Wiener criterion for relaxed Dirichlet problem relative to Riemannian p-homogeneous Dirichlet forms
We give a Wiener criterion for the relaxed Dirichlet problem relative to a Riemannian p-homogeneous Dirichlet form.
Gespeichert in:
| Datum: | 2008 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут прикладної математики і механіки НАН України
2008
|
| Schriftenreihe: | Український математичний вісник |
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/124293 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Wiener criterion for relaxed Dirichlet problem relative to Riemannian p-homogeneous Dirichlet forms / M. Biroli, F. Dal Fabbro, S. Marchi // Український математичний вісник. — 2008. — Т. 5, № 1. — С. 1-15. — Бібліогр.: 22 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-124293 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1242932025-02-09T20:21:51Z Wiener criterion for relaxed Dirichlet problem relative to Riemannian p-homogeneous Dirichlet forms Biroli, M. Dal Fabbro, F. Marchi, S. We give a Wiener criterion for the relaxed Dirichlet problem relative to a Riemannian p-homogeneous Dirichlet form. The first author has been supported by the MIUR research project n. 2005010173. 2008 Article Wiener criterion for relaxed Dirichlet problem relative to Riemannian p-homogeneous Dirichlet forms / M. Biroli, F. Dal Fabbro, S. Marchi // Український математичний вісник. — 2008. — Т. 5, № 1. — С. 1-15. — Бібліогр.: 22 назв. — англ. 1810-3200 2001 MSC. 31C25, 35B65, 35J70. https://nasplib.isofts.kiev.ua/handle/123456789/124293 en Український математичний вісник application/pdf Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We give a Wiener criterion for the relaxed Dirichlet problem relative to a Riemannian p-homogeneous Dirichlet form. |
| format |
Article |
| author |
Biroli, M. Dal Fabbro, F. Marchi, S. |
| spellingShingle |
Biroli, M. Dal Fabbro, F. Marchi, S. Wiener criterion for relaxed Dirichlet problem relative to Riemannian p-homogeneous Dirichlet forms Український математичний вісник |
| author_facet |
Biroli, M. Dal Fabbro, F. Marchi, S. |
| author_sort |
Biroli, M. |
| title |
Wiener criterion for relaxed Dirichlet problem relative to Riemannian p-homogeneous Dirichlet forms |
| title_short |
Wiener criterion for relaxed Dirichlet problem relative to Riemannian p-homogeneous Dirichlet forms |
| title_full |
Wiener criterion for relaxed Dirichlet problem relative to Riemannian p-homogeneous Dirichlet forms |
| title_fullStr |
Wiener criterion for relaxed Dirichlet problem relative to Riemannian p-homogeneous Dirichlet forms |
| title_full_unstemmed |
Wiener criterion for relaxed Dirichlet problem relative to Riemannian p-homogeneous Dirichlet forms |
| title_sort |
wiener criterion for relaxed dirichlet problem relative to riemannian p-homogeneous dirichlet forms |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2008 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/124293 |
| citation_txt |
Wiener criterion for relaxed Dirichlet problem relative to Riemannian p-homogeneous Dirichlet forms / M. Biroli, F. Dal Fabbro, S. Marchi // Український математичний вісник. — 2008. — Т. 5, № 1. — С. 1-15. — Бібліогр.: 22 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT birolim wienercriterionforrelaxeddirichletproblemrelativetoriemannianphomogeneousdirichletforms AT dalfabbrof wienercriterionforrelaxeddirichletproblemrelativetoriemannianphomogeneousdirichletforms AT marchis wienercriterionforrelaxeddirichletproblemrelativetoriemannianphomogeneousdirichletforms |
| first_indexed |
2025-11-30T11:06:20Z |
| last_indexed |
2025-11-30T11:06:20Z |
| _version_ |
1850213172547420160 |
| fulltext |
Український математичний вiсник
Том 5 (2008), № 1, 1 – 15
Wiener criterion for relaxed Dirichlet problem
relative to Riemannian p-homogeneous
Dirichlet forms
Marco Biroli, Florangela Dal Fabbro, Silvana Marchi
(Presented by A. E. Shishkov)
Abstract. We give a Wiener criterion for the relaxed Dirichlet problem
relative to a Riemannian p-homogeneous Dirichlet form.
2001 MSC. 31C25, 35B65, 35J70.
Key words and phrases. Dirichlet Forms, Relaxed Dirichlet Problem,
Wiener criterion.
Introduction
The Wiener criterion for a relaxed Dirichlet problem has been firstly
investigated in an Euclidean framework for linear ellptic coercive op-
erators with bounded measurable coefficients by Dal Maso and Mosco,
[18, 19]. The result of Dal Maso and Mosco has been generalized to the
case of the subelliptic p-Laplace operator with p > 1 (i.e. constructed
by means of vector fields satisfying an Hörmander condition), also when
we have a source term in a suitably defined Kato class, in [4, 5, 11]. The
Kato class of measures relative to a Riemannian bilinear Dirichlet form
has been introduced in [10] and the definition has been refined in [6],
where Schrödinger problems relative to a Riemannian bilinear Dirichlet
form with a potential in the Kato class have been investigated.
A generalization of the definition of strongly local Dirichlet forms to
the p-homogeneous case has been given in [7,13,14]. The definition of Rie-
mannian (p-homogeneous) Dirichlet form is given in [15] where the local
regularity and the Harnack inequality for the harmonic functions is stud-
ied; moreover the Kato class relative to a Riemannian (p-homogeneous)
Received 29.11.2007
The first author has been supported by the MIUR research project n. 2005010173.
ISSN 1810 – 3200. c© Iнститут математики НАН України
2 Wiener criterion...
Dirichlet form has been defined in [5] where the local regularity and Har-
nack inequality have been proved for the harmonics of a Schrödinger type
problem with a potential in Kato class. We recall also that a Wiener cri-
terion at the boundary for harmonic functions relative to a Riemannian
(p-homogeneous) Dirichlet form has been proved in [7].
In the present paper we are interested in the Wiener criterion for the
solutions of the relaxed Dirichlet problem relative to a Riemannian (p-
homogeneous) Dirichlet form. The interest of relaxed Dirichlet problems
is twofold:
(1) From the Wiener criterion for relaxed Dirichlet problems a Wiener
criterion for regular point of the boundary follows (at least for
boundary data derived from functions in the domain of the form
relative to the entire space in consideration)
(2) The class of relaxed Dirichlet problems is closed for Γ-convergence
and in particular the Γ-limits of Dirichlet problems in open sets
with holes and homogeneous Dirichlet condition on the boundary
of holes are relaxed Dirichlet problems.
The paper is organized in sections. In the second section we give the
definition and the main properties of p-homogeneous Dirichlet forms in
the general and in the Riemannian case. In the third section we give the
definition of Kato class relative to a Riemannian p-homogeneous Dirichlet
form. In the fourth section we introduce the relaxed Dirichlet problem
and the relative capacity. In the fifth section we give our main result
concerning the Wiener criterion for a relaxed Dirichlet problem and finally
in the sixth section we give a sketch of the proof of our result.
1. Dirichlet Functionals and Forms
For the definition and properties of bilinear Dirichlet forms we refer to
the book [20] and for the Riemannian case to the paper [9]. We observe
that in the nonlinear case we do not have an extension of Beuerling–
Deny decomposition formula then we try to define directly a strongly
local form.
Firstly we describe the notion of strongly local p-homogeneous Dirich-
let form, p > 1, as given in [13,14].
We start with the notion of Dirichlet functional. 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,+∞], p > 1, be a l.s.c. strictly convex functional with
domain D, i.e. D = {v ∈ Lp(X,m) : Φ(v) < +∞}, such that Φ(0) = 0.
M. Biroli, F. Dal Fabbro, S. Marchi 3
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 Banach space with
respect to the norm ‖ · ‖D and the following estimate holds
c1‖v‖
p
D ≤ Φ1(v) = Φ(v) +
∫
X
|v|p dm ≤ c2‖v‖
p
D
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.
(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)
The functional Φ satisfying the assumptions (H1), (H2), (H3) is called a
Dirichlet functional.
We recall that (under the above assumptions) we can define a Cho-
quet capacity cap(E). Moreover we can also define in a natural way
the quasi-continuity of a function and prove that every function in D0 is
quasi-continuous and is defined quasi-everywhere (i.e. up to sets of zero
capacity), [14].
The assumptions (H1), (H2), (H3) have a global character; now we
will recall the definition of strongly local Dirichlet functional with a ho-
mogeneity degree p > 1. Let Φ satisfy (H1), (H2), (H3); we say that Φ
is a strongly local Dirichlet functional with a homogeneity degree p > 1
if the following conditions hold:
(H4) Φ has the following representation onD0: Φ(u) =
∫
X α(u)(dx) where
α is a non-negative bounded Radon measure depending on u ∈ D0, which
does not charge sets of zero capacity. We say that α(u) is the energy
(measure) of our functional. The energy α(u) (of our functional) is convex
with respect to u in D0 in the space of measures , i.e. if u, v ∈ D0 and
t ∈ [0, 1] then α(tu+(1−t)v) ≤ tα(u)+(1−t)α(v), and it is homogeneous
of degree p > 1, i.e. α(tu) = |t|pα(u), ∀u ∈ D0, ∀ t ∈ R.
Moreover the following closure property holds: if un → u in D0 and
α(un) converges to χ in the space of measures then χ ≥ α(u).
(H5) α is of strongly local type, i.e. if u, v ∈ D0 and u− v = const on an
open set A we have α(u) = α(v) on A.
4 Wiener criterion...
(H6) α(u) is of Markov type, if β ∈ C1(R) is such that β′(t) ≤ 1 and
β(0) = 0 and u ∈ D∩C0(X), then β(u) ∈ D∩C0(X) and α(β(u)) ≤ α(u)
in the space of measures.
Let Φ(u) =
∫
X α(u)(dx) be a strongly local Dirichlet functional with
domain D0. 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 M is the space of Radon measures
on X) uniformly for u, v in a compact set of D0, where µ(u, v) is defined
on D0 ×D0 and is linear in v. We say that a(u, v) =
∫
X µ(u, v)(dx) is a
strongly local p-homogeneous Dirichlet form.
We observe that (H3) is a consequence of (H1), (H2), (H4)–(H6). The
strong locality property allow us to define the domain of the form with
respect to an open set O, denoted by D0[O] and the local domain of the
form with respect to an open set O, denoted by Dloc[O]. We recall that,
given an open set O in X we can define a Choquet capacity cap(E;O)
for a set E ⊂ E ⊂ O with respect to the open set O. Moreover the sets
of zero capacity are the same with respect to O and to X.
We summarize in the following Proposition the main properties of a
strongly local p-homogeneous Dirichlet form
Proposition 1.1. Let a(u, v) =
∫
X µ(u, v)(dx) be a (p-homogeneous,
strongly local) Dirichlet form. For any u, v belonging to Dloc[X] ∩
L∞(X,m) we have
(i) µ(u, v) is homogeneous of degree p− 1 in u and linear in v
(ii) for any a ∈ R
+
|µ(u, v)| ≤ α(u+ v) ≤ 2p−1a−pα(u) + 2p−1ap(p−1)α(v)
(iii) µ(u, u) = pα(u)
(iv) (Leibnitz rule on the second argument) for any v, w ∈ Dloc[X] ∩
L∞(X,m) we have vw ∈ Dloc[X] ∩ L∞(X,m) and
µ(u, vw) = vµ(u,w) + wµ(u, v)
(v) (Schwarz inequality) For any f ∈ Lp′(X,α(u)) and g ∈Lp(X,α(v)),
u, v ∈ Dloc[X] and 1/p + 1/p′ = 1, fg is integrable with respect to
M. Biroli, F. Dal Fabbro, S. Marchi 5
the absolute variation of µ(u, v) and ∀ a ∈ R
+
∫
X
|fg| |µ(u, v)|(dx)
≤ 2p−1a−p
∫
X
|f |p
′
α(u)(dx) + 2p−1ap(p−1)
∫
X
|g|pα(v)(dx)
(vi) (Chain rule) If u, v ∈ Dloc[X] ∩ L∞(X,m) and g ∈ C1(R) with g′
bounded on R, then g(u), g(v) belong to Dloc[X] ∩ L∞(Ω,m) and
µ(g(u), v) = |g′(u)|p−2g′(u)µ(u, v),
µ(u, g(v)) = g′(v)µ(u, v)
(vii) (Truncation rule) For every u and v in Dloc[X] we have u+, v+ ∈
Dloc[X]
µ(u+, v) = 1{u>0}µ(u, v),
µ(u, v+) = 1{v>0}µ(u, v)
(where we denote again by u and v the quasi-continuous represen-
tative of u).
Assume now that a quasi-distance d is given on X. We denote by
B(x, r) the (open) ball for the distance d with center x and radius r. We
assume that
(H8) the topology induced by d is equivalent to the original topology
of X. Moreover, given a compact subset K of X, there exist constants
c1 > 0, ν ≥ 1 such that for every x ∈ K and every 0 < r ≤ r0 we have
m(B(x, r)) ≤ c1m(B(x, s))
(r
s
)ν
(1.1)
(Duplication property)
Remark 1.1. (a) If we assume that for every x and y in X with x 6= y
there exists a function ϕ in D0 ∩ C0(X) with L∞(X,m) energy density
such that ϕ(x) 6= ϕ(y), then
d(x, y) = sup{ϕ(x) − ϕ(y)},
where the sup is on the set
{ϕ ∈ D0 ∩ C0(X), µ(ϕ) ≤ m on X},
if finite, is a distance on X such that µ(d) ≤ m.
6 Wiener criterion...
(b) Under the assumption (H8) X is a space of homogeneous type,
[17]. We also observe that the following property
0 < m(B(x, 2r)) ≤ c0m(B(x, r))
(where x belongs to a compact set K, 0 < r ≤ 2r0) implies the duplica-
tion property in (H8) .
The following assumption (H9) gives a relation between the metric,
the measure on X and the measure valued map α.
(H9) We assume that, given a compact subset K of X, there exist con-
stants c2 > 0 and k ≥ 1 such that for every x ∈ K and every 0 < r ≤ r0
the following Poincaré inequality of order p holds
∫
B(x,r)
|u− ur|
pm(dx) ≤ c2r
p
∫
B(x,kr)
α(u)(dx) (1.2)
for every u ∈ Dloc[B(x, kr)], where
ur = [m(B(x, r))]−1
∫
B(x,kr)
um(dx).
Let us assume p < ν. Under the above assumptions the following
Sobolev inequality holds
(
1
m(B(x, r))
∫
B(x,r)
|u|p
∗
m(dx)
)
p
p∗
≤ c
(
rp
m(B(x, r))
∫
B(x,kr)
α(u)(dx)
)
+
(
1
m(B(x, r))
∫
B(x,r)
|u|pm(dx)
)
(1.3)
where x ∈ K, 0 < r < r0 and p∗ = pν
ν−p and c depending only on c0 and
c2. If ν ≥ p, then (1.3) holds again where p⋆ is any finite positive number
greater than p. Moreover from (1.3) we have the compact embedding of
the space D0(B(x, r)) into Lp(B(x, r),m), see [8,12] for the bilinear case
and [21] for the general case. A Dirichlet functional on a quasi-metric
space X with a quasi-distance d, for which (H1)–(H6), (H7), (H8) hold,
is called a Dirichlet–Poincaré functional. A Dirichlet–Poincaré functional
M. Biroli, F. Dal Fabbro, S. Marchi 7
Φ(u) = =
∫
X α(u)(dx) on the space X endowed with a distance d, such
that d ∈ Dloc[X] and α(d) ≤ m in the measures, is called a Riemannian–
Dirichlet functional (for an example of distance satisfying the above as-
sumptions see Remark 2.1). The corresponding Dirichlet forms (if they
exist i.e. if (H7) also holds) are called respectively a Dirichlet–Poincaré
form or a Riemannian–Dirichlet form.
Remark 1.2. If u ∈ D0[B(x, r)] the Poincaré and Sobolev inequalities
on B(x, r) holds without the presence of the term
1
m(B(x, r))
∫
B(x,r)
|u|pm(dx)
Consider a Riemannian–Dirichlet functional
∫
α(u)(dx), we denote
by d the distance on X. Let ψ be a C1-function decreasing and such
that ψ = 1 on (0, s), ψ = 0 on [t, r0], 0 < s < t < r0, 0 ≤ ψ ≤ 1,
ψ′ ≤ c
(t−s) ; taking into account that α(d) ≤ m and using the chain rule
we can prove that ψ(d(x, ·)) is a “cut off ” function between the balls
B(x, s) and B(x, t) with the same properties as in the classical Euclidean
frame.
Proposition 1.2. Given any two concentric balls B(x, s) and B(x, t)
with 0 < s < t < r0 there exists a function ϕ ∈ D0∩C0(X) such that 0 ≤
ϕ ≤ 1, ϕ(y) = 1 for y ∈ B(x, s), suppϕ ⊂ B(x, t) and α(ϕ) ≤ c
(t−s)pm,
where c is any fixed constant with c > 1.
Remark 1.3. As a consequence of the assumptions on X and d and of
the Poincaré inequality we have the following estimate on the capacity of
a ball [7]: for every fixed compact set K there exists positive constants
c4 and c5 such that
c4
m(B(x, r))
rp
≤ p− cap(B(x, r), B(x, 2r)) ≤ c5
m(B(x, r))
rp
where x ∈ K and 0 < 2r < r0.
Examples of (p-homogeneous) Riemannian–Dirichlet forms are:
(a) The forms relative to a subelliptic p-Laplacian also in the weighted
case
(b) The form (if it exists) relative to the p-energy on a measure metric
space, whose corresponding Sobolev space satisfies on the assump-
tions onD0 (such a form exists if the corresponding Sobolev space is
uniformly convex), and in particular the form relative the p-energy
on a Cheeger type metric structure [16].
8 Wiener criterion...
2. The Kato class and the relaxed Dirichlet problem
We give now the notion of Kato class of measures relative to a Rie-
mannian p-homogeneous Dirichlet form. In [5] the Kato class was defined
in the case of the subelliptic p-Laplacian and in [6] the following definition
of Kato class relative to a Riemannian p-homogeneous Dirichlet form has
been given:
Definition 2.1. Let λ be a Radon measure. We say that λ is in the
Kato space K(X) if
lim
r→0
Λ(r) = 0 (2.1)
where
Λ(r) = sup
x∈X
r
∫
0
(
|λ|(B(x, ρ))
m(B(x, ρ))
ρp
)1/(p−1) dρ
ρ
Let Ω ⊂ X be an open set; K(Ω) is defined as the space of Radon mea-
sures λ on Ω such that the extension of λ by 0 out of Ω is in K(X).
In [6] the properties of the space K(Ω) are investigated. In particular
it is proved that if Ω is a relatively compact open set of diameter R̄
2 , then
‖λ‖K(Ω) := Λ(R̄)p−1
is a norm on K(Ω) and we can prove, as in [4] for the bilinear case, that
K(Ω) endowed with this norm is a Banach space. Moreover in the above
paper it is proved that K(Ω) is contained in D
′
[Ω], where D
′
[Ω] denotes
the dual of D0[Ω].
Let a(u, v) =
∫
X µ(u, v)(dx) be a Riemannian p-homogeneous Dirich-
let form of domain D0 and let Ω be a r.c. open set in X. We denote by
σ a Borel (positive) measure on Ω, that does not charge sets of zero ca-
pacity. Let g be a continuous function on the closure of Ω, which belongs
also to D0[Ω] and λ a measure in the Kato class (relative to Ω).
Definition 2.2. The function u ∈ Dloc[Ω]∩Lp
loc(Ω, σ) is a local solution
of the relaxed Dirichlet problem relative to µ, Ω, σ, g and λ if u − g ∈
Lp
loc(Ω, σ) and
∫
Ω
µ(u, v)(dx) +
∫
Ω
|u− g|p−2(u− g) v σ(dx) =
∫
Ω
v λ(dx) (2.2)
for any v ∈ D0[Ω]∩Lp(Ω, σ) with compact support in Ω (we observe that
the condition (u− g) in Lp
loc(Ω, σ) can be imposed due to the fact that u
is q.e defined on every compact subset of Ω).
M. Biroli, F. Dal Fabbro, S. Marchi 9
We introduce now a notion of σ-capacity related to our relaxed Dirich-
let problem:
Definition 2.3. We say that a Borel subset E of an open subset B ⊂ Ω
is σ-admissible (with respect to B) if there exists a function v ∈ D0[B]
such that (w − 1) ∈ Lp(B, σ|E), where σ|E is the restriction of σ to E.
If E is not σ-admissible, then we define capσ(E,B) = +∞. If E is
σ-admissible, then we define
p− capσ(E,B) = min
{
∫
B
α(v)(dx) +
∫
B
|v − 1|pσ|E(dx)
}
where the minimum is taken on the set
{v ∈ D0(B); (v − 1) ∈ Lp(B;σ|E)}
The function vE which realizes the minimum is called the σ-potential of
E relative to Ω.
Remark 2.1. Let ω be an open set with closure contained in Ω and
define σω as the measure defined by
σω(E) = m(E) if cap(E ∩ ωc) = 0
σω(E) = +∞ otherwise
Let x0 ∈ ∂ω such that B(x0, 2r) ⊂ Ω. Then B(x0, r) is admissible in
B(x0, 2r) with respect to σω and we have
capσω
(B(x0, r);B(x0, 2r)) = cap(ωc ∩B(x0, r);B(x0, 2r))
3. The Wiener criterion for the relaxed Dirichlet problem
At first we give the definition of regular point for the relaxed Dirichlet
problem
Definition 3.1. A point x0 ∈ Ω is a regular point if every local solution
u of the relaxed Dirichlet problem relative to a neighbourhood of x0 in Ω
and an arbitrary g, λ satisfying the condition required in Definition 2.2,
is continuous at x0 and u(x0) = g(x0).
Remark 3.1. The regularity of a point x0 for (2.2) does not depend on
Ω, g, λ.
We now give the definition of Wiener point:
10 Wiener criterion...
Definition 3.2. A point x0 ∈ Ω is a regular point if
R
∫
0
δ(ρ)
1
p−1
dρ
ρ
= +∞
where
δ(ρ) = δ(ρ;x0) =
capσ(B(x0, ρ), B(x0, 2ρ))
cap(B(x0, ρ), B(x0, 2ρ))
We are now in position to state the main result of this paper:
Theorem 3.1. The point x0 is regular if and only if x0 is a Wiener
point.
Remark 3.2. Using the same notations of Remark 2.1 we have that in
the case σ = σω it is equivalent for a point x0 ∈ ∂ω to be a regular
(Wiener) point for the relaxed Dirichlet problem relative to σω or to be
a regular (Wiener) point of the boundary for the Dirichlet problem in ω.
The proof is easy if g ∈ D0[Ω], but it is enough to prove the equivalence
for g = 0.
4. Sketch of the proof of Theorem 3.1
We begin by the proof of the sufficient part of our criterion
First we prove that a suitable truncation of a solution u of the relaxed
Dirichlet problem is a subsolution of the Dirichlet problem:
Proposition 4.1. Let λ be a Radon measure in Ω such that λ ∈ D
′
[Ω],
and let u be a local solution of (2.2). Then
∫
Ω
µ((u∓ k)±, v)(dx) ≤
∫
Ω
v|λ|(dx)
∀ v ∈ D0[Ω], v ≥ 0 a.e. in Ω, where g± ≤ k in Ω.
We think that the result of above Proposition has an interest in itself.
We observe now that we may assume without loss of generality g(x0)
= 0.
The second step is to prove that the result follows from an inequality
for the energy of a suitable function.
Let x0 ∈ Ω. Assume that u ∈ D0[Ω] ∩ Lp
loc(Ω, σ) is a local weak
solution of (2.2). Let r ≤ 3R
4 , B(x0, 2R) ⊆ Ω. From Proposition 4.1
M. Biroli, F. Dal Fabbro, S. Marchi 11
uk := (u − k)+, where k ≥ supB(x0,2r) g, is a local weak subsolution of
the relaxed Dirichlet problem with σ = 0, that is it satisfies
∫
B(x0,2r)
µ(uk, ϕ)(dx) ≤
∫
B(x0,2r)
ϕ|λ|(dx)
∀ϕ ∈ D0[B(x0, 2r)], ϕ ≥ 0 a.e. in B(x0, 2r). Then it is locally bounded
in B(x0, 2r) (see [6]). Let us define
M(r) = sup
B(x0,r)
uk
Let ξ(r) be a positive increasing function such that ξ(r) → 0 when r → 0
and suppose
(κ1) ξ(r)−pΛ(r) → 0 when r → 0 if p ≥ 2
(κ2) ξ(r)−2Λ(r) → 0 when r → 0 if 1 < p < 2
For example we can choose ξ(r) = Λ(r)
1−ǫ
p if p ≥ 2 and ξ(r) = Λ(r)
1−ǫ
2 if
1 < p < 2, when 1 − ǫ > 0. Let us observe that we will suppose r small
enough to have ξ(r) ≤ 1.
Let
v =
1
M(r) − uk + ξ(r)
Proposition 4.2. Let p ∈ (1, ν], r ≤ r0
8k2 and η ∈ D0[B(x0,
r
2)] ∩
L∞(B(x0,
r
2)) with α(η) ≤ c
rp a.e. in Ω, for a positive constant c. Then
there exists a constant C > 0 dependent only on Ω, p but independent on
x0, r such that
rp
m(B(x0, r))
∫
Ω
α(ηv−1)(dx)
+
rp
m(B(x0, r))
∫
Ω
|v−1 −M(r) − ξ(r)|pηpσ(dx)
≤ C [M(r) + ξ(r)]
{
[
M(r) −M
(r
2
)
+ ξ(r)
]p−1
+ Σ(r)(p−1)
}
where
Σ(r)p−1 := C
(
|λ|(B(x0, r))
1∨(p−1)
p + Λ(r)p−1
)
≤ C
(
Λ(r)
[1∨(p−1)](p−1)
p + Λ(r)p−1
)
12 Wiener criterion...
We prove now that the result follows from Proposition 4.2. From
Proposition 4.2 we obtain the following inequality
M(
r
2
) ≤
[
1 − C
− 1
p−1 δ
(r
2
)
1
p−1
]
M(r) + ξ(r) + Σ(r)
From the above inequality and a well known iteration result [22] the
sufficient part of our Wiener criterion follows. We can also obtain an
estimate on the rate of convergence of u(x) to g(x0) as x converges to
x0. In particular if g is Hölder continuous, Λ(r) ≤ Crγ then the rate of
convergence is of the type |x− x0|
τ , with 0 < τ < γ suitable.
Then to prove the sufficient part of our Wiener criterion is enough to
prove the inequality in Proposition 4.2.
The proof of Proposition 4.2 is divided in different steps. In the
first step choosing as test function ηp( 1
w )(p−1) where w = v−1 and η ∈
D0[B(x0, r)] ∩ L
∞(B(x0, r)) with η = 1 in B(x0,
3
4r) and α(η) ≤ cr−pm
for a positive constant c we prove that
∫
B(x0,r)
α(lg(w))(dx) =
∫
B(x0,r)
α(lg(v))(dx) ≤ C
m(B(x0, r))
rp
From the above inequality we obtain that there are constants C and σ0
such that for |σ̃| ≤ σ0, and 0 < r < 3
4kr0
(
1
m(B(x0, r))
∫
B(x0,r)
vσ̃m(dx)
)(
1
m(B(x0, r))
∫
B(x0,r)
v−σ̃m(dx)
)
≤ C
The second step is the proof of a weak Harnack inequality for v. We
have that v is a subsolution of the problem with σ = 0. Then, using the
estimate in [6] we obtain
[
M(r) −M
(r
2
)
+ ξ(r)
]−q
≤
C
m(B(x0, 3r/4))
∫
B(x0,3r/4)
vqm(dx) + C
[
ξ(r)−(p∨2)Λ(r)
]q
for any q > 0. If we take in particular 0 < q ≤ σ0, from the above
inequalities we have
1
m(B(x0, 3r/4))
∫
B(x0,3r/4)
v−qm(dx) ≤ C
[
M(r) −M
(r
2
)
+ ξ(r)
]q
M. Biroli, F. Dal Fabbro, S. Marchi 13
whenever 0 < q ≤ σ0 . We have taken into account that
[
M(r) −M
(r
2
)
+ ξ(r)
]q [
ξ(r)−(p∨2)Λ(r)
]q
≤
1
2
The third step is the proof that the above inequality holds also for
q > σ0. Let τ < 0 such that p(τ + 1) > 1. Let β = τp + p − 1. Let us
observe that β is positive. We use as test function
ϕ = ηpψ ≥ 0
where η ∈ D0(B(x0, r)) ∩ L
∞(B(x0, r)), η ≥ 0 and
ψ =
(
vβ −
( 1
M(r) + ξ(r)
)β)
and we obtain
∫
B(x0,r)
α(ηvτ )(dx) ≤ K(τ)
[
∫
B(x0,r)
vpτα(η)(dx) + Σ(r)
]
where K(τ) ≃ |τ |p + β−p and ξ(r)−(p∨2)Λ(r)|λ|(B(x0, r)) =: Σ(r). The
Sobolev inequality and a finite iteration of Moser type gives the result.
In the last step we conclude the proof choosing as test function ϕ =
ηpuk where η ∈ D0[B(x0,
r
2)] ∩ L∞(B(x0,
r
2) with α(η) ≤ c
rpm for a
positive constant c and using for a suitable choice of the exponents in the
Hölder inequality.
We have now to sketch the proof of the necessary part. The proof
is by contradiction. For a given σ we consider the σ-potential B(x0, R)
in B(x0, 2R) denoted by vR and we denote wR = vR + 1. Let x0 be a
regular point such that
2R
∫
0
(
p− capσ(B(x0, ρ), B(x0, 2ρ))
p− cap(B(x0, ρ), B(x0, 2ρ))
)
1
p−1 dρ
ρ
< +∞
Using this relation we obtain
wR(x0) ≥
3
4
The above relation gives a contradiction with the assumption of the reg-
ularity of x0, which implies that wR(x0) = 0 with continuity. We remark
that the proof follows the lines of the proof of the necessity part of Wiener
criterion at the boundary in [7] and of the one given in the subelliptic
case in [11]
14 Wiener criterion...
References
[1] M. Biroli, Nonlinear Kato measures and nonlinear subelliptic Schrödinger prob-
lems // Rend. Acc. Naz. Scie. detta dei XL, Memorie di Matematica e Appli-
cazioni, 21 (1997), 235–252.
[2] M. Biroli, Weak Kato measures and Schrödinger problems for a Dirichlet form //
Rend. Acc. Naz. Scie. detta dei XL, Memorie di Matematica e Applicazioni, 24
(2000), 235–252.
[3] M. Biroli, Schrödinger type and relaxed Dirichlet problems for the subelliptic p-
Laplacian // Potential Analysis, 15 (2001), 1–16.
[4] M. Biroli, Nonlinear p-homogeneous Dirichlet forms on nonreflexive Banach
spaces // Rend. Acc. Naz. Scie. detta dei XL, Memorie di Matematica e Ap-
plicazioni, 29 (2005), 235–252.
[5] M. Biroli, S. Marchi, Harnack inequality for Schrödinger problem relative to
strongly local Riemannian p-homogeneous forms with a potential in the Kato class,
Boundary Value Problems, Special Issue “Harnack Estimates and Local Behavior
of degenerate and singular Partial Differential Equations”, 2007, Art. ID 24806,
pp. 19.
[6] M. Biroli, S. Marchi, Oscillation estimates relative to p-homogeneuous forms and
Kato measures data // Le Matematiche, 51 (II) (2006), 335–361.
[7] M. Biroli, S. Marchi, Wiener criterion at the boundary related to p-homogeneous
strongly local Dirichlet forms // Le Matematiche, 52 (2007), in print.
[8] M. Biroli and U. Mosco, Sobolev inequalities on homogeneous spaces // Potential
Analysis, 4 (1995), 311–324.
[9] M. Biroli, U. Mosco, A Saint Venant type principle for Dirichlet forms on dis-
continuous media // Ann. Mat. Pura e Appl., 169 (IV) (1995), 125–181.
[10] M. Biroli, U. Mosco, Kato spaces for Dirichlet forms // Potential Analysis, 10
(1999), 1–16.
[11] M. Biroli, N. Tchou, Relaxed Dirichlet problem for the subelliptic p-Laplacian //
Ann. Mat. Pura e Appl., 129 (IV), 39–64.
[12] M. Biroli, S. Tersian, On the existence of nontrivial solutions to a semilinear
equation relative to a Dirichlet form, Ist. Lombardo, Accad. Sci. Lett. Rend. A,
31 (1997), 151–168.
[13] M. Biroli, G. P. Vernole, Strongly Local Nonlinear Dirichlet Functionals,
Ukrainian Math. Bullettin, 1, 2004, pp. 485-500.
[14] M. Biroli, G.P. Vernole, Strongly Local Nonlinear Dirichlet Functionals and
Forms // Advances in Mathematical Sciences and Applications, 15 (2005), 655–
682.
[15] M. Biroli, G. P. Vernole, Harnack inequality for harmonic functions relative to
a nonlinear p-homogeneous Riemannian Dirichlet form // Nonlinear Analysis
TMA, 64 (2005), 51–68.
[16] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces //
Geom. Func. Anal., 9 (1999), 428–517.
[17] R. R. Coifman and G. Weiss, Analyse harmonique non commutative sur certaines
espaces homogénes, Lectures Notes in Math., 242, Springer-Verlag, 1971.
[18] G. Dal Maso, U. Mosco, Wiener criteria and energy decay for relaxed Dirichlet
problems // Arch. Rat. Mech. An., 95 (1986), 345–387.
M. Biroli, F. Dal Fabbro, S. Marchi 15
[19] G. Dal Maso, U. Mosco, Wiener’s criterion and Γ-convergence // J. Appl. Math.
Opt.,15 (1987), 15–63.
[20] M. Fukushima, Y. Oshima, M. Takeda, Dirichlet forms and Markov processes,
W. De Gruyter and Co., 1994.
[21] J. Maly, U. Mosco, Remarks on measure-valued Lagrangians on homogeneous
spaces // Ricerche Mat., 48 (1999), 217–231.
[22] U. Mosco, Wiener criterion and potential estimates for the obstacle problem //
Indiana Un. Math. J., 36 (1987), 455–494.
Contact information
Marco Biroli Department of Mathematics “F. Brioschi”,
Politecnico di Milano,
Piazza Leonardo da Vinci 32,
20133 Milano,
Italy
E-Mail: marco.biroli@polimi.it
Florangela Dal
Fabbro
Department of Mathematics “F. Brioschi”,
Politecnico di Milano,
Piazza Leonardo da Vinci 32,
20133 Milano,
Italy
E-Mail: forangela.dal_fabbro@polimi.it
Silvana Marchi Department of Mathematics,
Università di Parma,
Parco delle Scienze 53/A,
Parma
Italy
E-Mail: silvana.marchi@unipr.it
|