows from the inequality
Φs(v) ≥ s
∫
X
|v|pm(dx)
for every v ∈ D0.
Consider now the property (b). Let E1 and E2 be open sets. The
property holds if at least one of the sets {v ∈ D0, v ≥ 1 a. e. on E1} or
{v ∈ D0, v ≥ 1 a. e. on E2} is empty. In the other cases the property
follows from the relation
{v ∈ D0, v ≥ 1 a. e. on E2} ⊂ {v ∈ D0, v ≥ 1 a. e. on E1}.
In the general case the result follows from the fact that E2 ⊂ O with O
open set implies E1 ⊂ O.
Consider the property (c). Let E1 and E2 be open sets, we observe
that if u ≥ 1 a. e. on E1 and v ≥ 1 a. e. on E2 then u ∨ v ≥ 1 a. e.
on E1 ∪ E2 and u ∧ v a. e. on E1 ∩ E2; then, if the sets {v ∈ D0, v ≥
1 a. e. on E1} and {v ∈ D0, v ≥ 1 a. e. on E2} are not empty, property
(c) follows from the assumption (H3). Moreover property (c) holds if one
of the sets {v ∈ D0, v ≥ 1 a. e. on E1} or {v ∈ D0, v ≥ 1 a. e. on E2} is
empty.
Consider now the general case. We have easily that the property
holds if capΦ(E1) = +∞ or capΦ(E2) = +∞. Consider now the case
where capΦ(E1) and capΦ(E2) are both finite. Then for every ǫ > 0
there exists two open sets O1 and O2 such that Ei ⊂ Oi and
capΦ(Oi) ≤ capΦ(Ei) + ǫ
for i = 1, 2. We have
capΦ(E1 ∪ E2) + capΦ(E1 ∩ E2) ≤ capΦ(O1 ∪O2) + capΦ(O1 ∩O2)
≤ capΦ(O1) + capΦ(O2) ≤ capΦ(E1) + capΦ(E2) + 2ǫ.
Since ǫ > 0 is arbitrary we have the result.
Consider now the property (d). Let En be open subsets of Ω and
E = ∪+∞
n=1En; from the monotonicity property we have that
capΦ(E) ≥ lim
n→+∞
capΦ(En) (2.1)
then, if lim
n→+∞
capΦ(En) = +∞, the property (d) holds.
Let now lim
n→+∞
capΦ(En) < +∞.
M. Biroli 491
Consider at first the case where capΦ(E) < +∞). There exists eEn and
eE potentials of En and E; from Proposition 2.1 we have eEn is increasing
with respect to n a. e. and that eEn ≤ eE a. e., then eEn converges in
Lp(X,m) to ẽE , with
Φs(ẽE) ≤ lim
n→+∞
capΦ(En)
(the limit in the right hand side exists finite since capΦ(En) is increasing
and bounded in n).
We observe that ẽE ≥ 1 a. e. on E and ẽE ∈ D0 then
capΦ(E) ≤ Φs(ẽE) ≤ lim
n→+∞
capΦ(En).
Then from (2.1) we have the result. Consider now the case capΦ(E) =
+∞. Assume lim
n→+∞
capΦ(En) < +∞. There exists eEn potentials of En
and we have eEn is increasing with respect to n a. e. The sequence eEn
is bounded in Lp(X,m) then we can assume that eEn strongly converges
in Lp(X,m) to ẽ (we use here the monotone convergence property) and
Φ(ẽ) ≤ lim
n→+∞
capΦ(En) = M < +∞, so we have that ẽ is in D0 and
ẽ ≥ 1 a. e. on E, then capΦ(E) ≤ Φs(ẽ) < +∞ We have a contradiction,
then the present case can not appear. Consider now the general case.
from the monotonicity property we have that
capΦ(E) ≥ lim
n→+∞
capΦ(En) (2.2)
then, if lim
n→+∞
capΦ(En) = +∞, the property (d) holds.
Let now lim
n→+∞
capΦ(En) < +∞; for every ǫ > 0 there exists an open
set On such that En ⊂ On and capΦ(On) − ǫ ≤ capΦ(En) ≤ capΦ(On),
moreover we can assume the sequence On as increasing. We have
capΦ(E) ≤ capΦ(∪nOn) = lim
n→+∞
capΦ(On) ≤ lim
n→+∞
capΦ(En) + ǫ.
Since ǫ > 0 is arbitrary we have the result. The property (e) is an easy
consequence of properties (c) and (d).
We give now the notion of quasi-continuity:
Definition 2.1. Let u be a function defined on X, we say that u is
quasi-continuous (with respect to Φ) if for every ǫ > 0 there exists a
set Eǫ ⊂ Ω such that capΦ(Eǫ) ≤ ǫ and the restriction of u to Ecǫ is
continuous, moreover we can assume Eǫ open.
We also have to deal with the notion of quasi-uniform convergence.
492 Strongly Local Nonlinear Dirichlet Functionals
Definition 2.2. Let un be a sequence of functions defined on X we say
that un converges to a function u quasi-uniformly (with respect to Φ) if
for every ǫ > 0 there exists a set Eǫ such that capΦ(Eǫ) ≤ ǫ and the
restriction of the sequence un to Ecǫ converges uniformly to u on Ecǫ ,
moreover we can assume Eǫ open.
Proposition 2.3. Let u ∈ D0, then there is ũ quasi-continuous such
that ũ = u a. e., moreover ũ is uniquely determined up to sets of zero
capacity.
Proof. Let u ∈ D0 there exists a sequence un ∈ D0 ∩ C0(X) such that
un converges in D0 to u. We can choose un such that un converges to u
a. e. and
||un − un+1||D0 ≤ 2−n.
From (H3) we have |un − un+1| ∈ D0, then
capΦ
({
|un − un+1| > 2
− n
2p
})
≤ Φ1
( |un − un+1|
2−
n
2p
)
≤ c3
||un − un+1||pD0
2−
n
2
≤ c32
−n(p− 1
2
).
Denote
Oq = ∪n≥q
{
|un − un+1| > 2
− n
2p
}
.
From Proposition 2.2 (e) we have
capΦ(Oq) ≤
∑
n≥q
2−n(p− 1
2
) ≤ c42
−q(p− 1
2
)
and
|um − un| ≤ c52
− n
2p
on Ocq, where m ≥ n. The sets Oq are decreasing in q. Then un converges
uniformly to ũ, which coincides with u a.e., on Ocq, so the restriction of u
to Ocq is continuous. The quasi-continuity of u easily follows. Moreover
ũ is defined on the set ∪qOcq, which is such that capΦ(X − ∪qOcq) = 0,
moreover ũ = u a. e. on ∪qOcq and then on X.
We say that ũ is the quasi-continuous representative of u and in the
following we identify u ∈ D0 with his quasi-continuous representative
considering u as defined up to sets of zero capacity.
Lemma 2.1. Let u be in D0. We have
capΦ({u > ǫ}) ≤ c
||u||pD0
ǫp
,
M. Biroli 493
where ǫ > 0 is arbitrary and the set {u > ǫ} is defined up to sets of
capacity zero.
Proof. Let un ∈ D ∩ C0(X) such that the sequence un converges to u
in D0. Let ǫ, σ > 0 be arbitrary; as in Proposition 2.3 there exists Eσ
with capΦ(Eσ) ≤ σ such that (at least after extraction of subsequences)
we have that un converges to u uniformly on X − Eσ. Then we there
exists nǫ,σ such that for n ≥ nǫ,σ we have |un − u| ≤ ǫ
2 on X − Eσ and
||un − u||D0 ≤ σ. We have
{u > ǫ} ⊂
{
un >
ǫ
2
}
∪ Eσ,
where n ≥ nǫ,σ. Then from Proposition 2.2 we obtain
capΦ
(
{u > ǫ}
)
≤ σ +
Φs(un)
( ǫ2)p
≤ σ + cp2
||un||pD0
( ǫ2)p
≤ σ + cp2
(
||u||D0 + σ
)p
( ǫ2)p
.
Let σ → 0, then
capΦ
(
{u > ǫ}
)
≤ 2pcp2
||u||pD0
ǫp
Proposition 2.4. Let un be a sequence in D0 converging in D0 (with
the norm ||.||D0) to u; then there exists a subsequence converging quasi-
uniformly. Moreover there exists a subsequence converging to u up to a
set of zero capacity.
Proof. Let u be the limit of un in D0. We observe that there exists a
subsequence, again denoted by un, which converges a. e. to u. Moreover
up to extraction of subsequences we may assume
||un − un+1||D0 ≤ 2−n.
We observe that from (H3) we have |un − un+1| ∈ D0. From Lemma 2.1
we obtain
capΦ
({
|un − un+1| > 2−
n
2p
})
≤ c3
||un − un+1||pD0
2−
n
2
≤ c32
−n(p− 1
2
).
Denote
Eq = ∪n≥q
{
|un − un+1| > 2
− n
2p
}
.
494 Strongly Local Nonlinear Dirichlet Functionals
From Proposition 2.2 (e) we have
capΦ(Eq) ≤
∑
n≥q
2−n(p− 1
2
) ≤ c42
−q(p− 1
2
)
and
|um − un| ≤ c52
− n
2p
on Ecq , where m ≥ n ≥ q. The sets Eq is decreasing in q, then un con-
verges uniformly to u on Ecq . We observe that there is Oq open containing
Eq such that
capΦ(Oq) ≤ c42
−q(p− 2
3
)
and we have that un converges uniformly to u on Ocq.
We say that a property holds quasi-everywhere (q. e.) if the property
holds up to sets of zero capacity.
Proposition 2.5. Let u ∈ D0 then u is a measurable function with
respect to every positive Radon measure ν, which does not charge sets of
zero capacity.
Proof. There exists a sequence un ∈ D ∩ C0(X) converging to u in D0.
The functions un are measurable with respect to ν and by Proposition 2.4
un converges to u q. e. (at least after extraction of subsequences). Then
we obtain the result.
The following property follows immediately from the definition of ca-
pacity.
Proposition 2.6. The capacities capΦ,s, s > 0, are mutually equivalent;
moreover if
Φ(u) ≥ c
∫
|u|pm(dx)
for a constant c > 0, then capΦ,0 is equivalent to every capacity capΦ,s
with s > 0.
We are now in position to give the definition of quasi-open set:
Definition 2.3. A set E is quasi-open (for the capacity capΦ) if for
every ǫ > 0 there exists a set Aǫ such that capΦ(Aǫ) ≤ ǫ and E ∪ Aǫ is
open.
The following result is an immediate consequence of Proposition 2.3:
Proposition 2.7. Let u ∈ D0; the set Es = {u > s} (defined up to sets
of zero capacity) is quasi-open.
M. Biroli 495
3. The potentials and the capacity measure
We fix in this section s = 1 but the results hold for any s > 0.
Theorem 3.1. Let E be a set in X then
capΦ(E) = inf{Φ1(v); v ∈ D0 v ≥ 1 q. e. on E}.
Proof. Denote
cap′
Φ(E) = inf{Φ1(v); v ∈ D0 v ≥ 1 q. e. on E}.
We prove at first that
cap′
Φ(E) ≤ capΦ(E). (3.1)
If capΦ(E) = +∞ the relation (3.1) holds. Otherwise for every ǫ > 0
there exists an open set O containing E such that capΦ(E)+ǫ ≥ capΦ(O).
Let eO be the potential of O; we have
capΦ(E) + ǫ ≥ capΦ(O) = Φ1(eO) (3.2)
and eO ≥ 1 a. e. then q. e. on O. Since eO ≥ 1 q. e. on E we have
Φ1(eO) ≥ cap′
Φ(E). (3.3)
We now prove that
capΦ(E) ≤ cap′
Φ(E). (3.4)
If cap′
Φ(E) = +∞ the relation (3.4) holds. Otherwise for every ǫ > 0
there exists u ∈ D0 such that cap′
Φ(E) + ǫ ≥ Φ1(u) and ≥ 1 q. e. on E .
Since u is quasi-continuous and u ≥ 1 q. e. on E, for every σ > 0 there
exists an open set O such that the restriction of u to X ∨O is continuous
and capΦ(O) ≤ σ. Denote
U = {x;u(x) ≥ 1 − ǫ} ∪O.
The set U is open; moreover, since capΦ(O) ≤ ǫ, there exists w ∈ D0
such that w ≥ 1 a. e. on O and Φ1(w) ≤ 2σ. Let
z =
( 1
1 − σ
u
)
∨ w.
We have z ≥ 1 q. e. on U then on E and z ∈ D0; we obtain
capΦ(E) ≤ capΦ(U) ≤ Φ1(z) ≤ Φ1
( 1
1 − ǫ
u
)
+ Φ1(w).
Since σ > 0 is arbitrary and since Φ1 is continuous on D0, we obtain
capΦ(E) ≤ Φ1(u) ≤ cap′
Φ(E) + ǫ.
Since σ > 0 is arbitrary , we obtain (3.4).
496 Strongly Local Nonlinear Dirichlet Functionals
We now prove that the inf in the Theorem 3.1 is really a minimum:
Theorem 3.2. Let E be a set of finite capacity in X then
capΦ(E) = min{Φ1(v); v ∈ D0 v ≥ 1 q. e. on E}.
The minimum point eE ∈ D0 is unique; we call eE the potential of E.
Assume that Φ has a subdifferential ∂Φ : D0 → D′
0, where D′
0 denotes the
dual of D0; then eE is the unique solution of the variational inequality
〈∂Φ(u), v − u〉 +
∫
X
|u|p−1sign(u)(v − u) m(dx) ≥ 0
∀ v ∈ K, u ∈ K
where 〈., .〉 denotes the duality between D′
0 and D0 and
K = {v ∈ D0 v ≥ 1 q. e. on E} ⊂ D0.
Proof. It is enough to prove that the convex set K is closed in D0. Let
vn be a sequence in K such that vn → v0 in D0. From Proposition 2.4
we have, at least after extraction of subsequences, that vn → v0 q. e. so
we have also v0 ≥ 1 q. e. on E then v0 ∈ K.
Lemma 3.1. Let v be a function in C0(X) with support K; then there
exists a sequence vn ∈ D ∩ C0(X) such that the support of every vn is
contained in K and the sequence vn converges to v uniformly on X.
Proof. We can assume, without loss of generality v positive. Let O be
the set where v > 0, then O is open and K is the closure of O. By
Remark 2.2. and the assumption (H3) there exists a positive function
vO such that vO ∈ D ∩C0(X) and vO = 1 on O, 0 ≤ vO ≤ 1 everywhere.
From (H2)there exists a sequence of positive functions vn in D ∩ C0(X)
uniformly convergent to v. We can assume without loss of generality
that |vn − v| ≤ 1
n . Let ṽn = (vn − 1
nvO)+ then ṽn has support contained
in K, moreover the sequence ṽn converges uniformly to v on X.
Proposition 3.1. Let g be a positive functional in D′
0; then there exists
a positive Radon measure γ (that does not charge sets of zero capacity)
such that
〈g, v〉 =
∫
v γ(dx)
for every v ∈ D0.
M. Biroli 497
Proof. Consider a positive function v ∈ D ∩ C0(X) with support con-
tained in the compact set K. Let eK be the potential of K we have
veK = v then
0 ≤ 〈g, v〉 ≤ 〈g, veK〉 ≤ 〈g, eK〉M,
where M = sup v. Then if v ∈ D ∩ C0(X) (without assumptions on
positivity) we have
|〈g, v〉| ≤ 2〈g, eK〉M
Using the previous lemma we have that there exists a measure γ such
that
〈g, v〉 =
∫
v γ(dx) (3.5).
for every v ∈ D ∩ C0(X).
Let O be a relatively compact open set by Remark 2.2 there exists
a sequence vn ∈ D ∩ C0(X) such that supp(vn) ⊂ Ō, 0 ≤ vn ≤ 1 and
lim
n→+∞
vn = 1 everywhere on O. Let eO be the potential of O, we have
vneO = vn then
∫
vn γ(dx) = 〈g, vn〉 = 〈g, vneO〉 ≤ 〈g, eO〉 ≤ c2||g||D′
0
capΦ(O) (3.6).
Passing to the limit in (3.6) as n→ +∞ (by the dominated convergence
theorem) we obtain
γ(O) ≤ c2||g||D′
0
capΦ(O) (3.7).
From (3.7) it follows that every set of zero capacity contained in a rela-
tively compact open set has zero γ measure. The space X can be covered
by a numerable union of relatively compact open sets; then by (e) Propo-
sition 2.2 we obtain that γ does not charge sets of zero capacity.
Let now v ∈ D0; there exists a sequence vn in D ∩ C0(X) such that
vn converges to v in D0. We have that, at least after extraction of
subsequences, vn converges to v q. e. then γ a. e. By the Fatout lemma
we have
∫
v γ(dx) ≤ lim inf
n→+∞
∫
vn γ(dx) = lim inf
n→+∞
〈g, vn〉 = 〈g, v〉.
We have also
vn ≤ v + |vn − v| (3.8)
q. e., so γ a. e. Then
〈g, v〉 = lim inf
n→+∞
〈g, vn〉 = lim inf
n→+∞
∫
vn γ(dx)
498 Strongly Local Nonlinear Dirichlet Functionals
≤
∫
v γ(dx) + lim inf
n→+∞
∫
|vn − v| γ(dx)
≤
∫
vγ(dx) + lim inf
n→+∞
〈g, |vn − v|rangle =
∫
vγ(dx),
where we use the previous inequality. So
〈g, v〉 =
∫
vγ(dx).
An easy consequence of Proposition is the following result:
Theorem 3.3. Let the assumptions of Theorem hold and let E be a set
of finite capacity and eE its potential; then there exists a positive Radon
measure γE ∈ D′
0 such that
∂Φ(u) + |u|p−1sign(u) = γE .
The measure γE is called the capacitary measure of E and its support is
contained in E.
Assume now that
Φ(u) =
∫
α(u)(dx),
where α is a positive Radon measure defined for u ∈ D0 and assume that
for every u, v ∈ D0 we have
lim
t→0
α(u+ tv) − α(u)
t
= µ(u, v)
in the weak⋆ topology of M, where µ is linear in v. Then the functional
Φ has a Gateaux derivative on D0 with values in D′
0 defined by
〈Φ′(u), v〉 =
∫
µ(u, v)(dx).
Assume also that the following locality assumption holds: let u = cst on
supp(v), u, v ∈ D0, then µ(u, v) = 0.
Proposition 3.2. Let the above assumptions hold and that the conditions
in Section 2 hold for s = 0. Denote by eE the potential of the set E for
the capacity capΦ,0; then we have γE = 0 on the interior of E (where γE
is the capacitary measure of E with respect to the capacity capΦ,0).
M. Biroli 499
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500 Strongly Local Nonlinear Dirichlet Functionals
Contact information
Marco Biroli Dipartimento di Matematica Politecnico di
Milano
Piazza Leonardo da Vinci 32
20133 Milano, Italy
E-Mail: marbir@mate.polimi.it
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