1 Introduction
A weakly monotone function in an open set Ω⊂ℝn, n≥2, is, loosely speaking, a Sobolev function that satisfies the minimum and maximum principles in a weak sense. Precisely, a function u∈Wloc1,1(Ω) is called weakly monotone if,
for every open set Ω′⊂⊂Ω, and every m,M∈ℝ such that m≤M and
(u-M)+-(m-u)+∈W01,1(Ω′),
one has that
m≤u≤M a.e. in Ω′.
Here, the subscript + stands for positive part.
The notion of weak monotonicity was introduced by Manfredi in [20], where he provided a new direct approach to the regularity theory of maps with finite distortion, and of maps in classes defined in terms of integrability properties of the adjugates of their gradients, which play a role in nonlinear elasticity (see [4]). Earlier proofs of continuity properties of these maps, contained in [11] and [30], made use of the notion of topological degree.
A key idea in [20] is to exploit the fact that the components of these maps are weakly monotone functions, and that any such function is continuous, or at least continuous outside a set of a certain capacity zero, provided that a sufficiently large power of the modulus of its gradient is integrable. Specifically, assume that u is a weakly monotone function from Wloc1,p(Ω) for some p≥1. If p>n, then u is continuous (irrespective of whether it is weakly monotone or not), by the Sobolev embedding theorem. Hence, it is monotone in the classical sense introduced by Lebesgue in his study of the Dirichlet problem in the plane [19]. The advance of [20, Theorem 1] amounts to showing that
if p=n, then u is continuous,
and that
(1.1)if p>n-1, then u is continuous outside a set of Cp,1-capacity zero.
Weakly monotone functions, further investigated in [21, 22], also arise in the regularity theory of elliptic partial differential equations as well. For instance, as pointed out in [12] and [17], weak solutions to p-Laplacian-type elliptic equations, with possibly degenerating ellipticity, turn out to be weakly monotone.
The result of [20] has paved the way to investigations on pointwise properties of weakly monotone functions in more general classes of Sobolev-type spaces. In particular, weakly monotone functions from Orlicz–Sobolev spaces are focused in the monograph [14], and in the papers [13, 16]. A motivation for these studies is the analysis of maps of bounded distortion from the local Orlicz–Sobolev space Wloc1,A(Ω) for some Young function A which need not be of power type. These contributions pointed out that continuity of a weakly monotone function is guaranteed even if it belongs to an Orlicz–Sobolev space slightly larger than Wloc1,n(Ω), namely if A(t) grows slightly more slowly than tn near infinity. Precisely, [16, Proposition 2.7] states that, if
(1.2)∫∞A(t)tn+1𝑑t=∞
and there exists ε>0 such that the function
(1.3)t↦A(t)tn-1+ε is increasing,
then any weakly monotone function from Wloc1,A(Ω) is continuous. The same conclusion, with (1.3) replaced by a somewhat stronger condition of a similar nature, is proved in [14, Theorem 7.5.1]. Furthermore, information on its (local) modulus of continuity is provided in [16]. Assumptions of a different kind ensuring
the continuity of weakly monotone functions in Orlicz–Sobolev spaces can also be found in [10].
Condition (1.2) amounts to imposing an appropriate degree of integrability of the gradient of the weakly monotone functions in question, and is an indispensable requirement. On the other hand, assumption (1.3) has an essentially technical nature. In fact, a close inspection of the proof of [16] reveals that (1.3) is basically needed to deduce certain properties of Orlicz–Sobolev functions from their analogues in the theory of standard Sobolev spaces.
In the present paper, we suggest some variants in the approach of [20], [14] and [16], that call into play peculiar Orlicz space techniques and results. This enables us to drop condition (1.3), and to establish the everywhere continuity of weakly monotone functions from the space Wloc1,A(Ω) under a single assumption, in the spirit of (1.2), but with A replaced by a closely related Young function depending also on n, that will be denoted by An-1. Namely, our condition reads
(1.4)∫∞An-1(t)tn+1𝑑t=∞,
and it also implies the local uniform continuity of any weakly monotone function in Wloc1,A(Ω), with an explicit modulus of continuity depending only on A and n.
The function An-1 appears in a sharp Poincaré-type inequality for the oscillation of Orlicz–Sobolev functions on the (n-1)-dimensional unit sphere in ℝn. A definition of An-1 can be found in Section 3, where the main results of this paper are stated. Here, let us just mention that, if n=2, assumption (1.4)
coincides with (1.2), since A1=A. This shows that condition (1.3) is actually irrelevant in the results of [14, 16] in this case. When n≥3, the function An-1 is equivalent to A (and hence conditions (1.2) and (1.4) again agree) in any customary situation, but, in general, A may grow slightly faster than An-1 near infinity. A characterization of those Young functions A which are equivalent to An-1 is given by property (3.16).
This is the content of Theorem 3.7, which enhances the results of [14, 16], since the pair of conditions (1.2)–(1.3) implies (1.4),
whereas Young functions can be exhibited that fulfill (1.4), but not (1.3) – see Proposition 5.2. This shows that Theorem 3.7 is applicable in circumstances where the available results in the literature may fail. Moreover, the results of [14, 16] can be recovered as a consequence of Theorem 3.7. In fact, condition (1.2) can be shown to be sufficient for the continuity of weakly monotone functions from Wloc1,A(Ω) when coupled with the additional condition (3.16), that is slightly less demanding than (1.3) – see Corollary 3.10.
Under a weaker assumption than (1.4) – a counterpart of the assumption p>n-1 appearing in (1.1) for classical Sobolev spaces – we prove
in Theorem 3.1 that every weakly monotone function from Wloc1,A(Ω) is locally bounded in Ω, and differentiable (and hence continuous) a.e. in Ω. The assumption in question is only needed when n≥3, and takes the form
(1.5)∫∞(tA(t))1n-2𝑑t<∞.
If n=2, the same conclusion holds whatever A is.
The same assumptions
ensure that every weakly monotone function from Wloc1,A(Ω) is indeed continuous outside an exceptional set of vanishing Orlicz capacity, defined in terms of A and n. This is the subject of Theorem 3.3, that not only extends, but as well enhances property (1.1) even in the case when Wloc1,A(Ω)=Wloc1,p(Ω). Remark 3.5 is devoted to this observation. Having Theorem 3.3 at disposal, an estimate for the size of the exceptional set in terms of Hausdorff measures, depending on A and n, is established in Theorem 3.7.
2 Orlicz and Orlicz–Sobolev spaces
The notion of Orlicz space relies upon that of Young function.
A function A:[0,∞)→[0,∞] is called a Young function if it is
convex, non-constant in (0,∞), and vanishes at 0. Any function fulfilling these properties
has the form
(2.1)A(t)=∫0ta(r)𝑑r for t≥0,
for some non-decreasing, left-continuous function a:[0,∞)→[0,∞] which is neither identically 0, nor infinity. Observe that the function
t↦A(t)t is
non-decreasing,
and
(2.2)A(t)≤a(t)t≤A(2t) for t≥0.
Furthermore, if k≥1, then
(2.3)kA(t)≤A(kt) for t≥0,
and hence
(2.4)kA-1(t)≥A-1(kt) for t≥0.
Here, A-1 denotes the (generalized) right-continuous inverse of A. The Young conjugate A~ of A is
defined by
A~(t)=sup{st-A(s):s≥0} for t≥0.
The alternative notation A~
will also be adopted instead of A~ whenever convenient. Note
the representation formula
A~(t)=∫0ta-1(r)𝑑r for t≥0,
where a-1 denotes the (generalized) left-continuous inverse of
the function a. Observe that A~~=A.
If, for instance, A(t)=tpp for some p∈(1,∞), then
A~(t)=tp′p′,
where p′=pp-1 is the Hölder conjugate of p.
A property to be used in what follows tells us that, if A is a Young function and q∈(1,∞), then
(2.5)the function t↦A(t)tq is increasing if and only if the function t↦A~(t)tq′ is decreasing.
An application of equation
(2.2) with A replaced by A~ yields
A~(t)≤a-1(t)t≤A~(2t) for t≥0.
Moreover,
t≤A-1(t)A~-1(t)≤2t for t≥0.
A Young function A is said to satisfy the Δ2-condition
near infinity – briefly, A∈Δ2 near infinity – if it is
finite-valued and there exist constants C>2 and t0≥0 such
that
(2.6)A(2t)≤CA(t) for t≥t0.
Owing to equation (2.2), condition (2.6) turns out to be equivalent to the existence of constants C>0 and t0>0 such that
(2.7)a(2t)≤Ca(t) for t≥t0.
The function A is said to satisfy the ∇2-condition
near infinity – briefly, A∈∇2 near infinity – if there
exist constants C>2 and t0≥0 such that
A(2t)>CA(t) for t≥t0.
One has that
(2.8)A∈∇2 near infinity if and only if the function t↦A(t)t1+ε is increasing for t≥t0,
for some constants
ε>0 and t0≥0.
Let us also note that
(2.9)A∈Δ2 near infinity if and only if A~∈∇2 near infinity.
A Young function A is said to
dominate another Young function B near infinity if there exist
constants C>0 and t0≥0 such that
B(t)≤A(Ct) for t≥t0.
The functions A and B are called equivalent near infinity if
they dominate each other near infinity.
If any of the above properties is satisfied with t0=0, then it is
said to hold globally, instead of just near infinity.
Now, let E be a measurable subset of ℝn.
We denote by ℳ(E) the space of real-valued measurable functions on E. The notation ℳ+(E) is adopted for the subset of nonnegative functions in ℳ(E). The subscript “+” will be used with an analogous meaning when attached to the notation of other function spaces.
The Orlicz space LA(E) built upon a
Young function A is the Banach function space of those functions
u∈ℳ(E) for which the
Luxemburg norm
∥u∥LA(E)=inf{λ>0:∫EA(|u|λ)𝑑x≤1}
is finite. In particular, LA(E)=Lp(E) if A(t)=tp for some p∈[1,∞), and LA(E)=L∞(E) if A(t)=0 for t∈[0,1) and A(t)=∞ for t∈[1,∞).
The Hölder-type inequality
∥v∥LA~(E)≤supu∈LA(E)∫E|uv|𝑑x∥u∥LA(E)≤2∥v∥LA~(E)
holds for every u∈LA(E) and v∈LA~(E).
Denote by |E| the Lebesgue measure of E, and assume that |E|<∞. Then
LA(E)→LB(E)
if and only if the Young function A dominates the Young function B near
infinity. Here, and in what follows,
the arrow “→” stands for continuous embedding. In
particular,
(2.10)LA(E)=LB(E) (up to equivalent norms) if and only if A and B are equivalent near infinity.
Given an open set Ω⊂ℝn and a Young function A,
the Orlicz–Sobolev space W1,A(Ω) is defined as
W1,A(Ω)={u∈LA(Ω):u is weakly differentiable, and |∇u|∈LA(Ω)}.
The space W1,A(Ω), equipped with the norm given by
∥u∥W1,A(Ω)=∥u∥LA(Ω)+∥∇u∥LA(Ω)
for u∈W1,A(Ω),
is a Banach space. The space of those functions u∈ℳ(Ω) such that u∈W1,A(Ω′) for every bounded open set Ω′⊂⊂Ω will be denoted by Wloc1,A(Ω).
We refer the reader to the monographs [18, 26, 27] for a comprehensive treatment of the topics of this section.
3 Main results
We begin our analysis with a condition on Young functions A ensuring the local boundedness, as well as the continuity and differentiability almost everywhere, of weakly monotone functions from the Orlicz–Sobolev space Wloc1,A(Ω). Its formulation involves the Young function An-1 associated with A and n by
(3.1)An-1(t)={A(t)if n=2,(tn-1n-2∫t∞A~(r)r1+n-1n-2𝑑r)~if n≥3,
for t≥0. The integral on the right-hand side of (3.1) is convergent if and only if
the function A fulfills condition (1.5) – see, e.g., [7, Lemma 4.1]. This condition
will always come into play when dealing with the function An-1 for n≥3.
As mentioned in Section 1, the function An-1 arises in a Poincaré-type inequality for the oscillation of functions from Orlicz–Sobolev spaces on (n-1)-dimensional spheres, and has a crucial role in the results to be presented. Let us notice that the function An-1(t) is always dominated by A(t) near infinity, and, if n≥3, it is equivalent to A(t) whenever the latter grows faster than the function tn-1 in a suitable sense – see equation (3.16) below.
Correction added on April 28, 2019 after online publication: “always dominates” was changed to “is always dominated by”.
In what follows, the notation ⨍Br(x)⋯𝑑z stands for 1|Br(x)|∫E⋯𝑑z, where Br(x) denotes the ball, centered at x∈ℝn, with radius r>0. Furthermore, we set
essoscBr(x)u=esssupBr(x)u-essinfBr(x)u
for u∈ℳ(Ω).
Theorem 3.1.
Let A be a Young function. Assume that either n=2, or n≥3 and A fulfills condition (1.5). Let An-1 be the Young function defined by (3.1).
Let u∈Wloc1,A(Ω) be a weakly monotone function.
Then u∈Lloc∞(Ω), and there exists a constant c=c(n) such that
(3.2)essoscBr(x)u≤crAn-1-1(⨍B2r(x)A(|∇u|)𝑑z)
whenever B2r(x)⊂⊂Ω. Moreover, there exists a representative of u that is differentiable a.e. in Ω.
Remark 3.2.
The a.e. differentiability of weakly differentiable functions u∈Wloc1,A(Ω) under assumption (1.5) can also be derived from [25, Theorem 1.2],
via an inclusion relation between Orlicz and Lorentz spaces established in [15]. Here, we present a self-contained proof, that just relies upon Orlicz spaces techniques, and contains some preliminary steps of use for our subsequent results.
Of course, in particular Theorem 3.1 tells us that any function u as in its statement has a representative which is continuous outside a set of Lebesgue measure zero.
More precise information about the set of points of continuity of any weakly monotone function u∈Wloc1,A(Ω) can in fact be provided under the same assumptions on A and n. It turns out that any such function has a representative whose restriction to the complement in Ω of an exceptional set of (suitably defined) vanishing capacity is continuous. This is the content of Theorem 3.3 below.
The relevant capacity generalizes the Cp,1-capacity associated with Riesz potential spaces, and depends on the Young function A and on the dimension n of the ambient space ℝn of Ω. It can be defined as follows.
Let Ψ:[0,∞)→[0,∞) be a continuous function.
Consider the Riesz-type operator defined as
IΨf(x)=∫ℝnf(y)|x-y|nΨ(1|x-y|)𝑑y for x∈ℝn
for f∈ℳ+(ℝn).
The associated
CΨ,1-capacity of a set E⊂ℝn is given by
(3.3)CΨ,1(E)=inf{∫ℝnf(x)𝑑x:f∈ℳ+(ℝn),IΨf(x)≥1for x∈E}.
Note that, with the choice Ψ(t)=tα, where α∈(0,n), the operator IΨ reproduces the classical Riesz potential Iα given by
Iαf(x)=∫ℝnf(y)|x-y|n-α𝑑y for x∈ℝn.
Hence,
the CΨ,1-capacity agrees with the standard Cα,1-capacity associated with the operator Iα (see [1]).
Theorem 3.3.
Let A be a Young function. Assume that either n=2, or n≥3 and A fulfills condition (1.5). Let An-1 be the Young function defined by (3.1). Assume that σ:[0,∞)→[0,∞) is a continuous function such that
(3.4)∫∞An-1(λt)tσ(t)An-1(t)𝑑t=∞ for every λ>0.
Let Ψ:[0,∞)→[0,∞) be the function given by
(3.5)Ψ(t)=σ(t)An-1(t) for t≥0.
Then every weakly monotone function u∈Wloc1,A(Ω)
admits a representative whose restriction to the complement in Ω of an exceptional set of CΨ,1-capacity zero is continuous.
When the function A∈Δ2 near infinity, the function An-1 enjoys the same property – see Proposition 4.3, Section 4. Hence, An-1(λt) and An-1(t) are bounded by each other for large t, up to positive multiplicative constants depending on A, n and λ. Assumption (3.4) then takes a simpler form in this special case. This is stated in the following corollary.
Corollary 3.4.
Let A be a Young function such that A∈Δ2 near infinity. Assume that either n=2, or n≥3 and A fulfills condition (1.5). Let
σ:[0,∞)→[0,∞) be a continuous function such that
(3.6)∫∞dttσ(t)=∞,
and
let Ψ be the function defined as in (3.5).
Then every weakly monotone function u∈Wloc1,A(Ω)
admits a representative whose restriction to the complement in Ω of an exceptional set of CΨ,1-capacity zero is continuous. In particular, condition (3.6) holds with σ(t)=1, and hence u enjoys this property outside a set of CAn-1,1-capacity zero, where An-1 is the Young function defined by (3.1).
Remark 3.5.
Corollary 3.4 not only recovers, but somewhat allows for improvements of property (1.1) proved in [20] in the case when A(t)=tp for some p∈(n-1,n). Indeed, it tells us that any weakly monotone function in Wloc1,p(Ω), with p in this interval, admits a representative that is continuous outside a subset of Ω having Cσ(t)tp,1-capacity zero, for any function σ fulfilling (3.6). Possible choices of σ are, for instance, σ(t)=log(1+t), σ(t)=log(1+t)log(1+log(1+t)), etc.
Let us also notice that, since no restriction is imposed on A if n=2, in this case Theorems 3.1 and 3.3 also hold if A(t)=t, namely when Wloc1,A(Ω)=Wloc1,1(Ω). In particular, this shows that the result (1.1) of [20] is still valid for the endpoint value p=1 when n=2.
Capacities can be dismissed in the description of the size of the exceptional set of possible discontinuity points of a weakly monotone function in Wloc1,A(Ω), with A as in Theorem 3.3. This size can be estimated in terms of Hausdorff measures. Given a continuous, increasing function h:[0,∞)→[0,∞) such that h(0)=0, the classical h-Hausdorff measure
ℋh(⋅)(E) of a set E⊂ℝn is defined as
(3.7)ℋh(⋅)(E)=limε→0+inf{∑j=1∞h(diam(Kj)):E⊂⋃j=1∞Kj,diam(Kj)≤ε}.
If h(t)=tβ for some β>0, then ℋh(⋅)
agrees (up to a multiplicative constant) with the standard
β-dimensional Hausdorff measure ℋβ.
Note that we may assume, without loss of generality, that
(3.8)r↦h(r)rn is a non-increasing function.
Indeed, [1, Proposition 5.1.8] tells us what follows. If lim infr→0+h(r)rn=0, then ℋh(⋅)(E)=0 for every set E⊂ℝn. If lim infr→0+h(r)rn>0, then there exists another continuous increasing function h¯ such that (3.8) is satisfied with h replaced by h¯, and moreover ℋh(⋅)(E) and ℋh¯(⋅)(E) are bounded by each other for every set E⊂ℝn, up to multiplicative constants independent of E.
Our estimate of the exceptional set of weakly monotone Orlicz–Sobolev functions via Hausdorff measures involves a function Ψ satisfying the assumptions of Theorem 3.3, and such that
(3.9)t↦tnΨ(1t) is a
non-decreasing function decaying to 0 as t→0+.
Clearly, the latter assumption ensures that the derivative d(-1snΨ(1s)) defines a positive measure on (0,∞).
Theorem 3.6.
Let A be a Young function. Assume that either n=2, or n≥3 and A fulfills condition (1.5). Assume that the function h is as above, and let Ψ be as
in Theorem 3.3. Assume, in addition, that Ψ satisfies condition (3.9). If
(3.10)∫0h(s)d(-1snΨ(1s))<∞,
then every weakly monotone function u∈Wloc1,A(Ω)
admits a representative whose restriction to the complement in Ω of an exceptional set of Hh(⋅)-measure zero is continuous.
We are ready to state the main result of this paper, concerning the everywhere continuity of Orlicz–Sobolev weakly monotone functions. It asserts that,
if assumption (1.5) is properly strengthened, then any weakly monotone function from Wloc1,A(Ω) has a representative which is continuous in the whole of Ω. The assumption to be imposed is (1.4),
with An-1 defined by (3.1). In fact, under assumption (1.4), any weakly monotone function from Wloc1,A(Ω) is
locally uniformly continuous with a modulus of continuity depending only on A and n. This modulus of continuity
ω:[0,∞)→[0,∞) is defined as
(3.11)ω(r)=rB-1(r-n) for r>0,
where
B:[0,∞)→[0,∞) is the function given by
(3.12)B(t)=tn∫0tAn-1(s)s1+n𝑑s for t>0.
The space of functions in Ω that are locally uniformly continuous, with a modulus of continuity not exceeding ω, will be denoted by 𝒞locω(⋅)(Ω).
Observe that, since we are dealing with properties of functions from the local
Orlicz–Sobolev space Wloc1,A(Ω), the function A can be modified, if
necessary, near 0 in such a way that
(3.13)∫0An-1(t)t1+n𝑑t<∞.
Owing to property (2.10), a modification of this kind leaves the space Wloc1,A(Ω) unchanged. With condition (3.13) in force, the function B is well defined. Also, it can be verified that it is a Young function, whence its inverse B-1 is well defined as well. Observe that different modifications of B near 0 result in the same space 𝒞locω(⋅)(Ω).
Theorem 3.7.
Let n≥2, and let A be a Young function fulfilling condition (1.4),
with An-1 defined by (3.1).
Then every weakly monotone function u∈Wloc1,A(Ω)
admits a continuous representative. Moreover,
u∈Clocω(⋅)(Ω), where ω is given by (3.11).
Remark 3.8.
If n=2, then A1=A, and hence
assumption (1.4) agrees with
∫∞A(t)t3𝑑t=∞,
namely with (1.2). On the other hand, if n≥3, assumption (1.4)
is equivalent to
(3.14)∫∞t1-nn-2(∫t∞A~(s)s1+n-1n-2𝑑s)1-n𝑑t=∞.
Indeed, by [7, Lemma 4.1], condition (1.4)
is equivalent to
∫∞(sAn-1~(s))n-1𝑑s=∞,
and, in view of definition (1.5), the latter coincides with (3.14).
Under the additional assumption that A∈Δ2 near infinity, condition (1.4) can be reformulated, for n≥3, in a form which only involves A, and avoids explicit reference to An-1. This fact is a consequence of [5, Lemma 3.3], and is enucleated in the next result.
Corollary 3.9.
Let n≥3. Assume that A is a Young function satisfying the Δ2-condition near infinity, and such that
(3.15)∫∞(tA(t))2n-2(∫t∞(sA(s))1n-2𝑑s)-n𝑑t=∞.
Then every weakly monotone function u∈Wloc1,A(Ω)
admits a continuous representative. Moreover,
u∈Clocω(⋅)(Ω), where ω is given by (3.11).
Let us next mention a standard situation when assumption (1.4) reduces to (1.2), even for n≥3. Suppose, for simplicity, that A is finite-valued. This is of course the only nontrivial case, since Wloc1,A(Ω)=Wloc1,∞(Ω) if A jumps to infinity, and every function in the latter space is locally Lipschitz continuous. When n≥3,
the function An-1 is equivalent to A if and only if its lower Boyd index at infinity, defined as
i(A)=limλ→∞log(lim inft→∞A(λt)A(t))logλ,
satisfies
(3.16)i(A)>n-1
– see [29, Lemma 2.3] and [8, Proposition 4.1] for more details. Thus, when assumption (3.16) is in force, conditions (1.4) and (1.2) are equivalent. As a consequence, the following corollary of Theorem 3.7 holds. Note that, since assumption (1.3) implies (3.16), this corollary recovers, in particular, the results of [14] and [16] on the continuity of weakly monotone functions recalled in Section 1. Furthermore, the modulus of continuity takes a somewhat simpler form than (3.11), since, in this case, An-1 can be replaced by A in (3.12).
Corollary 3.10.
Let A be a Young function fulfilling condition (1.2). Assume that either n=2, or n≥3 and condition (3.16) holds. Then every weakly monotone function u∈Wloc1,A(Ω)
admits a continuous representative. Moreover, u∈Clocω¯(⋅)(Ω), where the function ω¯:[0,∞)→[0,∞) is defined as
ω¯(r)=rB¯-1(r-n) for r>0,
and
B¯:[0,∞)→[0,∞) is the function given by
B¯(t)=tn∫0tA(s)s1+n𝑑s for t>0.
4 Proofs of the main results
The proof of Theorem 3.1 requires a preliminary result on smooth approximation of functions from an Orlicz–Sobolev space W1,A(Ω). Such an approximation is not possible in norm, unless A satisfies the Δ2-condition. However, convolution with a sequence of smooth kernels provides us with an approximating sequence whose Dirichlet integrals associated with A converge to the corresponding Dirichlet integral of the limit function.
Lemma 4.1.
Assume that |Ω|<∞. Let A be a Young function, and let u∈W01,1(Ω) be
such that
∫ΩA(|∇u|)𝑑x<∞.
Let {uk} be
a sequence of convolutions of u with mollifiers ϱk, namely uk:Rn→R,
uk=u*ϱk,
where ϱk∈C0∞(Rn), suppϱk⊂B1/k(0), ϱk≥0 and ∫Rnϱk𝑑x=1 for k∈N.
Then {uk}⊂C∞(Rn) and
(up to subsequences)
(4.1)limk→∞uk=u at every Lebesgue point of u,
(4.2)limk→∞∇uk=∇u at every Lebesgue point of ∇u,
(4.3)limk→∞∫EA(|∇uk|)𝑑x=∫EA(|∇u|)𝑑x for every measurable set E⊂Ω.
Proof.
Properties (4.1) and (4.2) are classical. As far as
(4.3) is concerned, given any measurable set F⊂E, the Hardy–Littlewood inequality ensures that
(4.4)∫FA(|∇uk|)𝑑x≤∫0|F|A(|∇uk|*)𝑑r,
where the asterisk “*” stands for decreasing rearrangement. Moreover, by
a rearrangement inequality for convolutions [24],
(4.5)∫0τ|∇uk|*𝑑s≤∫0τ|∇u|*𝑑s∫0τϱk*𝑑s+τ∫τ∞|∇u|*ϱk*𝑑s≤∫0τ|∇u|*𝑑s∫0τϱk*𝑑s+τ|∇u|*(τ)∫τ∞ϱk*𝑑s≤∫0τ|∇u|*(τ)𝑑s∫0∞ϱk*𝑑s=∫0τ|∇u|*𝑑s∫ℝnϱk𝑑x=∫0τ|∇u|*𝑑s for τ≥0.
Note that, in the last but one inequality, we have made use of the fact that ∫0∞ϱk*𝑑s=∫ℝnϱk𝑑x=1 for k∈ℕ. Inequality (4.5), via [3, Proposition 2.1], tells us that
(4.6)∫0|F|A(|∇uk|*)𝑑s≤∫0|F|A(|∇u|*)𝑑s
for k∈ℕ.
Coupling inequalities (4.4) and (4.6) yields
(4.7)∫FA(|∇uk|)𝑑x≤∫0|F|A(|∇u|*)𝑑s
for k∈ℕ.
Inequality (4.7) entails that the sequence {A(|∇uk|)} is equi-integrable over E, since the right-hand side of (4.7) approaches 0 as |F|→0, inasmuch as
∫0|F|A(|∇u|*)𝑑s≤∫0|Ω|A(|∇u|*)𝑑s=∫ΩA(|∇u|)𝑑x.
Hence, (4.3) follows via (4.2) and Vitali’s
convergence theorem.
∎
We are now ready to prove Theorem 3.1.
Proof of Theorem 3.1.
Since the statement has a local nature, on replacing, if necessary, Ω with a bounded open subset and multiplying u by a smooth cut-off function compactly supported in Ω, we may suppose, without loss of generality, that u∈W1,A(Ω)∩W01,1(Ω). Moreover, we assume, for the time being, that
(4.8)∫ΩA(|∇u|)𝑑z<∞.
Let uk be the sequence appearing in Lemma 4.1. Given any
x0∈Ω, let R>0 be such that BR(x0)⊂⊂Ω, and let r∈(0,R). By [14, Lemma 7.4.1], for every δ>0 and any Lebesgue points x,y∈Br(x0) of u, there exists k¯=k¯(x,y,δ,r,R) such that
(4.9)|uk(x)-uk(y)|≤2δ+oscSτ(x0)uk
if k≥k¯ and τ∈[r,R]. Here, Sτ(x0) denotes the (n-1)-dimensional sphere in ℝn centered at x0, with radius τ, and
oscSτ(x0)uk=supSτ(x0)uk-infSτ(x0)uk.
An Orlicz–Sobolev
Poincaré-type inequality on the (n-1)-dimensional sphere Sτ(x0)
(see [5, Theorem 4.1], see also [2, 6, 23, 31] for related results) tells us that, if either n=2, or n≥3 and (1.5) holds, then
(4.10)oscSτ(x0)uk≤CτAn-1-1(τ1-n∫Sτ(x0)A(|∇uk|)𝑑ℋn-1)
for some constant C=C(n), and for τ>0.
Thanks to (4.9) and (4.10),
(4.11)1Cτ|uk(x)-uk(y)|≤2δCτ+An-1-1(τ1-n∫Sτ(x0)A(|∇uk|)𝑑ℋn-1),
if k≥k¯ and τ∈[r,R].
Given α∈(0,1), inequality (4.11)
can be rewritten as
αCτ|uk(x)-uk(y)|≤2αδCτ(1-α)(1-α)+αAn-1-1(τ1-n∫Sτ(x0)A(|∇uk|)𝑑ℋn-1).
Hence, by the convexity of the
function An-1,
(4.12)An-1(αCτ|uk(x)-uk(y)|)≤(1-α)An-1(2αδCτ(1-α))+ατ1-n∫Sτ(x0)A(|∇uk|)𝑑ℋn-1
if k≥k¯, and τ∈[r,R].
Now, fix any Lebesgue point t∈(r,R) for the function t↦∫St(x0)A(|∇u|)𝑑ℋn-1. Note that this
function belongs to L1(0,R), since ∫BR(x0)A(|∇u|)𝑑z<∞, owing to assumption (4.8). Given any number ε>0 such that (t-ε,t+ε)⊂(r,R), multiply through by τn-1 inequality (4.12), and integrate over (t-ε,t+ε) to obtain
(4.13)∫t-εt+ετn-1An-1(αCτ|uk(x)-uk(y)|)𝑑τ≤2ε(1-α)Rn-1An-1(2αδCr(1-α))+α∫t-εt+ε∫Sτ(x0)A(|∇uk|)𝑑ℋn-1𝑑τ.
Passing to the limit as k→∞ in inequality (4.13), making use of equations
(4.1) and (4.3), and then passing to the limit as
δ→0 yield
(4.14)∫t-εt+ετn-1αAn-1(α|u(x)-u(y)|Cτ)𝑑τ≤∫t-εt+ε∫Sτ(x0)A(|∇u|)𝑑ℋn-1𝑑τ.
Since An-1 is a Young function, the function
An-1(α)α is increasing in α. One can then pass to the limit as α→1-, and make use of the
monotone convergence theorem in the integral on the left-hand side
of (4.14) to deduce that inequality (4.14) continues to hold for α=1. On dividing through by 2ε the
resulting inequality, and letting ε→0, we conclude
that
(4.15)tn-1An-1(|u(x)-u(y)|Ct)≤∫St(x0)A(|∇u|)𝑑ℋn-1
for all Lebesgue points x,y∈Br(x0) of u, and for a.e. t∈[r,R].
Next, observe that
the function
(4.16)t↦tn-1An-1(1t) is decreasing.
Indeed, property (4.16) is equivalent to the fact that the function
(4.17)t↦An-1(t)tn-1 is increasing.
If n=2, then (4.17) just holds because A1=A, a Young function. If n≥3, then by (2.5) property (4.17) is in turn equivalent to the fact that the function
(4.18)t↦An-1~(t)t(n-1)′ is decreasing.
Property (4.18) trivially holds, since
An-1~(t)t(n-1)′=∫t∞A~(r)r1+n-1n-2𝑑r for t>0.
Equation (4.16) is this verified.
Next, assume that 2r<R. On integrating (4.15) over (r,2r) and making use of (4.16)
we deduce that
r(2r)n-1An-1(|u(x)-u(y)|2Cr)≤∫r2rtn-1An-1(|u(x)-u(y)|Ct)𝑑t≤∫r2r∫St(x0)A(|∇u|)𝑑ℋn-1𝑑t
≤∫02r∫St(x0)A(|∇u|)𝑑ℋn-1𝑑t=∫B2r(x0)A(|∇u|)𝑑z<∞,
whence (3.2) follows, thanks to property (2.4) with A replaced by An-1.
Next, let u^:Ω→(-∞,∞] be the function defined by
(4.19)u^(x)=lim supr→0+⨍Br(x)u(z)𝑑z for x∈Ω.
By the Lebesgue differentiation theorem,
u(x)=u^(x) for a.e. x∈Ω.
Thereby, u^ is a representative of u.
Given any x,y∈Br(x0), let ρ,σ>0 be such that Bρ(x)⊂Br(x0) and Bσ(y)⊂Br(x0). Owing to inequality (3.2),
(4.20)|⨍Bρ(x)udz-⨍Bσ(y)udz|≤CrAn-1-1(⨍B2r(x0)A(|∇u|)dz).
Taking the lim sup in (4.20) first as ρ→0+, and then as σ→0+ tells us that
(4.21)|u^(x)-u^(y)|≤CrAn-1-1(⨍B2r(x0)A(|∇u|)𝑑z).
Inequality (4.21) ensures that u^ is continuous at every Lebesgue point of the function A(|∇u|), and hence a.e. in Ω. Moreover, an application of (4.21) with y=x0 and r=|x-x0| yields
|u^(x)-u^(x0)||x-x0|≤CAn-1-1(⨍B2|x-x0|(x0)A(|∇u|)𝑑z).
Thus,
lim supx→x0|u^(x)-u^(x0)||x-x0|<∞
if x0 is any Lebesgue point of the function A(|∇u|). Hence, the differentiability of u^ a.e. in Ω follows, via a classical result by Stepanoff [28].
Finally, if condition (4.8) does not hold, then, however, it does hold with u replaced with λu for a suitable λ>0. The above argument then applies to λu, and hence the a.e. differentiability of u still follows. As for inequality (3.2), it trivially continues to hold even if (4.8) fails, since its right-hand side is infinite in this case.
∎
The content of the next lemma is a basic property of the capacity defined by (3.3), to be used in the proof of Theorem 3.3.
Lemma 4.2.
Let Ψ:[0,∞)→[0,∞) be a continuous function, and let CΨ,1 be the capacity defined by (3.3).
If f∈L+1(Rn), then
(4.22)CΨ,1({IΨf=∞})=0.
Proof.
Given f∈L+1(ℝn), we have that
CΨ,1({IΨf=∞})≤CΨ,1({IΨf>λ})=CΨ,1({IΨ(fλ)>1})≤1λ∫ℝnf(x)dx
for every λ>0. Hence, equation (4.22) follows on
letting λ go to infinity.
∎
Proof of Theorem 3.3.
On replacing, if necessary, u by λu for a suitable λ>0, we may assume, without loss of generality, that condition (4.8) is in force.
Given x0∈Ω, let R>0 be such that BR(x0)⊂⊂Ω and let r∈(0,R).
Inequality
(4.15) implies that
(4.23)tn-1An-1(essoscBr(x0)uCt)≤∫St(x0)A(|∇u|)𝑑ℋn-1
for a.e. t∈[r,R].
Multiplying through inequality (4.23)
by 1tnσ(1t)An-1(1t), and
integrating the resulting inequality over (r,R) yield
(4.24)∫rR1tσ(1t)An-1(essoscBr(x0)utC)An-1(1t)𝑑t≤∫rR1tnσ(1t)An-1(1t)∫St(x0)A(|∇u|)𝑑ℋn-1𝑑t≤∫0R1tnσ(1t)An-1(1t)∫St(x0)A(|∇u|)𝑑ℋn-1𝑑t.
A change of variables to polar coordinates tells us that
(4.25)∫0R1tnσ(1t)An-1(1t)∫St(x0)A(|∇u|)𝑑ℋn-1𝑑t=∫BR(x0)A(|∇u|)|x-x0|nσ(1|x-x0|)An-1(1|x-x0|)𝑑x=IΨ(A(|∇u|)χBR(x0)),
where χBR(x0) denotes the characteristic function of BR(x0).
By Lemma 4.2, there exists a set E⊂Ω such that CΨ,1(E)=0, and IΨ(A(|∇u|)χBR(x0)) is finite for every x0∈Ω∖E.
We claim that
(4.26)limr→0+essoscBr(x0)u=0
for any such x0. Assume, by contradiction, that (4.26)
fails, and hence essoscBr(x0)u≥λ for some λ>0, and for every r∈(0,R).
Equations (4.24) and (4.25) imply that
C′≥∫rR1tσ(1t)An-1(essoscBr(x0)utC)An-1(1t)𝑑t≥∫rR1tσ(1t)An-1(λtC)An-1(1t)𝑑t
for some constant C′. Passing to the limit as r→0+ leads to
a contradiction, owing to assumption (3.4). Equation (4.26) is thus established.
Since the functions
r↦essinfBr(x0)u and r↦esssupBr(x0)u
are monotone in r,
they admit limits as r→0+,
which, by Theorem 3.1, are finite and, owing to
(4.26),
agree in Ω∖E.
In particular, the representative u^, defined by (4.19), satisfies the equality
(4.27)u^(x)=limr→0+essinfBr(x)u=limr→0+esssupBr(x)u for every x∈Ω∖E.
It is easily verified that the function
x↦limr→0+essinfBr(x)u
is lower-semicontinuous in Ω. Hence, by (4.27), u^ is lower-semicontinuous in Ω∖E. Similarly, the function
x↦limr→0+esssupBr(x)u
is upper-semicontinuous in Ω, and, by (4.27) again, u^ is also upper-semicontinuous in Ω∖E. Altogether, we have shown that u^ is continuous in Ω∖E. The proof is complete.
∎
As mentioned in Section 3, Corollary 3.4 follows from Theorem 3.3 via the next result.
Proposition 4.3.
Let n≥2, and let A be a Young function satisfying the Δ2-condition near infinity. Then the Young function An-1, defined by (3.1), satisfies the Δ2-condition near
infinity as well.
Proof.
If n=2, then A1=A, and there is nothing to prove. Assume that n≥3. Owing to (2.9), one has that
A~∈∇2 near infinity. Thus, by (2.8), there exists
ε0>0 such that, if 0<ε<ε0, then
the function A~(t)t-1-ε is increasing for large t.
Hence, if 0<ε<min{ε0,1n-2},
then
∫t∞A~(r)r1+n-1n-2𝑑r≥A~(t)t1+ε∫tdrrn-1n-2-ε∞=A~(t)(n-1n-2-1-ε)tn-1n-2 for large t.
As a consequence,
(4.28)lim inft→∞tdAn-1~dt(t)An-1~(t)=lim inft→∞t(n-1n-2tn-1n-2-1∫t∞A~(r)r1+n-1n-2𝑑r-A~(t)t)tn-1n-2∫t∞A~(r)r1+n-1n-2𝑑r=n-1n-2-lim supt→∞A~(t)tn-1n-2∫t∞A~(r)r1+n-1n-2𝑑r≥1+ε.
By (4.28), for every ε∈(0,min{ε0,1n-2})
the function An-1~(t)t-1-ε is increasing for large t. Hence, by (2.8),
An-1~∈∇2 near infinity, whence, thanks to (2.9),
An-1∈Δ2 near infinity. ∎
The link between the generalized capacities CΨ,1 and the classical Hausdorff measures ℋh(⋅) is analyzed in [32]. It provides us with a key tool in deriving Corollary 3.6 from Theorem 3.3.
Proof of Corollary 3.6.
Let E be the set where the representative u^ of u, exhibited in the proof of Theorem 3.3, is not continuous. Being the complement in Ω of the set where the limit of the integral averages of u exists, the set E is Borel measurable. Thus, for every k∈ℕ, the set Ek=E∩Bk(0) is a bounded Borel set.
Given k∈ℕ, let f∈L+1(ℝn) be such that IΨf(x)≥1 for x∈Ek, and let μ
be any Borel measure, supported in Ek, such that μ(Ek)=1. Thus,
1=μ(Ek)≤∫ℝnIΨf(x)𝑑μ(x)=∫ℝnIΨμ(y)f(y)𝑑y≤∥IΨμ∥L∞(ℝn)∥f∥L1(ℝn).
Hence, from the very definition of CΨ,1-capacity,
CΨ,1(Ek)≥1∥IΨμ∥L∞(ℝn).
Since, by Theorem 3.3, CΨ,1(Ek)=0 for every k∈ℕ, one has that ∥IΨμ∥L∞(ℝn)=∞ for every μ as above. This piece of information, combined with assumption (3.10), implies, via [32, Theorem
2], that
ℋh(⋅)(Ek)=0
for every k∈ℕ. Next, recall that, although ℋh(⋅) is just an outer measure, it is a measure when restricted to the class of Borel sets. Thus, since Ek is an increasing sequence of Borel sets such that E=⋃k=1∞Ek,
ℋh(⋅)(E)=limk→∞ℋh(⋅)(Ek)=0.∎
It only remains to prove Theorem 3.7.
Proof of Theorem 3.7.
As in the proof of Theorem 3.1, we may suppose, without loss of generality, that u∈W1,A(Ω)∩W01,1(Ω). Assume, for the time being, that u fulfills condition (4.8) as well. It is clear from the proof of inequality (4.15) that this inequality also holds with u replaced by each approximating function uk, defined as in the proof of Theorem 3.1.
Integrating over (r,R) the inequality obtained from this replacement
yields
(4.29)∫rRtn-1An-1(|uk(x)-uk(y)|Ct)𝑑t≤∫Br(x0)A(|∇uk|)𝑑z<∞,
provided that BR(x0)⊂⊂Ω, 0<r<t<R and x,y∈Br(x0).
Assumption (1.4) is equivalent to
(4.30)∫0tn-1An-1(1t)𝑑t=∞.
As a consequence of (4.29) and (4.30), the sequence uk is equi-continuous in Ω.
Thus, by Ascoli–Arzelá’s theorem, the sequence {uk} converges
to a continuous function u¯, which agrees with u a.e. in Ω.
Denote by ω(r) the modulus of continuity of u¯ in Br(x0).
Inequality (4.29)
implies, via Fatou’s lemma and equation (4.3), that
(4.31)∫rRtn-1An-1(ω(r)Ct)𝑑t≤∫BR(x0)A(|∇u|)𝑑z.
From an application of inequality (4.31) with r=R2, and property (2.3) applied with A replaced by An-1, we deduce that, if ω(R2)≥1, then
ω(R2)∫R2Rtn-1An-1(1Ct)𝑑t≤∫R2Rtn-1An-1(ω(R2)Ct)𝑑t≤∫BR(x0)A(|∇u|)𝑑z.
Hence,
ω(R2)≤max{1,∫BR(x0)A(|∇u|)𝑑z∫R2Rtn-1An-1(1Ct)𝑑t}.
Thus, if 0<r≤R2,
(4.32)∫R∞tn-1An-1(ω(r)Ct)𝑑t=(ω(r)C)n∫0ω(r)RCAn-1(s)sn+1𝑑s≤(ω(R2)C)n∫0ω(R2)RCAn-1(s)sn+1𝑑s<∞,
where the equality holds by a change of variable in the integral, and the last inequality by equation (3.13).
Combining inequalities (4.31) and (4.32) tells us that
(4.33)∫r∞tn-1An-1(ω(r)Ct)𝑑t≤∫BR(x0)A(|∇u|)𝑑z+(ω(R2)C)n∫0ω(R2)RCAn-1(s)sn+1𝑑s if 0<r≤R2.
Denote by C′=C′(x0,R,u,A,n) the quantity on the right-hand side of (4.33), and set C′′=max{C,C′}, where C is the constant appearing in (4.33). Thus,
(ω(r)C′′)n∫0ω(r)rC′′An-1(s)sn+1𝑑s=∫r∞tn-1An-1(ω(r)C′′t)𝑑t≤C′′ if 0<r≤R2,
or, equivalently,
B(ω(r)rC′′)≤C′′rn if 0<r≤R2,
where B is given by (3.12). Hence,
ω(r)≤C′′rB-1(C′′r-n)≤C′′max{C′′,1}rB-1(r-n) if 0<r≤R2,
where the last inequality holds by property (2.4) applied to the Young function B. As a consequence, u∈𝒞locω(⋅)(Ω).
If assumption (4.8) is dropped, the same argument, applied with
u replaced by λu for a suitable constant λ>0, yields the same conclusion.
∎
5 Examples
5.1 A customary example
Estimates in Lloc∞(Ω) for weakly monotone Orlicz–Sobolev functions in the spirit of Theorem 3.1, as well as information on the capacity of the exceptional set and on its Hausdorff measure, such as that provided in Theorems 3.3 and 3.7, seem to be missing in literature.
In Example 5.1 below, we illustrate these results in
a model instance of Orlicz–Sobolev spaces Wloc1Lplogα(Ω) built upon Young functions of power-logarithmic type. Let us emphasize that, as already pointed out in Remark 3.5, even the classical result (1.1) on the capacity of the singular set of weakly monotone functions is refined by Theorem 3.3, when the latter is specialized to standard Sobolev spaces Wloc1,p(Ω) corresponding to plain power-type Young functions.
For those exponents p and α that guarantee the continuity everywhere of weakly monotone functions from Wloc1Lplogα(Ω), we recover in Example 5.1 the results of [14, Chapter 6]. We also compare the modulus of continuity of a weakly monotone function in Wloc1Lplogα(Ω) given by Theorem 3.7 with that of an arbitrary function from the same Orlicz–Sobolev space, and derive some interesting conclusions.
Example 5.1.
Assume that A(t)=tplogα(c+t), where p>1, α∈ℝ, and c is a positive constant so large that A is a Young function. Let us denote by Wloc1Lplogα(Ω) the local Orlicz–Sobolev space associated with A.
From Corollaries 3.9 and 3.10 one deduces that every weakly monotone function u∈Wloc1Lplogα(Ω) has a representative that is everywhere continuous in Ω in any of the following cases:
(5.1){p>n and α∈ℝ,p=n and α≥-1.
Furthermore, from Theorem 3.7 one infers what follows.
If p>n and α∈ℝ, then
(5.2)u∈𝒞locr1-nplog-αp(1r)(Ω).
If p=n and α>-1, then
(5.3)u∈𝒞loclog-α+1n(1r)(Ω).
If p=n and α=-1, then
(5.4)u∈𝒞loc(loglog)-1n(1r)(Ω).
Equations (5.1)–(5.4) recover results from [14, Chapter 6].
Let us notice that, if p>n and α∈ℝ, then the modulus of continuity of a weakly monotone function u given by (5.2) coincides with that of any function u∈Wloc1Lplogα(Ω) – see [6] or [9, equation (6.13)] – and, in particular, with that given by the classical embedding theorem by Morrey, if α=0. Thus, being a weakly monotone does not provide a function with a better modulus of continuity in these cases.
By contrast, if p=n and α>n-1, then any function u∈Wloc1Lnlogα(Ω) is still continuous, but one has just that
u∈𝒞loclog-α-n+1n(1r)(Ω).
This is of course a weaker property than (5.3). Weak monotonicity thus turns out to improve the quality of the modulus of continuity of Orlicz–Sobolev functions in borderline situations, a phenomenon that cannot be appreciated in the less fine scale of the standard Sobolev spaces.
Assume next that (5.1) fails, but any of the following conditions holds:
{p=n and α<-1, n-1<p<n and α∈ℝ,either n≥3, p=n-1 and α>n-2, or n=2, p=1 and α≥0 .
Then, by Theorem 3.1, any weakly monotone function u∈Wloc1Lplogα(Ω) is locally bounded, and differentiable a.e. in Ω.
Moreover, on denoting by E the exceptional set of discontinuity points of u, from an application of Corollary 3.3 and Theorem 3.7 one can derive the following conclusions.
If p=n and α<-1, then
𝒞tnlogα+1t,1(E)=0 and ℋlog-γ(1s)(E)=0 for every γ>-α.
If either n-1<p<n and α∈ℝ, or n=2, p=1 and α≥0, then
𝒞tplogα+1t,1(E)=0 and ℋsn-plog-γ(1s)(E)=0 for every γ>1-α.
If n≥3, p=n-1 and α>n-2, then
𝒞tn-1logα+3-nt,1(E)=0 and ℋsn-plog-γ(1s)(E)=0 for every γ>n-2-α.
5.2 Augmenting the existing literature: A non-standard example
Since condition (1.3) implies (3.16), Corollary 3.10 recovers the results of [14] and [16]. The objective of Proposition 5.2 below is to demonstrate that assumption (1.4) of Theorem 3.7 is actually weaker than the pair of assumptions (1.2)–(1.3).
This shows that Theorem 3.7 can be applied to deduce the continuity of weakly monotone Orlicz–Sobolev functions in certain situations where the criterion of [14] and [16] fails.
Proposition 5.2.
Let n≥2. Then there exist Young functions A for which condition (1.4) holds, whereas (1.3) fails.
Let us point out that the Young functions that will be exhibited in the proof of Proposition 5.2 satisfy, in addition, the Δ2-condition. Thus, they fulfill condition (1.4) in the equivalent form (3.15) appearing in Corollary 3.9.
Proof of Proposition 5.2.
Let A be a piecewise affine Young function of the form (2.1). Thus, there exists an increasing sequence {tk} of nonnegative numbers tk, with t0=0 and
(5.5)limk→∞tk=∞,
and an increasing sequence
{mk} of nonnegative numbers mk such that
a(t)=mk for t∈(tk,tk+1),
for k∈ℕ.
Hence,
(5.6)A(t)=∑h=0k-1mh(th+1-th)+mk(t-tk) if t∈[tk,tk+1),
for k∈ℕ.
The conclusion will follow if we show that, given any q>1, the sequences {tk} and {mk} can be chosen in such a way that the function A satisfies condition (1.4), and the function
(5.7)t↦A(t)tq is not increasing.
Condition (5.7) holds if there exists α∈(1,q) such that
(5.8)a(tk-)tk-αA(tk)=0
for k∈ℕ, where we have set a(tk-)=limt→tk-a(t).
The following formulas can be derived
from equations
(5.6) and (5.8), via an induction argument:
(5.9)tk+1=αt1(m1-m0)(αm2-m1)⋯(αmk-1-mk-2)(αmk-mk-1)(α-1)km1m2⋯mk
=αt1(α-1)k(1-m0m1)(α-m1m2)⋯(α-mk-2mk-1)(α-mk-1mk)
and
(5.10)tk+1-tk=αt1(m1-m0)(αm2-m1)⋯(αmk-1-mk-2)(mk-mk-1)(α-1)km1m2⋯mk
=αt1(α-1)k(1-m0m1)(α-m1m2)⋯(α-mk-2mk-1)(1-mk-1mk)
for k∈ℕ.
Let us define
the sequence {mk} in such a way that
(5.11)βmk=αmk-mk-1
for k∈ℕ, with β∈(α-1,α) to be fixed later. Hence,
(5.12)mk=m0(α-β)k,
and
mk-mk-1=(β-α+1)m0(α-β)k
for k∈ℕ.
By (5.9),
(5.13)tk+1=αt1(m1-m0)βk-1m1(α-1)k
for k∈ℕ, whence (5.5) holds. Furthermore, by (5.10),
(5.14)tk+1-tk=αt1(m1-m0)(β-α+1)βk-2m1(α-1)k
for k∈ℕ.
Assume first that n=2. We claim that choosing α∈(1,min{q,2}) and β=1 in (5.11) yields a function A fulfilling condition (1.4), which, in this case, reads
(5.15)∫t1∞A(t)t3𝑑t=∞.
To verify our claim, observe that, owing to property (2.2), equation (5.15) is equivalent to
(5.16)∫t1∞a(t)t2𝑑t=∞.
On the other hand, by (5.12), (5.13) and (5.14), with β=1,
∫t1∞a(t)t2𝑑t=∑k=1∞∫tktk+1mkt2𝑑t=m0m1t1(m1-m0)2-αα-1∑k=1∞1=∞,
whence (5.16) follows.
Assume next that n≥3. Let us preliminarily observe that any function A defined by (5.6), with mk obeying (5.12), satisfies the Δ2-condition. Owing to the equivalence of equations (2.6) and (2.7), in order to verity this assertion it suffices to show that there exists a constant c such that
(5.17)a(2t)≤ca(t) for t≥0.
Given t>0, let k∈ℕ be the index satisfying
t∈[tk,tk+1),
whence 2t∈[2tk,2tk+1). If we prove that there exists a constant c such that
(5.18)a(2tk+1)≤ca(tk)
for k∈ℕ,
then inequality (5.17) will follow, inasmuch as
a(2t)≤a(2tk+1)≤ca(tk)≤ca(t) for t>0.
Given k∈ℕ, denote by j=j(k) the index fulfilling
(5.19)2tk+1∈[tj,tj+1).
Thanks to equation (5.13), condition (5.19) is equivalent to the inequalities
(5.20)αt1(m1-m0)m1(α-1)(βα-1)j-2≤2αt1(m1-m0)m1(α-1)(βα-1)k-1<αt1(m1-m0)m1(α-1)(βα-1)j-1.
On setting b=βα-1, equation (5.20) is in turn equivalent to
j-2≤logb2+k-1≤j-1,
whence
j≤1+logb2+k≤j+1≤[γ]+k+2,
where γ=1+logb2, and [γ] stands for the integer part of γ. Therefore
a(2tk+1)≤a(tj+1)≤a(t[γ]+k+2)=m[γ]+k+2=m0(α-β)[γ]+k+2=mk(α-β)[γ]+2=a(tk)(α-β)[γ]+2
for k∈ℕ. We conclude that inequality (5.18), and hence (5.17), holds with c=1(α-β)[γ]+2.
Since A satisfies the Δ2-condition, by [5, Lemma 3.3] condition (1.4) is equivalent to (3.15). By property (2.2), the latter is in turn equivalent to
(5.21)∫∞1a(t)2n-2(∫t∞1a(s)1n-2𝑑s)-n𝑑t=∞.
We shall show that (5.21) holds, provided that β is chosen in such a way that α-β is sufficiently small. Indeed, if k∈ℕ and t∈[tk,tk+1), then, by (5.12) and (5.14),
(5.22)∫t∞dsa(s)1n-2≤∫tk∞dsa(s)1n-2=∑h=k∞∫thth+1dsa(s)1n-2=∑h=k∞th+1-thmh1n-2=C∑h=k∞[β(α-β)1n-2α-1]h=C′[β(α-β)1n-2α-1]k
for suitable constants C and C′, provided that α-β is sufficiently small.
From inequality (5.22) and equations (5.12) and (5.14) one then infers that
∫t1∞1a(t)2n-2(∫t∞1a(s)1n-2𝑑s)-ndt=∑k=1∞∫tktk+11a(t)2n-2(∫t∞1a(s)1n-2𝑑s)-n𝑑t≥C∑k=1∞tk+1-tkmk2n-2[β(α-β)1n-2α-1]-nk=C′∑k=1∞[(α-1β)n-11α-β]k
for suitable positive constants C and C′. Since the last series diverges if α-β is small enough, equation (5.21) follows.
∎
References
[1]
D. R. Adams and L. I. Hedberg,
Function Spaces and Potential Theory,
Springer, Berlin, 1996.
10.1007/978-3-662-03282-4Search in Google Scholar
[2]
A. Alberico and A. Cianchi,
Differentiability properties of Orlicz–Sobolev functions,
Ark. Math. 43 (2005), 1–28.
10.1007/BF02383608Search in Google Scholar
[3]
A. Alvino, P.-L. Lions and G. Trombetti,
On optimization problems with prescribed rearrangements,
Nonlinear Anal. 13 (1989), 185–220.
10.1016/0362-546X(89)90043-6Search in Google Scholar
[4]
J. Ball,
Convexity conditions and existence theorems in nonlinear elasticity,
Arch. Ration. Mech. Anal. 63 (1977), 337–403.
10.1007/BF00279992Search in Google Scholar
[5]
M. Carozza and A. Cianchi,
Smooth approximation of Orlicz–Sobolev maps between Riemannian manifolds,
Potential Anal. 45 (2016), 557–578.
10.1007/s11118-016-9558-xSearch in Google Scholar
[6]
A. Cianchi,
Continuity properties of functions from Orlicz–Sobolev spaces and embedding theorems,
Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 23 (1996), 575–608.
Search in Google Scholar
[7]
A. Cianchi,
Optimal Orlicz–Sobolev embeddings,
Rev. Mat. Iberoam. 20 (2004), 427–474.
10.4171/RMI/396Search in Google Scholar
[8]
A. Cianchi and V. Musil,
Optimal domain spaces in Orlicz–Sobolev embeddings,
Indiana Univ. Math. J., to appear.
10.1512/iumj.2019.68.7649Search in Google Scholar
[9]
A. Cianchi and M. Randolfi,
On the modulus of continuity of weakly differentiable functions,
Indiana Univ. Math. J. 60 (2011), 1939–1973.
10.1512/iumj.2011.60.4441Search in Google Scholar
[10]
T. Futamura and Y. Mizuta,
Continuity of weakly monotone Sobolev functions,
Adv. Math. Sci. Appl. 15 (2005), 571–585.
Search in Google Scholar
[11]
V. M. Gol’dstein and S. K. Vodop’yanov,
Quasiconformal mappings and spaces of functions with generalized first derivatives,
Sibirsk. Mat. Zh. 17 (1976), 515–531.
Search in Google Scholar
[12]
J. Heinonen, T. Kilpelainen and O. Martio,
Nonlinear Potential Theory of Degenerate Elliptic Equations,
Oxford University Press, Oxford, 1993.
Search in Google Scholar
[13]
T. Iwaniec, P. Koskela and J. Onninen,
Mappings of finite distortion: Monotonicity and continuity,
Invent. Math. 144 (2001), 507–531.
10.1007/s002220100130Search in Google Scholar
[14]
T. Iwaniec and G. Martin,
Geometric Function Theory and Nonlinear Analysis,
Clarendon Press, Oxford, 2001.
10.1093/oso/9780198509295.001.0001Search in Google Scholar
[15]
J. Kauhanen, P. Koskela and J. Maly,
On functions with derivatives in a Lorentz space,
Manuscripta Math. 100 (1999), 87–101.
10.1007/s002290050197Search in Google Scholar
[16]
J. Kauhanen, P. Koskela, J. Maly, J. Onninen and X. Zhong,
Mappings of finite distortion: Sharp Orlicz-conditions,
Rev. Mat. Iberoam. 19 (2003), 857–872.
10.4171/RMI/372Search in Google Scholar
[17]
P. Koskela, J. J. Manfredi and E. Villamor,
Regularity theory and traces of A-harmonic functions,
Trans. Amer. Math. Soc. 348 (1996), 755–766.
10.1090/S0002-9947-96-01430-4Search in Google Scholar
[18]
M. A. Krasnosel’skii and J. B. Rutickii,
Convex Functions and Orlicz Spaces,
P. Noordhoff, Groningen, 1961.
Search in Google Scholar
[19]
H. Lebesgue,
Sur le probleme de Dirichlet,
Rend. Circ. Mat. Palermo 24 (1907), 371–402.
10.1007/BF03015070Search in Google Scholar
[20]
J. J. Manfredi,
Weakly monotone functions,
J. Geom. Anal. 4 (1994), 393–402.
10.1007/BF02921588Search in Google Scholar
[21]
J. J. Manfredi and E. Villamor,
Traces of monotone Sobolev functions,
J. Geom. Anal. 6 (1996), 433–434.
10.1007/BF02921659Search in Google Scholar
[22]
J. J. Manfredi and E. Villamor,
Traces of monotone Sobolev functions in weighted Sobolev spaces,
Illinois J. Math. 45 (2001), 403–422.
10.1215/ijm/1258138347Search in Google Scholar
[23]
V. G. Maz’ya,
Sobolev Spaces with Applications to Partial Differential Equations,
Springer, Berlin, 2011.
10.1007/978-3-642-15564-2Search in Google Scholar
[24]
R. O’Neil,
Convolution operators in L(p,q) spaces,
Duke Math. J. 30 (1963), 129–142.
10.1215/S0012-7094-63-03015-1Search in Google Scholar
[25]
J. Onninen,
Differentiability of monotone Sobolev functions,
Real Anal. Exchange 26 (2000), 761–772.
10.2307/44154076Search in Google Scholar
[26]
M. M. Rao and Z. D. Ren,
Theory of Orlicz Spaces,
Marcel Dekker, New York, 1991.
Search in Google Scholar
[27]
M. M. Rao and Z. D. Ren,
Applications of Orlicz Spaces,
Marcel Dekker, New York, 2002.
10.1201/9780203910863Search in Google Scholar
[28]
V. Stepanoff,
Sur les conditions de la differentielle totale,
Mat. Sb. 32 (1925), 511–526.
Search in Google Scholar
[29]
J.-O. Strömberg,
Bounded mean oscillation with Orlicz norms and duality of Hardy spaces,
Indiana Univ. Math. J. 28 (1979), 511–544.
10.1090/S0002-9904-1976-14231-0Search in Google Scholar
[30]
V. Sverak,
Regularity properties of deformations with finite energy,
Arch. Ration. Mech. Anal. 100 (1988), 105–127.
10.1007/BF00282200Search in Google Scholar
[31]
G. Talenti,
An embedding theorem,
Partial Differential Equations and the Calculus of Variations. Essays in Honor of Ennio De Giorgi,
Birkhäuser, Boston (1989), 919–924.
10.1007/978-1-4615-9831-2_18Search in Google Scholar
[32]
S. J. Taylor,
On the connexion between Hausdorff measures and generalized capacity,
Proc. Cambridge Philos. Soc. 57 (1961), 524–531.
10.1017/S0305004100035581Search in Google Scholar
