Ahlfors Theorems for Differential Forms

Abstract

Some counterparts of theorems of Phragmén-Lindelöf and of Ahlfors are proved for differential forms of 𝒲𝒯-classes.

1. 𝒲𝒯-Forms

This paper is continuation of the earlier work [1], where the main topic was to examine the connection between quasiregular (qr) mappings and so-called 𝒲𝒯-classes of differential forms. We first recall some basic notation and terminology from [1].

Let ℳ be a Riemannian manifold of class 𝐶³, dimℳ=𝑛, with or without boundary, and let 𝑤∈𝐿^𝑝_loc(ℳ),deg𝑤=𝑘,0≤𝑘≤𝑛,𝑝>1, (1.1) be a weakly closed differential form on ℳ, that is, for each form 𝜑∈𝑊¹_𝑞,loc(ℳ)1,deg𝜑=𝑘+1,𝑝+1𝑞=1, (1.2) with a compact supp𝜑 in ℳ and such that supp𝜑∩𝜕ℳ=∅; we have _ℳ𝑤,𝛿𝜑𝑑𝑣=0. (1.3) Here 𝛿𝜑=(−1)^𝑘⋆⁻¹𝑑⋆𝜑, 𝑘=deg𝜑, and ⋆𝛼 is the orthogonal complement of a differential form 𝛼 on a Riemannian manifold ℳ.

A weakly closed form 𝑤 of the kind (1.1) is said to be of the class 𝒲𝒯₁ on ℳ if there exists a weakly closed differential form 𝜃∈𝐿^𝑞_loc(ℳ)1,deg𝜃=𝑛−𝑘,𝑝+1𝑞=1, (1.4) such that almost everywhere on ℳ we have 𝜈₀|𝜃|^𝑞≤⟨𝑤,∗𝜃⟩ (1.5) for some constant 𝜈₀>0.

The differential form (1.1) is said to be of the class 𝒲𝒯₂ on ℳ if there exists a differential form (1.4) such that almost everywhere on ℳ𝜈₁|𝑤|^𝑝≤⟨⟩,|𝜃|𝑤,∗𝜃≤𝜈₂|𝑤|^𝑝−1, (1.6)(1.7) for some constants 𝜈₁,𝜈₂>0.

Theorem 1.1. 𝒲𝒯₂⊂𝒲𝒯₁

For a proof see [1].

The following partial integration formula for differential forms is useful [1].

Lemma 1.2. Let 𝛼∈𝑊¹_𝑝,loc(ℳ) and 𝛽∈𝑊¹_𝑞(ℳ) be differential forms, deg𝛼+deg𝛽=𝑛−1, 1/𝑝+1/𝑞=1, 1≤𝑝, 𝑞≤∞, and let 𝛽 have a compact support supp𝛽⊂ℳ. Then 𝑑𝛼∧𝛽=(_ℳ−1)^deg𝛼+1_ℳ𝛼∧𝑑𝛽. (1.8) In particular, the form 𝛼 is weakly closed if and only if 𝑑𝛼=0 a.e. on ℳ.

Let 𝒜 and ℬ be Riemannian manifolds of dimensions dim𝒜=𝑘, dimℬ=𝑛−𝑘, 1≤𝑘<𝑛, and with scalar products ⟨⋅,⋅⟩_𝐴, ⟨⋅,⋅⟩_𝐵, respectively. The Cartesian product 𝒩=𝒜×ℬ has the natural structure of a Riemannian manifold with the scalar product ⟨⋅,⋅⟩=⟨⋅,⋅⟩_𝒜+⟨⋅,⋅⟩_ℬ. (1.9) We denote by 𝜋∶𝒜×ℬ→𝒜 and 𝜂∶𝒜×ℬ→ℬ the natural projections of the manifold 𝒩 onto submanifolds.

If 𝑤_𝒜 and 𝑤_ℬ are volume forms on 𝒜 and ℬ, respectively, then the differential form 𝑤_𝒩=𝜋^∗𝑤_𝒜∧𝜂^∗𝑤_ℬ is a volume form on 𝒩.

Let 𝑦₁,…,𝑦_𝑘 be an orthonormal system of coordinates in 𝐑^𝑘, 1≤𝑘≤𝑛. Let 𝒜 be a domain in 𝐑^𝑘, and let ℬ be an (𝑛−𝑘)-dimensional Riemannian manifold. We consider the manifold 𝒩=𝒜×ℬ.

2. Boundary Sets

Below we introduce the notions of parabolic and hyperbolic type of boundary sets on noncompact Riemannian manifolds and study exhaustion functions of such sets. We also present some illuminating examples.

Let ℳ be an 𝑛-dimensional noncompact Riemannian manifold without boundary. Boundary sets on ℳ are analogies to prime ends due to Carathéodory (cf. e.g., [2]).

Let {𝒰_𝑘}, 𝑘=1,2,… be a collection of open sets 𝒰_𝑘⊂ℳ with the following properties:(i)for all 𝑘=1,2,…,  𝒰_𝑘+1⊂𝒰_𝑘,(ii)⋂^∞_𝑘=1𝒰_𝑘=∅.

A sequence with these properties will be called a chain on the manifold ℳ.

Let {𝒰^′_𝑘}, {𝒰_𝑘^′′} be two chains of open sets on 𝑀. We will say that the chain 𝒰^′_𝑘 is contained in the chain {𝒰_𝑘^′′}, if for each 𝑚≥1 there exists a number 𝑘(𝑚) such that for all 𝑘>𝑘(𝑚) we have 𝒰^′_𝑘⊂𝒰_𝑚^′′. Two chains, each of which is contained in the other one, are called equivalent. Each equivalence class 𝜉 of chains is called a boundary set of the manifold ℳ. To define 𝜉 it is enough to determine at least one representative in the equivalence class. If the boundary set 𝜉 is defined by the chain {𝒰_𝑘}, then we will write 𝜉≍{𝒰_𝑘}.

A sequence of points 𝑚_𝑘∈ℳ converges to 𝜉 if for some (and, therefore, all) chain {𝒰_𝑘}∈𝜉 the following condition is satisfied: for every 𝑘=1,2,… there exists an integer 𝑛(𝑘) such that 𝑚_𝑛∈𝒰_𝑘 for all 𝑛>𝑛(𝑘). A sequence (𝑚_𝑛) lies off a boundary set 𝜉≍{𝒰_𝑘}, if for every 𝑘=1,2,… there exists a number 𝑛(𝑘) such that for all 𝑛>𝑛(𝑘)𝑚_𝑛∉𝒰_𝑘.

A boundary set 𝜉≍{𝒰_𝑘} is called a set of ends of the manifold ℳ if each of {𝒰_𝑘} has a compact boundary 𝜕𝒰_𝑘. If in addition each of the sets 𝑈_𝑘 is connected, then 𝜉≍{𝑈_𝑘} is called an end of the manifold ℳ.

2.1. Types of Boundary Set

Let 𝐷 be an open set on ℳ and let 𝐴,𝐵⊂𝐷 be closed subsets in 𝐷 such that 𝐴∩𝐵=∅. Each triple (𝐴,𝐵;𝐷) is called a condenser on ℳ.

We fix 𝑝≥1. The 𝑝-capacity of the condenser (𝐴,𝐵;𝐷) is defined by cap_𝑝(𝐴,𝐵;𝐷)=inf_𝐷∇𝜑^𝑝∗𝟙_ℳ, (2.1) where the infimum is taken over the set of all continuous functions of class 𝑊¹_𝑝,loc(𝐷) such that 𝜑(𝑚)|_𝐴=0, 𝜑(𝑚)|_𝐵=1. It is easy to see that for a pair (𝐴,𝐵;𝐷) and (𝐴₁,𝐵₁;𝐷) with 𝐴₁⊂𝐴, 𝐵₁⊂𝐵 we have cap_𝑝𝐴₁,𝐵₁;𝐷≤cap_𝑝(𝐴,𝐵;𝐷). (2.2)

Let 𝐴, 𝐵 be compact in 𝐷. A standard approximation method shows that cap_𝑝(𝐴,𝐵;𝐷) does not change if one restricts the class of functions in the variational problem (2.1) to Lipschitz functions equal to 0 and 1 in the sets 𝐴 and 𝐵, respectively.

Let {𝒰_𝑘} be an arbitrary chain on a manifold ℳ. We fix a subdomain 𝐻⊂⊂ℳ. If 𝑘 is sufficiently large, the intersection 𝐻∩𝒰_𝑘=∅ and we consider the condenser (𝐻,𝒰_𝑘;ℳ). Then it is clear that for 𝑘=1,2,…cap_𝑝𝐻,𝒰_𝑘;ℳ≥cap_𝑝𝐻,𝒰_𝑘+1;ℳ. (2.3) We will say that the chain {𝒰_𝑘} on ℳ has 𝑝-capacity zero, if for every subdomain 𝐻⊂⊂ℳ we have lim_𝑘→∞cap_𝑝𝐻,𝒰_𝑘;ℳ=0. (2.4)

We will say that a boundary set 𝜉 is of 𝑝-parabolic type if every chain {𝒰_𝑘}≍𝜉 is of 𝑝-capacity zero. A boundary set 𝜉 is of 𝛼-hyperbolic type if at least one of the chains {𝒰_𝑘}∈𝜉 is not of 𝑝-parabolic type.

Let 𝒰_𝑘^∞_𝑘=1,𝒰_𝑘⊂𝒰_𝑘+1,^∞_𝑘=1𝒰_𝑘=ℳ (2.5) be an arbitrary exhaustion of the manifold ℳ by subdomains {𝒰_𝑘}. The manifold ℳ is of 𝑝-parabolic or 𝑝-hyperbolic type depending on the 𝑝-parabolicity or 𝑝-hyperbolicity of the boundary set {ℳ⧵𝒰_𝑘}.

It is well-known, see [3], that a noncompact Riemannian manifold ℳ without boundary is of 𝑝-parabolic type if and only if every solution of the inequality div_ℳ|∇𝑢|^𝑝−2∇𝑢≥0 (2.6) which is bounded from above is a constant.

The classical parabolicity and hyperbolicity coincides with 2-parabolicity and 2-hyperbolicity, respectively. Therefore whenever we refer to parabolic or hyperbolic type (of a manifold or a boundary set) we mean 2-parabolicity or 2-hyperbolicity.

Example 2.1. The space 𝐑^𝑛 is of 𝑝-parabolic type for 𝑝≥𝑛 and 𝑝-hyperbolic type for 𝑝<𝑛.

We next present a proposition that provides a convenient method of verifying the 𝑝-parabolicity and 𝑝-hyperbolicity of boundary sets.

Lemma 2.2 (see [4]). Let 𝜉 be a boundary set on ℳ. If for a chain {𝒰_𝑘≍𝜉} and for a nonempty open set 𝐻₀⊂⊂ℳ the condition (2.4) holds, then the boundary set 𝜉 is of 𝑝-parabolic type.

2.2. 𝐴-Solutions

Let ℳ be a Riemannian manifold and let 𝐴∶𝑇(ℳ)⟶𝑇(ℳ) (2.7) be a mapping defined a.e. on the tangent bundle 𝑇(ℳ). Suppose that for a.e. 𝑚∈ℳ the mapping 𝐴 is continuous on the fiber 𝑇_𝑚, that is, for a.e. 𝑚∈ℳ the function 𝐴(𝑚,⋅)∶𝜉∈𝑇_𝑚→𝑇_𝑚 is defined and continuous; the mapping 𝑚→𝐴_𝑚(𝑋) is measurable for all measurable vector fields 𝑋 (see [5]).

Suppose that for a.e. 𝑚∈ℳ and for all 𝜉∈𝑇_𝑚 the inequalities 𝜈₁𝜉^𝑝≤(,𝐴(𝜉,𝐴𝑚,𝜉)𝑚,𝜉)≤𝜈₂𝜉^𝑝−1 (2.8) hold with 𝑝>1 and for some constants 𝜈₁,𝜈₂>0. It is clear that we have 𝜈₁≤𝜈₂.

We consider the equation div𝐴𝑚,∇𝑓=0. (2.9) Solutions to (2.9) are understood in the weak sense, that is, 𝐴-solutions are 𝑊¹_𝑝,loc-functions satisfying the integral identity _ℳ∇𝜃,𝐴𝑚,∇𝑓∗𝟙_ℳ=0 (2.10) for all 𝜃∈𝑊¹_𝑝(ℳ) with a compact support supp𝜃⊂ℳ.

A function 𝑓 in 𝑊¹_𝑝,loc(ℳ) is an 𝐴-subsolution of (2.9) in ℳ if div𝐴𝑚,∇𝑓≥0 (2.11) weakly in ℳ, that is, _ℳ∇𝜃,𝐴𝑚,∇𝑓∗𝟙_ℳ≤0 (2.12) whenever 𝜃∈𝑊¹_𝑝(ℳ), is nonnegative with a compact support in ℳ.

A basic example of such an equation is the 𝑝-Laplace equation div∇𝑓^𝑝−2∇𝑓=0. (2.13)

3. Exhaustion Functions

Below we introduce exhaustion and special exhaustion functions on Riemannian manifolds and give illustrating examples.

3.1. Exhaustion Functions of Boundary Sets

Let ℎ∶ℳ→(0,ℎ₀), 0<ℎ₀≤∞, be a locally Lipschitz function. For arbitrary 𝑡∈(0,ℎ₀) we denote by 𝐵_ℎ(𝑡)={𝑚∈ℳ∶ℎ(𝑚)<𝑡},Σ_ℎ(𝑡)={𝑚∈ℳ∶ℎ(𝑚)=𝑡} (3.1) the ℎ-balls and ℎ-spheres, respectively.

Let ℎ∶ℳ→𝐑 be a locally Lipschitz function such that: there exists a compact 𝐾⊂ℳ such that |∇ℎ(𝑥)|>0 for a.e. 𝑚∈ℳ⧵𝐾. We say that the function ℎ is an exhaustion function for a boundary set Ξ of ℳ if for an arbitrary sequence of points 𝑚_𝑘∈ℳ, 𝑘=1,2,… the function ℎ(𝑚_𝑘)→ℎ₀ if and only if 𝑚_𝑘→𝜉.

It is easy to see that this requirement is satisfied if and only if for an arbitrary increasing sequence 𝑡₁<𝑡₂<⋯<ℎ₀ the sequence of the open sets 𝑉_𝑘={𝑚∈ℳ∶ℎ(𝑚)>𝑡_𝑘} is a chain, defining a boundary set 𝜉. Thus the function ℎ exhausts the boundary set 𝜉 in the traditional sense of the word.

The function ℎ∶ℳ→(0,ℎ₀) is called the exhaustion function of the manifold ℳ if the following two conditions are satisfied(i)for all 𝑡∈(0,ℎ₀) the ℎ-ball 𝐵_ℎ(𝑡) is compact;(ii)for every sequence 𝑡₁<𝑡₂<⋯<ℎ₀ with lim_𝑘→∞𝑡_𝑘=ℎ₀, the sequence of ℎ-balls {𝐵_ℎ(𝑡_𝑘)} generates an exhaustion of ℳ, that is, 𝐵_ℎ𝑡₁⊂𝐵_ℎ𝑡₂⊂⋯⊂𝐵_ℎ𝑡_𝑘⊂⋯,_𝑘𝐵_ℎ𝑡_𝑘=ℳ. (3.2)

Example 3.1. Let ℳ be a Riemannian manifold. We set ℎ(𝑚)=dist(𝑚,𝑚₀) where 𝑚₀∈ℳ is a fixed point. Because |∇ℎ(𝑚)|=1 almost everywhere on ℳ, the function ℎ defines an exhaustion function of the manifold ℳ.

3.2. Special Exhaustion Functions

Let ℳ be a noncompact Riemannian manifold with the boundary 𝜕ℳ (possibly empty). Let 𝐴 satisfy (2.8) and let ℎ∶ℳ→(0,ℎ₀) be an exhaustion function, satisfying the following conditions: (a₁)there is ℎ^′>0 such that ℎ⁻¹([0,ℎ^′]) is compact and ℎ is a solution of (2.9) in the open set ℎ⁻¹((ℎ^′,ℎ₀));(a₂)for a.e. 𝑡₁,𝑡₂∈(ℎ^′,ℎ₀), 𝑡₁<𝑡₂, _Σℎ(𝑡2)∇ℎ||(∇ℎ,𝐴𝑥,∇ℎ)𝑑ℋ^𝑛−1=_Σℎ𝑡1∇ℎ||(∇ℎ,𝐴𝑥,∇ℎ)𝑑ℋ^𝑛−1. (3.3)

Here 𝑑ℋ^𝑛−1 is the element of the (𝑛−1)-dimensional Hausdorff measure on Σ_ℎ. Exhaustion functions with these properties will be called the special exhaustion functions of ℳ with respect to 𝐴. In most cases the mapping 𝐴 will be the 𝑝-Laplace operator (2.13).

Since the unit vector 𝜈=∇ℎ/|∇ℎ| is orthogonal to the ℎ-sphere Σ_ℎ, the condition (a₂) means that the flux of the vector field 𝐴(𝑚,∇ℎ) through ℎ-spheres Σ_ℎ(𝑡) is constant.

Suppose that the function 𝐴(𝑚,𝜉) is continuously differentiable. If (b₁)ℎ∈𝐶²(ℳ⧵𝐾) and satisfies (2.9), and (b₂)at every point 𝑚∈ℳ where 𝜕ℳ has a tangent plane 𝑇_𝑚(𝜕ℳ) the condition ⟨𝐴(𝑚,∇ℎ(𝑚)),𝜈⟩=0 (3.4)is satisfied where 𝜈 is a unit vector of the inner normal to the boundary 𝜕ℳ, then ℎ is a special exhaustion function of the manifold ℳ.

The proof of this statement is simple. Consider the open set ℳ𝑡₁,𝑡₂=𝑚∈ℳ∶𝑡₁<ℎ(𝑚)<𝑡₂,0<𝑡₁<𝑡₂<∞, (3.5) with the boundary 𝜕ℳ(𝑡₁,𝑡₂). Using the Stokes formula, we have for noncritical values 𝑡₁<𝑡₂ (for the definition of critical values of 𝐶^𝑘-functions see, e.g., [6, Part II, Chapter 2, Section 10]) _Σℎ(𝑡2)∇ℎ||(∇ℎ,𝐴𝑚,∇ℎ)𝑑ℋ^𝑛−1−_Σℎ𝑡1∇ℎ||(∇ℎ,𝐴𝑚,∇ℎ)𝑑ℋ^𝑛−1=_{𝜕ℳ(𝑡1,𝑡2)∪∪𝑖=1,2Σℎ(𝑡𝑖)}⟨𝜈,𝐴(𝑚,∇ℎ)⟩𝑑ℋ^𝑛−1=_{𝑡𝜕ℳ1,𝑡2}⟨𝜈,𝐴(𝑚,∇ℎ)⟩𝑑ℋ^𝑛−1=_{ℳ(𝑡1,𝑡2)}div𝐴(𝑚,∇ℎ)∗𝟙=0, (3.6) and (a₂) follows.

Example 3.2. We fix an integer 𝑘, 1≤𝑘≤𝑛, and set 𝑑_𝑘(𝑥)=^𝑘_𝑖=1𝑥²_𝑖^1/2. (3.7) It is easy to see that |∇𝑑_𝑘(𝑥)|=1 everywhere in 𝐑^𝑛⧵Σ₀ where Σ₀={𝑥∈𝐑^𝑛∶𝑑_𝑘(𝑥)=0}. We will call the set 𝐵_𝑘(𝑡)=𝑥∈𝐑^𝑛∶𝑑_𝑘(𝑥)<𝑡 (3.8) a 𝑘-ball and the set Σ_𝑘(𝑡)=𝑥∈𝐑^𝑛∶𝑑_𝑘(𝑥)=𝑡 (3.9) a 𝑘-sphere in 𝐑^𝑛.

We will say that an unbounded domain 𝐷⊂𝐑^𝑛 is 𝑘-admissible if for each 𝑡>inf_𝑥∈𝐷𝑑_𝑘(𝑥) the set 𝐷∩𝐵_𝑘(𝑡) has compact closure.

It is clear that every unbounded domain 𝐷⊂𝐑^𝑛 is 𝑛-admissible. In the general case the domain 𝐷 is 𝑘-admissible if and only if the function 𝑑_𝑘(𝑥) is an exhaustion function of 𝐷. It is not difficult to see that if a domain 𝐷⊂𝐑^𝑛 is 𝑘-admissible, then it is 𝑙-admissible for all 𝑘<𝑙<𝑛.

Fix 1≤𝑘<𝑛. Let Δ be a bounded domain in the (𝑛−𝑘)-plane 𝑥₁=⋯=𝑥_𝑘=0 and let 𝐷=𝑥=𝑥₁,…,𝑥_𝑘,𝑥_𝑘+1,…,𝑥_𝑛∈𝐑^𝑛∶𝑥_𝑘+1,…,𝑥_𝑛∈Δ (3.10) be a domain in 𝐑^𝑛.

The domain 𝐷 is 𝑘-admissible. The 𝑘-spheres Σ_𝑘(𝑡) are orthogonal to the boundary 𝜕𝐷 and therefore ⟨∇𝑑_𝑘,𝜈⟩=0 everywhere on the boundary. The function ℎ(𝑥)=log𝑑_𝑘(𝑥)𝑑,𝑝=𝑘,_{𝑘(𝑝−𝑛)/(𝑝−1)}(𝑥),𝑝≠𝑘, (3.11) is a special exhaustion function of the domain 𝐷. Therefore for 𝑝≥𝑘 the domain 𝐷 is of 𝑝-parabolic type and for 𝑝<𝑘  𝑝-hyperbolic type.

Example 3.3. Fix 1≤𝑘<𝑛. Let Δ be a bounded domain in the plane 𝑥₁=⋯=𝑥_𝑘=0 with a piecewise smooth boundary and let 𝐷=𝑥=𝑥₁,…,𝑥_𝑛∈𝐑^𝑛∶𝑥_𝑘+1,…,𝑥_𝑛∈Δ=𝐑^𝑛−𝑘×Δ (3.12) be the cylinder domain with base Δ.

The domain 𝐷 is 𝑘-admissible. The 𝑘-spheres Σ_𝑘(𝑡) are orthogonal to the boundary 𝜕𝐷 and therefore ⟨∇𝑑_𝑘,𝜈⟩=0 everywhere on the boundary, where 𝑑_𝑘 is as in Example 3.2.

Let ℎ=𝜙(𝑑_𝑘) where 𝜙 is a 𝐶²-function. We have ∇ℎ=𝜙^′∇𝑑_𝑘 and ^𝑛_𝑖=1𝜕𝜕𝑥_𝑖|∇ℎ|^𝑛−2𝜕ℎ𝜕𝑥_𝑖=^𝑘_𝑖=1𝜕𝜕𝑥_𝑖𝜙^′𝑛−1𝜕𝑑_𝑘𝜕𝑥_𝑖=(𝑛−1)𝜙^′𝑛−2𝜙^′′+𝑘−1𝑑_𝑘𝜙^′𝑛−1. (3.13) From the equation (𝑛−1)𝜙^′′+𝑘−1𝑑_𝑘𝜙^′=0 (3.14) we conclude that the function ℎ(𝑥)=𝑑_𝑘(𝑥)^{(𝑛−𝑘)/(𝑛−1)} (3.15)satisfies (2.13) in 𝐷⧵𝐾 and thus it is a special exhaustion function of the domain 𝐷.

Example 3.4. Let (𝑟,𝜃), where 𝑟≥0, 𝜃∈𝑆^𝑛−1(1), be the spherical coordinates in 𝑅^𝑛. Let 𝑈⊂𝑆^𝑛−1(1), 𝜕𝑈≠∅, be an arbitrary domain on the unit sphere 𝑆^𝑛−1(1). We fix 0≤𝑟₁<∞ and consider the domain 𝐷=(𝑟,𝜃)∈𝑅^𝑛∶𝑟₁<𝑟<∞,𝜃∈𝑈. (3.16) As mentioned above it is easy to verify that the given domain is 𝑛-admissible and the functionℎ𝑥|𝑥|(||)=log𝑟₁ (3.17)is a special exhaustion function of the domain 𝐷 for 𝑝=𝑛.

Example 3.5. Fix 1≤𝑛≤𝑝. Let 𝑥₁,𝑥₂,…,𝑥_𝑛 be an orthonormal system of coordinates in 𝐑^𝑛,1≤𝑛<𝑝. Let 𝐷⊂𝐑^𝑛 be an unbounded domain with piecewise smooth boundary and let ℬ be an (𝑝−𝑛)-dimensional compact Riemannian manifold with or without boundary. We consider the manifold ℳ=𝐷×ℬ.

We denote by 𝑥∈𝐷, 𝑏∈ℬ, and (𝑥,𝑏)∈ℳ the points of the corresponding manifolds. Let 𝜋∶𝐷×ℬ→𝐷 and 𝜂∶𝐷×ℬ→ℬ be the natural projections of the manifold ℳ.

Assume now that the function ℎ is a function on the domain 𝐷 satisfying the conditions (b₁), (b₂) and (2.13). We consider the function ℎ^∗=ℎ∘𝜋∶ℳ→(0,∞).

We have ∇ℎ^∗=∇(ℎ∘𝜋)=∇_𝑥ℎ∘𝜋,div∇ℎ^∗𝑝−2∇ℎ^∗=div|∇(ℎ∘𝜋)|^𝑝−2∇()∇ℎ∘𝜋=div_𝑥ℎ^𝑝−2∘𝜋∇_𝑥ℎ∘𝜋=^𝑛_𝑖=1𝜕𝜕𝑥_𝑖∇_𝑥ℎ^𝑝−2𝜕ℎ𝜕𝑥_𝑖∘𝜋. (3.18) Because ℎ is a special exhaustion function of 𝐷 we have div∇ℎ^∗𝑝−2∇ℎ^∗=0. (3.19)

Let (𝑥,𝑏)∈𝜕ℳ be an arbitrary point where the boundary 𝜕ℳ has a tangent hyperplane and let 𝜈 be a unit normal vector to 𝜕ℳ.

If 𝑥∈𝜕𝐷, then 𝜈=𝜈₁+𝜈₂ where the vector 𝜈₁∈𝐑^𝑘 is orthogonal to 𝜕𝐷 and 𝜈₂ is a vector from 𝑇_𝑏(ℬ). Thus ∇ℎ^∗,𝜈=∇_𝑥ℎ∘𝜋,𝜈₁=0, (3.20) because ℎ is a special exhaustion function on 𝐷 and satisfies the property (𝑏₂) on 𝜕𝐷. If 𝑏∈𝜕ℬ, then the vector 𝜈 is orthogonal to 𝜕ℬ×𝐑^𝑛 and ∇ℎ^∗,𝜈=∇_𝑥ℎ∘𝜋,𝜈=0, (3.21) because the vector (∇_𝑥ℎ)∘𝜋 is parallel to 𝐑^𝑛.

The other requirements for a special exhaustion function for the manifold ℳ are easy to verify.

Therefore, the function ℎ^∗=ℎ^∗(𝑥,𝑏)=ℎ∘𝜋∶ℳ⟶(0,∞) (3.22) is a special exhaustion function on the manifold ℳ=𝐷×ℬ.

Example 3.6. Let 𝒜 be a compact Riemannian manifold, dim𝒜=𝑘, with piecewise smooth boundary or without boundary. We consider the Cartesian product ℳ=𝒜×𝐑^𝑛, 𝑛≥1. We denote by 𝑎∈𝒜, 𝑥∈𝐑^𝑛 and (𝑎,𝑥)∈ℳ the points of the corresponding spaces. It is easy to see that the function ℎ()=|𝑥||𝑥|𝑎,𝑥log,𝑝=𝑛,_{(𝑝−𝑛)/(𝑝−1)},𝑝≠𝑛, (3.23) is a special exhaustion function for the manifold ℳ. Therefore, for 𝑝≥𝑛 the given manifold is of 𝑝-parabolic type and for 𝑝<𝑛𝑝-hyperbolic type.

Example 3.7. Let (𝑟,𝜃), where 𝑟≥0, 𝜃∈𝑆^𝑛−1(1), be the spherical coordinates in 𝐑^𝑛. Let 𝑈⊂𝑆^𝑛−1(1) be an arbitrary domain on the unit sphere 𝑆^𝑛−1(1). We fix 0≤𝑟₁<𝑟₂<∞ and consider the domain 𝐷=(𝑟,𝜃)∈𝐑^𝑛∶𝑟₁<𝑟<𝑟₂,𝜃∈𝑈 (3.24) with the metric 𝑑𝑠²_ℳ=𝛼²(𝑟)𝑑𝑟²+𝛽²(𝑟)𝑑𝑙²_𝜃, (3.25) where 𝛼(𝑟),𝛽(𝑟)>0 are 𝐶⁰-functions on [𝑟₁,𝑟₂) and 𝑑𝑙_𝜃 is an element of length on 𝑆^𝑛−1(1).

The manifold ℳ=(𝐷,𝑑𝑠²_ℳ) is a warped Riemannian product. In the case 𝛼(𝑟)≡1, 𝛽(𝑟)=1, and 𝑈=𝑆^𝑛−1 the manifold ℳ is isometric to a cylinder in 𝐑^𝑛+1. In the case 𝛼(𝑟)≡1, 𝛽(𝑟)=𝑟, and 𝑈=𝑆^𝑛−1 the manifold ℳ is a spherical annulus in 𝐑^𝑛.

The volume element in the metric (3.25) is given by the expression 𝑑𝜎_ℳ=𝛼(𝑟)𝛽^𝑛−1(𝑟)𝑑𝑟𝑑𝑆^𝑛−1(1). (3.26) If 𝜙(𝑟,𝜃)∈𝐶¹(𝐷), then the length of the gradient ∇𝜙 in ℳ takes the form ∇𝜙²=1𝛼²𝜙^′_𝑟²+1𝛽²∇_𝜃𝜙², (3.27) where ∇_𝜃𝜙 is the gradient in the metric of the unit sphere 𝑆^𝑛−1(1).

For the special exhaustion function ℎ(𝑟,𝜃)≡ℎ(𝑟) (2.13) reduces to the following form 𝑑1𝑑𝑟𝛼(𝑟)^𝑝−1ℎ^′_𝑟(𝑟)^𝑝−1𝛽^𝑛−1(𝑟)=0. (3.28) Solutions of this equation are the functions ℎ(𝑟)=𝐶₁^𝑟_𝑟1𝛼(𝑡)𝛽^{(𝑛−1)/(𝑝−1)}(𝑡)𝑑𝑡+𝐶₂ (3.29) where 𝐶₁ and 𝐶₂ are constants.

Because the function ℎ satisfies obviously the boundary condition (a₂) as well as the other conditions of Section 3.2, we see that under the assumption ^𝑟2𝛼(𝑡)𝛽^{(𝑛−1)/(𝑝−1)}(𝑡)𝑑𝑡=∞ (3.30) the function ℎ(𝑟)=^𝑟_𝑟1𝛼(𝑡)𝛽^{(𝑛−1)/(𝑝−1)}(𝑡)𝑑𝑡 (3.31) is a special exhaustion function on the manifold ℳ.

Theorem 3.8. Let ℎ∶ℳ→(0,ℎ₀) be a special exhaustion function of a boundary set 𝜉 of the manifold ℳ. Then(i)if ℎ₀=∞, the set 𝜉 is of 𝑝-parabolic type,(ii)if ℎ₀<∞, the set 𝜉 is of 𝑝-hyperbolic type.

Proof. Choose 0<𝑡₁<𝑡₂<ℎ₀ such that 𝐾⊂𝐵_ℎ(𝑡₁). We need to estimate the 𝑝-capacity of the condenser (𝐵_ℎ(𝑡₁),ℳ⧵𝐵_ℎ(𝑡₂);ℳ). We have cap_𝑝𝐵_ℎ𝑡₁,ℳ⧵𝐵_ℎ𝑡₂=𝐽;ℳ𝑡₂−𝑡₁^𝑝−1, (3.32) where 𝐽=_Σℎ(𝑡)|∇ℎ|^𝑝−1𝑑ℋ_ℳ^𝑛−1 (3.33) is a quantity independent of 𝑡>ℎ(𝐾)=sup{ℎ(𝑚)∶𝑚∈𝐾}. Indeed, for the variational problem (2.1) we choose the function 𝜑₀, 𝜑₀(𝑚)=0 for 𝑚∈𝐵_ℎ(𝑡₁), 𝜑₀(𝑚)=ℎ(𝑚)−𝑡₁𝑡₂−𝑡₁,𝑚∈𝐵_ℎ𝑡₂⧵𝐵_ℎ𝑡₁ (3.34) and 𝜑₀(𝑚)=1 for 𝑚∈ℳ⧵𝐵_ℎ(𝑡₂). Using the Kronrod-Federer formula [7, Theorem 3.2.22], we get cap_𝑝𝐵_ℎ𝑡₁,ℳ⧵𝐵_ℎ𝑡₂;ℳ≤_ℳ∇𝜑₀^𝑝∗𝟙_ℳ≤1𝑡₂−𝑡₁^𝑝𝑡₁^{<ℎ(𝑚)<𝑡}₂|∇ℎ(𝑚)|^𝑝∗𝟙_ℳ=^𝑡2_𝑡1𝑑𝑡_Σℎ(𝑡)|∇ℎ(𝑚)|^𝑝−1𝑑ℋ_ℳ^𝑛−1. (3.35)
Because the special exhaustion function satisfies (2.13) and the boundary condition (𝑎₂), one obtains for arbitrary 𝜏₁,𝜏₂, ℎ(𝐾)<𝜏₁<𝜏₂<ℎ_{0Σℎ(𝑡2)}|∇ℎ|^𝑝−1𝑑ℋ_ℳ^𝑛−1−_Σℎ(𝑡1)|∇ℎ|^𝑝−1𝑑ℋ_ℳ^𝑛−1=_Σℎ(𝑡2)|∇ℎ|^𝑝−2⟨∇ℎ,𝜈⟩𝑑ℋ_ℳ^𝑛−1−_Σℎ𝑡1|∇ℎ|^𝑝−2⟨∇ℎ,𝜈⟩𝑑ℋ_ℳ^𝑛−1=_{𝑡1(𝑚)<ℎ<𝑡2}div_ℳ|∇ℎ|^𝑝−2∇ℎ∗𝟙_ℳ=0. (3.36) Thus we have established the inequality cap_𝑝𝐵_ℎ𝑡₁,ℳ⧵𝐵_ℎ𝑡₂≤𝐽;ℳ𝑡₂−𝑡₁^𝑝−1. (3.37)
By the conditions, imposed on the special exhaustion function, the function 𝜑₀ is an extremal in the variational problem (2.1). Such an extremal is unique and therefore the preceding inequality holds in fact with equality. This conclusion proves (3.32).
If ℎ₀=∞, then letting 𝑡₂→∞ in (3.32) we conclude the parabolicity of the type of 𝜉. Let ℎ₀<∞. Consider an exhaustion {𝒰_𝑘} and choose 𝑡₀>0 such that the ℎ-ball 𝐵_ℎ(𝑡₀) contains the compact set 𝐾.
Set 𝑡_𝑘=sup_{𝑚∈𝜕𝒰𝑘}ℎ(𝑚). Then for 𝑡_𝑘>𝑡₀ we have cap_𝑝𝑈_𝑘0,𝒰_𝑘;ℳ≥cap_𝑝𝐵_ℎ𝑡₀,𝐵_ℎ𝑡_𝑘=𝐽;ℳ𝑡_𝑘−𝑡₀^𝑝−1, (3.38) hence liminf_𝑘→∞cap_𝑝𝑈_𝑘0,𝒰_𝑘≥𝐽;ℳℎ₀−𝑡₀^𝑝−1>0, (3.39) and the boundary set 𝜉 is of 𝑝-hyperbolic type.

4. Energy Integral

The fundamental result of this section is an estimate for the rate of growth of the energy integral of forms of the class 𝒲𝒯₂ on noncompact manifolds under various boundary conditions for the forms. As an application we get Phragmén-Lindelöf type theorems for the forms of this class and we prove some generalizations of the classical theorem of Ahlfors concerning the number of distinct asymptotic tracts of an entire function of finite order.

4.1. Boundary Conditions

Let ℳ be an 𝑛-dimensional Riemannian manifold with nonempty boundary 𝜕ℳ. We will fix a closed differential form 𝑤, deg𝑤=𝑘, 1≤𝑘≤𝑛, 𝑤∈𝐿^𝑝_loc(ℳ) of class 𝒲𝒯₁ and the complementary closed form 𝜃, deg𝜃=𝑛−𝑘, 𝜃∈𝐿^𝑞_loc(ℳ), satisfying the condition (1.4). We assume that there exists a differential form 𝑍∈𝑊¹_𝑝,loc with continuous coefficients for which 𝑑𝑍=𝑤.

Let ℎ∶ℳ→(0,ℎ₀) be an exhaustion function of ℳ. As mentioned before we let 𝐵_ℎ(𝜏) be an ℎ-ball and Σ_ℎ(𝜏) an ℎ-sphere.

4.2. Dirichlet Condition with Zero Boundary Values

We will say that the form 𝑍∈𝑊¹_𝑝,loc(with continuous coefficients and such that 𝑑𝑍=𝑤) satisfies Dirichlet's condition with zero boundary values on 𝜕ℳ if for every differential form 𝑣∈𝐿^𝑞_loc(ℳ), deg𝑣=𝑛−𝑘, and for almost every 𝜏∈(0,ℎ₀)𝑤∧𝑣+(_{𝐵ℎ(𝜏)}−1)^𝑘−1_{𝐵ℎ(𝜏)}𝑍∧𝑑𝑣=_Σℎ(𝜏)𝑍∧𝑣. (4.1)

In particular, the form 𝑍 satisfies the boundary condition (4.1), if its coefficients are continuous and if its support does not intersect with 𝜕ℳ, that is supp𝑍∩𝜕ℳ=∅,wheresupp𝑍={𝑚∈ℳ∶𝑍(𝑚)≠0}. (4.2)

If ℳ is compact then (4.1) takes the form 𝑤∧𝑣+(_ℳ−1)^𝑘−1_ℳ𝑍∧𝑑𝑣=0. (4.3)

4.3. Neumann Condition with Zero Boundary Values

We will say that a form 𝑍 satisfies Neumann's condition with zero boundary values, if for every differential form 𝑣∈𝑊¹_𝑝,loc(ℳ), deg𝑣=𝑘−1, and for almost every 𝜏∈(0,ℎ₀)_{𝐵ℎ(𝜏)}𝑑𝑣∧𝜃=_Σℎ(𝜏)𝑣∧𝜃. (4.4)

If ℳ is compact then (4.4) takes the form _ℳ𝑑𝑣∧𝜃=0. (4.5)

4.4. Mixed Zero Boundary Condition

We will say that a form 𝑍 satisfies mixed zero boundary condition if for an arbitrary function 𝜙∈𝐶¹(ℳ) and for almost every 𝜏∈(0,ℎ₀) we have 𝜙𝑤∧𝜃+(_{𝐵ℎ(𝜏)}−1)^𝑛−1_{𝐵ℎ(𝜏)}𝑍∧𝜃∧𝑑𝜙=_Σℎ(𝜏)𝜙𝑍∧𝜃. (4.6)

If ℳ is compact then (4.6) takes the form 𝜙𝑤∧𝜃+(_ℳ−1)^𝑛−1_ℳ𝑍∧𝜃∧𝑑𝜙=0. (4.7)

We assume that the form 𝑍∈𝐶²(intℳ)∩𝐶¹(𝜕ℳ) (4.8) has the property (4.1). On the basis of Stokes' formula (the standard Stokes formula with generalized derivatives) we conclude that for almost every 𝜏∈(0,ℎ₀)𝑑𝑍∧𝑣+(_{𝐵ℎ(𝜏)}−1)^𝑘−1_{𝐵ℎ(𝜏)}𝑍∧𝑑𝑣=_{𝜕𝐵ℎ(𝜏)}𝑍∧𝑣 (4.9) holds. Therefore we get _{𝜕𝐵ℎ(𝜏)⧵Σℎ(𝜏)}𝑍∧𝑣=0,∀𝑣∈𝑊¹_𝑞,loc(ℳ). (4.10) This implies that the restriction of 𝑍 onto the boundary 𝜕ℳ is the zero form, that is, 𝑍|_𝜕ℳ(𝑚)=0ateverypoint𝑚∈𝜕ℳ. (4.11)

We next clarify the geometric meaning of the condition (4.11). We assume that 𝑚∈𝜕ℳ is a point where the boundary 𝜕ℳ has a tangent plane 𝑇_𝑚(𝜕ℳ) and that the form 𝑍 satisfies the regularity condition (4.8) in some neighborhood of the point 𝑚.

Proposition 4.1. If a form 𝑍 is simple at a point 𝑚∈ℳ, then the condition (4.11) is fulfilled if and only if the form 𝑍 is of the form 𝑍=𝜔∧𝑑𝑥_𝑛, (4.12) where 𝜔 is a form, deg𝜔=deg𝑍−1.

Proof. We give an orthonormal system of coordinates 𝑥₁,…,𝑥_𝑛 at the point 𝑚 such that the hyperplane 𝑇_𝑚(𝜕ℳ) is given by the equation 𝑥_𝑛=0. Let deg𝑍=𝑙. Because the form 𝑍 is simple, we can represent it as follows 𝑍=^𝑙_𝑖=1𝑎_1,𝑖𝑑𝑥_𝑖+𝑎_1,𝑛𝑑𝑥_𝑛∧⋯∧^𝑙_𝑖=1𝑎_𝑙,𝑖𝑑𝑥_𝑖+𝑎_𝑙,𝑛𝑑𝑥_𝑛, (4.13) where 𝑎_𝑖,𝑗=𝑎_𝑖,𝑗(𝑚) are some constants. The condition (4.11) can now be rewritten as follows ^𝑙_𝑖=1𝑎_1,𝑖𝑑𝑥_𝑖∧⋯∧^𝑙_𝑖=1𝑎_𝑙,𝑖𝑑𝑥_𝑖=0, (4.14) and we easily obtain (4.12).
The proof of the converse implication is obvious.

We next clarify the geometric meaning of the Neumann condition (4.4). We fix the forms 𝑍,𝑣∈𝐶²(intℳ)∩𝐶¹(𝜕ℳ). (4.15) By Stokes' formula we have for almost every 𝜏∈(0,ℎ₀)_{𝜕𝐵ℎ(𝜏)}𝑣∧𝜃=_{𝐵ℎ(𝜏)}𝑑𝑣∧𝜃+(−1)^𝑘−1_{𝐵ℎ(𝜏)}𝑣∧𝑑𝜃. (4.16) Because the form 𝜃 is closed, condition (4.4) gives _Σℎ(𝜏)𝑣∧𝜃=0∀𝑣∈𝑊¹_𝑝,loc. (4.17) Therefore 𝜃|_𝜕ℳ(𝑚)=0 (4.18) at every point 𝑚∈𝜕ℳ.

Exactly in the same way we verify that the mixed zero boundary condition (4.6) is equivalent to the condition 𝑍∧𝜃|_𝜕ℳ(𝑚)=0 (4.19) at every point 𝑚∈𝜕ℳ.

Consider the case of quasilinear equations (2.9). Let 𝑚∈𝜕ℳ be a regular point and let 𝑥₁,…,𝑥_𝑛 be local coordinates in a neighborhood of this point. We have 𝜃=∗^𝑛_𝑖=1𝐴_𝑖𝑚,∇𝑓(𝑚)𝑑𝑥_𝑖=^𝑛_𝑖=1(−1)^𝑖−1𝐴_𝑖𝑚,∇𝑓(𝑚)𝑑𝑥₁∧⋯∧𝑑𝑥_𝑖∧⋯∧𝑑𝑥_𝑛. (4.20) We set 𝑍=𝑓. In the case (4.1) we choose 𝑣=𝜙𝜃 where 𝜃 is an arbitrary locally Lipschitz function. We obtain _{𝐵ℎ(𝜏)}𝜙𝑑𝑓∧𝜃+_{𝐵ℎ(𝜏)}𝑓𝑑𝜙𝜃=_Σℎ(𝜏)𝜙𝑓𝜃 (4.21) and further _{𝐵ℎ(𝜏)}^𝑛_𝑖=1𝜙𝑓_𝑥𝑖𝐴_𝑖𝑚,∇𝑓∗𝟙=_Σℎ(𝜏)𝜙𝑓𝜃,∀𝜙. (4.22) This condition characterizes generalized solutions of (2.9) with zero Dirichlet boundary condition on 𝜕ℳ.

On the other side, choosing in the case of the Neumann condition (4.4) for 𝑣 an arbitrary locally Lipschitz function 𝜙 we get for almost every 𝜏∈(0,ℎ)_{𝐵ℎ(𝜏)}∇𝜙,𝐴𝑚,∇𝑓∗𝟙=_Σℎ(𝜏)𝜙𝐴𝑚,∇𝑓,𝜈𝑑ℋ_ℳ^𝑛−1 (4.23) which characterizes generalized solutions of (2.9) with zero Neumann boundary conditions on 𝜕ℳ.

It is easy to see that at every point of the boundary we have (−1)^𝑖−1𝑑𝑥₁∧⋯∧𝑑𝑥_𝑖∧⋯∧𝑑𝑥_𝑛𝜕ℳ=cos𝜈,𝑥_𝑖𝑑ℋ_ℳ^𝑛−1, (4.24) where (𝜈,𝑥_𝑖) is the angle between the inner normal vector 𝜈 to 𝜕ℳ and the direction 0𝑥_𝑖; 𝑑ℋ_ℳ^𝑛−1 is the element of surface area on ℳ.

Thus, at a regular boundary point, the condition (4.18) is equivalent to the requirement 𝐴𝑚,∇𝑓(𝑚),𝜈=0. (4.25)

Using (4.11) we see that the condition (4.6) is equivalent to the traditional mixed boundary condition at regular boundary points.

4.5. Maximum Principle for 𝒲𝒯-Forms

Let ℳ be a compact Riemannian manifold with nonempty boundary, dimℳ=𝑛, and let 𝑣∈𝐿^𝑝_loc, deg𝑤=𝑘, 1≤𝑘≤𝑛, be a differential form of class 𝒲𝒯₁ on ℳ. Let 𝜃, deg𝜃=𝑛−𝑘, be a form complementary to the form 𝑤.

Theorem 4.2. Suppose that there exists a differential form 𝑍∈𝑊¹_𝑝,loc(ℳ), 𝑑𝑍=𝑤 on ℳ. If either (4.3) or (4.5) holds, then 𝜃≡0 on ℳ.

Proof. We assume that (4.3) holds and set 𝑣=𝜃. Then (4.3) yields _ℳ𝑤∧𝜃=0. (4.26) Because (−1)^{𝑘(𝑛−𝑘)}(−1)^{𝑘(𝑛−𝑘)}∗(𝑤∧𝜃)=(−1)^{𝑘(𝑛−𝑘)}∗⁻¹(𝑤∧𝜃)=∗⁻¹𝑤∧(−1)^{𝑘(𝑛−𝑘)}𝜃=∗⁻¹(𝑤∧∗(∗𝜃))=⟨𝑤,∗𝜃⟩, (4.27) we get _ℳ𝑤∧𝜃=_ℳ∗(𝑤∧𝜃)∗𝟙=_ℳ⟨𝑤,∗𝜃⟩∗𝟙. (4.28) Using (1.5) we deduce 0=_ℳ𝑤∧𝜃≥𝜈_0ℳ|𝜃|^𝑞∗𝟙. (4.29)
We assume that the boundary condition (4.5) holds. Choose 𝑣=𝑍. Then (4.5) gives_ℳ𝑤∧𝜃=0. (4.30) As above, we arrive at the inequality (4.29). This inequality implies that 𝜃≡0 on ℳ.

In order to illustrate Theorem 4.2 we consider the example of generalized solutions 𝑓∈𝑊¹_𝑝,loc(ℳ) of (2.9) under the condition 𝜈₀𝐴(𝑚,𝜉)^𝑝≤𝜉,𝐴(𝑚,𝜉) (4.31) for all 𝜉∈𝑇_𝑚(ℳ) with the constants 𝑝>1 and 𝜈₀>0.

Setting 𝑍=𝑓 we get

Corollary 4.3. Suppose that the manifold ℳ is compact and the boundary 𝜕ℳ is not empty. If the function 𝑓 satisfies the condition (4.22) or (4.23), then 𝑓≡𝑐𝑜𝑛𝑠𝑡 on ℳ.

5. Estimates for Energy Integral: Applications

This chapter is devoted to Phragmén-Lindelöf and Ahlfors theorems for differential forms.

5.1. Basic Theorem

Let ℳ be a noncompact Riemannian manifold, dimℳ=𝑛. We consider a class ℱ of differential forms 𝑍∈𝑊¹_𝑝,loc(ℳ), deg𝑍=𝑘−1, such that the form 𝑑𝑍=𝑤 satisfies the conditions (1.1) and is in the class 𝒲𝒯₂. Let 𝜃∈𝐿^𝑞_loc be a form satisfying the condition (1.4), complementary to 𝑤.

If the boundary 𝜕ℳ is nonempty then we will assume that the form 𝑍 satisfies on 𝜕ℳ some boundary condition 𝐵. In the case considered below such a boundary condition can be any of the conditions (4.1), (4.2), (4.4), and (4.6). We will denote by ℱ_𝐵(ℳ) the set of forms 𝑍, 𝑑𝑍∈𝒲𝒯₂, satisfying the boundary condition 𝐵 on ℳ. In particular, below we will operate with the classes of the ℱ_𝐷, ℱ₀, ℱ_𝑁, and ℱ_𝐷𝑁 forms corresponding to the boundary conditions (4.1), (4.2), (4.4), and (4.6), respectively.

We fix a locally Lipschitz exhaustion function ℎ∶ℳ→(0,ℎ₀), 0<ℎ₀≤∞. Let 𝜏∈(0,ℎ₀) and let 𝐵_ℎ(𝜏) be an ℎ-ball, and Σ_ℎ(𝜏) its boundary sphere as before.

We introduce a characteristic 𝜀(𝜏) setting 𝜀𝜏;ℱ_𝐵∫=inf^Σℎ(𝜏)|𝑤|^𝑝|∇ℎ|⁻¹𝑑ℋ_ℳ^𝑛−1∫_Σℎ(𝜏)⟨𝑍,∗𝜃⟩𝑑ℋ_ℳ^𝑛−1, (5.1) where the infimum is taken over all 𝑍∈ℱ_𝐵(ℳ), 𝑍≠0.

Some estimates of (5.1) are given in [8, 9].

Under these circumstances we have the following theorem.

Theorem 5.1. Suppose that the form 𝑍∈ℱ_𝐵(ℳ) satisfies one of the boundary condition (4.1), (4.4), or (4.6). Then for almost all 𝜏∈(0,ℎ₀) and for an arbitrary 𝜏₀ the following relation holds 𝑑𝐼(𝜏)𝑑𝜏exp−𝜈₁^𝜏_𝜏0𝜀𝑡;ℱ_𝐵𝑑𝑡≥0, (5.2) where 𝐼(𝜏)=_{𝐵ℎ(𝜏)}|𝑤|^𝑝∗𝟙. (5.3)
In particular, for all 𝜏₁<𝜏₂ we have 𝐼𝜏₁≤𝐼𝜏₂exp−𝜈₁^𝜏2_𝜏1𝜀(𝑡)𝑑𝑡. (5.4)

Proof. The Kronrod-Federer formula yields 𝐼(𝜏)=^𝜏₀𝑑𝑡_Σℎ(𝜏)|𝑤|^𝑝𝑑ℋ_ℳ^𝑛−1|∇ℎ| (5.5) and, in particular, the function 𝐼(𝜏) is absolutely continuous on closed intervals of (0,ℎ₀). Now it is enough to prove the inequality 𝑑𝐼(𝜏)𝑑𝜏≥𝜈₁𝐼(𝜏)𝜀(𝜏). (5.6) From (5.5) we have for almost every 𝜏∈(0,ℎ₀)𝑑𝐼(𝜏)=𝑑𝜏_Σℎ(𝜏)|𝑤|^𝑝𝑑ℋ_ℳ^𝑛−1|∇ℎ|. (5.7)
By (1.6) we obtain𝐼(𝜏)=_{𝐵ℎ(𝜏)}|𝑤|^𝑝∗𝟙≤𝜈₁⁻¹_{𝐵ℎ(𝜏)}⟨𝑤,∗𝜃⟩∗𝟙=𝜈₁⁻¹_{𝐵ℎ(𝜏)}𝑤∧𝜃=𝜈₁⁻¹_{𝐵ℎ(𝜏)}𝑑𝑍∧𝜃. (5.8) However, the form 𝑍 is weakly closed and satisfies one of the conditions (4.1), (4.2), or (4.4). Therefore for a.e. 𝜏∈(0,ℎ₀), _{𝐵ℎ(𝜏)}𝑑𝑍∧𝜃=_Σℎ(𝜏)𝑍∧𝜃. (5.9) Thus we get 𝐼(𝜏)≤𝜈₁⁻¹_Σℎ(𝜏)⟨𝑍,∗𝜃⟩𝑑ℋ_ℳ^𝑛−1. (5.10)
Further from (5.1) it follows that_Σℎ(𝜏)|𝑤|^𝑝𝑑ℋ_ℳ^𝑛−1|∇ℎ|≥𝜀𝜏;ℱ_{𝐵Σℎ(𝜏)}⟨𝑍,∗𝜃⟩𝑑ℋ_ℳ^𝑛−1. (5.11)
Combining the above inequalities we obtain 𝐼(𝜏)≤𝜈₁⁻¹𝜀𝜏;ℱ_{𝐵Σℎ(𝜏)}|𝑤|^𝑝𝑑ℋ_ℳ^𝑛−1|∇ℎ|. (5.12) This inequality together with the equality (5.7) yields 𝐼(𝜏)≤𝜈₁⁻¹𝜀𝜏;ℱ_𝐵𝑑𝐼(𝜏).𝑑𝜏 (5.13) We thus obtain the desired conclusion (5.6).

We will need also some other estimates of the energy integral. We now prove the first of these inequalities. Denote by ℱ(𝐵_ℎ(𝜏)) the set of all differential forms 𝑍₀∈𝐶¹𝐵_ℎ(𝜏),deg𝑍₀=𝑘−1,𝑑𝑍₀=0, (5.14) such that for almost every 𝜏∈(0,ℎ₀) and for an arbitrary Lipschitz function 𝜙 the following formula holds _Σℎ(𝜏)𝜙𝑍₀∧𝜃=_{𝐵ℎ(𝜏)}𝑑𝜙∧𝑍₀∧𝜃. (5.15)

Theorem 5.2. If the differential form 𝑍∈ℱ_𝐵(ℳ), 𝑑𝑍∈𝒲𝒯₂, satisfies the boundary condition (4.1), (4.4), or (4.6), then for all 𝜏₁<𝜏₂<ℎ₀ and for an arbitrary form 𝑍₀∈ℱ(𝐵_ℎ(𝜏₂)) the following relation holds 𝜈_{1𝐵ℎ(𝜏1)}|𝑑𝑍|^𝑝𝑝∗𝟙≤𝜏₂−𝜏_{1𝐵ℎ(𝜏2)⧵𝐵ℎ(𝜏1)}|∇ℎ|𝑍−𝑍₀∧𝜃∗𝟙. (5.16)

Proof. We consider the function 𝜙(𝑚)=1for𝑚∈𝐵_ℎ𝜏₁,𝜏₂−ℎ(𝑚)𝜏₂−𝜏₁for𝑚∈𝐵_ℎ𝜏₂⧵𝐵_ℎ𝜏₁,0for𝑚∈ℳ⧵𝐵_ℎ𝜏₂. (5.17) Suppose that the form 𝑍 satisfies the condition (4.1). Setting in (4.1) 𝑣=(𝜙)^𝑝𝑍∧𝜃 we get _{𝐵ℎ(𝜏2)}𝜙^𝑝𝑤∧𝜃+(−1)^𝑘−1_{𝐵ℎ(𝜏2)}𝑍∧𝑑𝜙^𝑝∧𝜃=0, (5.18) or _{𝐵ℎ(𝜏2)}𝜙^𝑝⟨𝑤,∗𝜃⟩∗𝟙=(−1)^𝑘𝑝_{𝐵ℎ(𝜏2)}𝜙^𝑝−1𝑍∧𝑑𝜙∧𝜃. (5.19)
The function (𝜙)^𝑝 is locally Lipschitz on 𝐵_ℎ(𝜏₂) and 𝜙|_Σℎ(𝜏2)=0. Thus by (5.15) we get _{𝐵ℎ(𝜏2)}𝑑𝜙^𝑝∧𝑍₀∧𝜃=_{𝐵ℎ(𝜏2)}𝑑𝜙^𝑝∧𝑍₀+∧𝜃_{𝐵ℎ(𝜏2)}𝜙^𝑝𝑑𝑍₀∧𝜃+_{𝐵ℎ(𝜏2)}𝜙^𝑝𝑍₀=∧𝑑𝜃_Σℎ(𝜏2)𝜙^𝑝𝑍₀∧𝜃=0. (5.20)
Hence we arrive at the relation_{𝐵ℎ(𝜏2)}𝜙^𝑝⟨𝑤,∗𝜃⟩∗𝟙=(−1)^𝑘𝑝_{𝐵ℎ(𝜏2)}𝜙^𝑝−1𝑍−𝑍₀∧𝑑𝜙∧𝜃, (5.21) which by (1.6) yields 𝜈_{1𝐵ℎ(𝜏2)}𝜙^𝑝|𝑤|^𝑝𝑝∗𝟙≤𝜏₂−𝜏_{1𝐵ℎ(𝜏2)}𝜙^𝑝−1𝑍−𝑍₀∧𝜃|∇ℎ|∗𝟙. (5.22)
Observing that 𝜙(𝑚)=1 for 𝑚∈𝐵_ℎ(𝜏₁) and 𝜙(𝑚)=0 for 𝑚∈ℳ⧵𝐵_ℎ(𝜏₂) we obtain𝜈_{1𝐵ℎ(𝜏1)}|𝑤|^𝑝𝑝∗𝟙≤𝜏₂−𝜏_{1𝐵ℎ(𝜏2)⧵𝐵ℎ(𝜏1)}𝜙^𝑝−1|∇ℎ|𝑍−𝑍₀∧𝜃∗𝟙. (5.23) Because |𝜙|≤1, the inequality (5.16) follows.
Let the form 𝑍 satisfy the condition (4.4). We choose 𝑣=(𝜙)^𝑝𝑍 and observe that𝑣|_Σℎ(𝜏2)=0. (5.24) Then we get _{𝐵ℎ(𝜏2)}𝜙^𝑝𝑤∧𝜃=−_{𝐵ℎ(𝜏2)}𝑑𝜙^𝑝∧𝑍∧𝜃=(−1)^𝑘𝑝_{𝐵ℎ(𝜏2)}𝜙^𝑝−1𝑍∧𝑑𝜙∧𝜃. (5.25) Further details of the proof in this case are similar to those carried out above.
We assume that the form 𝑍 satisfies the mixed boundary condition (4.6). Observing that𝜙^𝑝_Σℎ(𝜏2)=0, (5.26) we get _{𝐵ℎ(𝜏2)}𝜙^𝑝𝑤∧𝜃=(−1)^𝑛_{𝐵ℎ(𝜏2)}𝑍∧𝜃∧𝑑𝜙^𝑝=(−1)^𝑛−𝑘𝑝_{𝐵ℎ(𝜏2)}𝜙^𝑝−1𝑍∧𝑑𝜙∧𝜃. (5.27) Arguing as above we complete the proof of the theorem.

There is also an estimate for the energy integral which does not use the complementary form 𝜃 of 𝑑𝑍=𝑤. Such an estimate is given in the next theorem.

Theorem 5.3. If the form 𝑍, 𝑑𝑍∈𝒲𝒯₂(ℳ), satisfies on 𝜕ℳ one of the boundary conditions (4.1), (4.4), or (4.6), then for all 0<𝜏₁<𝜏₂<ℎ₀ and for an arbitrary form 𝑍₀∈ℱ(𝐵_ℎ(𝜏₂)) we have _{𝐵ℎ(𝜏1)}|𝑑𝑍|^𝑝∗𝟙≤𝑝𝜈₂𝜏₂−𝜏₁𝜈₁^𝑝_{𝐵ℎ(𝜏2)⧵𝐵ℎ(𝜏1)}|∇ℎ|^𝑝𝑍−𝑍₀^𝑝∗𝟙. (5.28)

Proof. We use the earlier established relation (5.22). We estimate the integral on the right hand side of (5.22). By (1.7) we get _ℳ𝜙^𝑝−1|∇ℎ|𝑍−𝑍₀∧𝜃∗𝟙≤_ℳ𝜙^𝑝−1|∇ℎ|𝑍−𝑍₀|𝜃|∗𝟙≤𝜈_2ℳ𝜙^𝑝−1|∇ℎ|𝑍−𝑍₀|𝑤|^𝑝−1∗𝟙≤𝜈_2ℳ|∇ℎ|^𝑝𝑍−𝑍₀^𝑝∗𝟙^1/𝑝_ℳ𝜙^𝑝|𝑤|^𝑝∗𝟙^{(𝑝−1)/𝑝}. (5.29) From (5.22) we get 𝜈₁𝜈₂^𝑝_ℳ|𝑤|^𝑝𝑝∗𝟙≤𝜏₂−𝜏₁^𝑝_ℳ|∇ℎ|^𝑝𝑍−𝑍₀^𝑝∗𝟙. (5.30) Using the facts that 𝜙=1 on 𝐵_ℎ(𝜏₁) and 𝜙=0 on ℳ⧵𝐵_ℎ(𝜏₂) we easily obtain (5.28).

5.2. Phragmén-Lindelöf Theorem

Let ℳ be an 𝑛-dimensional noncompact Riemannian manifold with or without boundary and let 𝑤∈𝒲𝒯₂ be a differential form as in (1.1), deg𝑤=𝑘, and 𝜃 its complementary form as in (1.4).

We assume that there exists a differential form 𝑍∈𝑊¹_𝑝,loc with 𝑑𝑍=𝑤. If the boundary 𝜕ℳ is nonempty, then we will assume that 𝑍 satisfies the boundary condition of Dirichlet (4.1), Neumann's condition (4.4), or the condition (4.6).

We fix a locally Lipschitz exhaustion function ℎ∶ℳ→(0,ℎ₀), 0<ℎ₀≤∞. Let, as above 𝐼(𝜏;𝑍)=_{𝐵ℎ(𝜏)}|𝑑𝑍|^𝑝∗𝟙, (5.31) and let 𝜇(𝜏;𝑍)=inf_{𝜏<ℎ(𝑚)<𝜏+1}|∇ℎ|𝑍−𝑍₀𝑚()∧𝜃∗𝟙,𝜏;𝑍=inf_{𝜏<ℎ(𝑚)<𝜏+1}|∇ℎ|^𝑝𝑍−𝑍₀^𝑝∗𝟙, (5.32) where the infimum is taken over all closed forms 𝑍₀, satisfying conditions (5.14), (5.15) on 𝐵_ℎ(𝜏).

The following theorem exhibits a generalization of the classical Phragmén-Lindelöf principle for holomorphic functions.

Theorem 5.4. Suppose that the form 𝑍, 𝑑𝑍∈𝒲𝒯₂(ℳ), satisfies one of the boundary conditions (4.1), (4.4) or (4.6). The following alternatives hold: either the form 𝑑𝑍=0 a.e. on the manifold ℳ, or for all 𝜏₀∈(0,ℎ₀) we have liminf_𝜏→ℎ0𝐼(𝜏;𝑍)exp−𝜈₁^𝜏_𝜏0𝜀𝑡;ℱ_𝐵𝑑𝑡>0;liminf_𝜏→ℎ0𝜇(𝜏;𝑍)exp−𝜈₁^𝜏_𝜏0𝜀𝑡;ℱ_𝐵𝑑𝑡>0,liminf_𝜏→ℎ0𝑚(𝜏;𝑍)exp−𝜈₁^𝜏_𝜏0𝜀𝑡;ℱ_𝐵𝑑𝑡>0. (5.33)(5.34)(5.35)

Proof. The property (5.33) follows readily from (5.4) and is presented here only for the sake of completeness.
By (5.4) and (5.16), for a.e. 𝜏∈(𝜏₀,ℎ₀) we have𝐼𝜏₀≤𝐼(𝜏)exp−𝜈₁^𝜏_𝜏0𝜀(𝑡)𝑑𝑡≤𝑝𝜈₁⁻¹_{𝐵ℎ(𝜏+1)⧵𝐵ℎ(𝜏)}|∇ℎ|𝑍−𝑍₀∧𝜃∗𝟙exp−𝜈₁^𝜏_𝜏0𝜀(𝑡)𝑑𝑡. (5.36) Therefore we get 𝐼𝜏₀≤𝑝𝜈₁⁻¹𝜇(𝜏;𝑍)exp−𝜈₁^𝜏_𝜏0𝜀(𝑡)𝑑𝑡. (5.37)
Analogously, using (5.28) we get 𝐼𝜏₀≤𝜈₂𝜈₁^𝑝𝑝^𝑝𝑚(𝜏;𝑍)exp−𝜈₁^𝜏_𝜏0𝜀(𝑡)𝑑𝑡. (5.38)
If we now assume that the form 𝑤≢0, then 𝐼(𝜏₀)>0 for some 𝜏₀∈(0,ℎ₀). From this there easily follow (5.34) and (5.35).

5.3. Integral of Energy and Allocation of Finite Forms

There is another application of the above estimates of energy integrals connected with a generalization of the classical Denjoy-Carleman-Ahlfors theorem about the number of different asymptotic tracts of an entire function of a given order. In the present case this theorem can be interpreted as a statement concerning the connection between the number of finite forms in the class 𝒲𝒯₂ defined on the manifold ℳ and the rate of growth of their energy integrals.

Let ℳ be an 𝑛-dimensional noncompact Riemannian manifold with or without boundary. We fix a locally Lipschitz exhaustion function ℎ∶ℳ→(0,ℎ₀), 0<ℎ₀≤∞, of the manifold ℳ.

We assume that there are 𝐿≥1 mutually disjoint domains 𝒪₁,𝒪₂,…,𝒪_𝐿 on ℳ such that 𝒪_𝑖∩𝜕ℳ=∅ if the boundary 𝜕ℳ is nonempty. We also assume that on each domain 𝒪_𝑖 there is given a differential form 𝑍_𝑖 with continuous coefficients and the properties:(i)deg𝑍_𝑖=𝑘−1, 𝑑𝑍≢0,(ii)𝑑𝑍_𝑖=𝑤∈𝒲𝒯₂ with structure constants 𝑝,𝜈₁,𝜈₂, independent of 𝑖=1,2,…,𝐿,(iii)𝑍_𝑖 satisfies on 𝜕𝒪_𝑖 the zero boundary condition (4.2).

We define a form 𝑍 on ℳ by setting 𝑍|_𝒪𝑖=𝑍_𝑖 and 𝑍=0 on ℳ⧵⋃^𝐿_𝑖=1𝒪_𝑖.

According to Theorem 4.2 each of the domains 𝒪_𝑖 has a noncompact closure. Then by Theorem 5.4 the "narrower" the intersection of the domains 𝒪_𝑖 with ℎ-spheres Σ_ℎ(𝑡) for 𝑡→∞, the higher is the rate of growth of the form 𝑍. Below we will consider the Denjoy-Carleman-Ahlfors theorem as a statement on the connection between the number 𝐿 of mutually disjoint domains 𝒪_𝑖 on ℳ₀ and the rate of growth of the energy of the form 𝑍 (or of the form 𝑍 itself) with respect to an exhaustion function ℎ(𝑚) of the manifold ℳ. We will prove that such a formulation of the problem contains, in particular, the classical Denjoy-Carleman-Ahlfors problem for holomorphic functions of the complex plane. In the case of harmonic functions of 𝐑^𝑛 see [10, 11] for the history of the problem.

We next introduce some necessary notation. We consider an open subset 𝐷⊂ℳ with a noncompact closure and we assume that the restriction of the form 𝑍 to 𝐷 satisfies condition (4.2).

The function ℎ|_𝐷∶𝐷→(0,∞) is an exhaustion function of 𝐷. We fix an ℎ-ball 𝐵_ℎ(𝜏). Considering the variational problem (5.1) for the class of forms 𝑍, satisfying the boundary condition (4.2) on 𝐷₁ we define the characteristic 𝜀(𝑡;𝐷)=𝜀𝑡;ℱ₀ (5.39) where ℱ₀ is defined in Section 5.1 and in (4.1).

Following [12] we introduce the 𝑁-mean 𝐸()1𝑡;𝑁=inf𝑁^𝑁_𝑘=1𝜀𝑡;𝐷_𝑘 (5.40) where the infimum is taken over all decompositions of 𝐷 into 𝑁 nonintersecting open sets 𝐷₁,𝐷₂,…,𝐷_𝑁 with noncompact closures.

We record the following simple result.

Lemma 5.5. Let 𝐷₁⊂𝐷₂ be arbitrary open subsets of ℳ with noncompact closures. Then 𝜀𝑡;𝐷₂≤𝜀𝑡;𝐷₁,𝜀(𝑡;ℳ)≤𝐸(𝑡;𝑁),𝑁≥1. (5.41)(5.42)

Proof. It is enough to observe that each pair of forms 𝑍,𝑍₀ admissible for the variational problem (5.1) for the set 𝐷₁ is also admissible for this problem for the set 𝐷₂.
From (5.41) we get (5.42).

We next derive a more general assertion about the monotonicity of 𝑁-means.

Lemma 5.6. For arbitrary 𝑁>1 we have 𝐸(𝑡;𝑁+1)≥𝐸(𝑡;𝑁). (5.43)

Proof. We consider an arbitrary family of open subsets {𝐷_𝑘}, 𝑘=1,2,…,𝑁+1, admissible for the infimum in (5.40). It is not difficult to see that 1𝑁+1^𝑁+1_𝑘=1𝜀𝑡;𝐷_𝑘=1𝑁+1^𝑁+1_𝑘=11𝑁^𝑁+1_{𝑗=1,𝑗≠𝑘}𝜀𝑡;𝐷_𝑗. (5.44) Because 1𝑁^𝑁+1_{𝑗=1,𝑗≠𝑘}𝜀𝑡;𝐷_𝑘≥𝐸(𝑡;𝑁), (5.45) we see that 1𝑁+1^𝑁+1_𝑘=1𝜀𝑡;𝐷_𝑘≥𝐸(𝑡;𝑁) (5.46) and the lemma is proved.

The next theorem provides a solution to the aforementioned problem concerning the connection between the number 𝐿 of finite forms on ℳ, and the rate of growth of the total energy of these forms or the sum of their 𝐿^𝑝-norms on an ℎ-ball 𝐵_ℎ(𝜏).

Theorem 5.7. Suppose that the manifold ℳ satisfies the properties listed in the beginning of this subsection and that for some 𝑁=1,2,…^ℎ0ℎ₀𝐸(𝑡,𝑁)𝑑𝑡=∞. (5.47) If the differential form 𝑍, 𝑑𝑍∈𝒲𝒯₂(ℳ), is such that liminf_{𝜏→ℎ0ℎ(𝑚)<𝜏}|𝑑𝑍|^𝑝∗𝟙exp−𝜈₁^𝜏_𝜏0𝐸(𝑡;𝑁)𝑑𝑡=0, (5.48) or liminf_{𝜏′,𝜏′′→ℎ00<𝜏′<𝜏′′<ℎ0𝜏′<ℎ(𝑚)<𝜏′′}|∇ℎ||𝑍∧𝜃|∗𝟙exp−𝜈₁^𝜏′_𝜏0𝐸(𝑡;𝑁)𝑑𝑡=0, (5.49) or liminf_{𝜏′,𝜏′′→ℎ00<𝜏′<𝜏′′<ℎ0𝜏′<ℎ(𝑚)<𝜏′′}|∇ℎ|^𝑝|𝑍|^𝑝∗𝟙exp−𝜈₁^𝜏′_𝜏0𝐸(𝑡;𝑁)𝑑𝑡=0, (5.50) then 𝐿<𝑁.

Proof. We assume that there exists 𝑁 mutually nonintersecting domains 𝒪₁,𝒪₂,…,𝒪_𝑁 on the set ℳ₀ and the forms 𝑍_𝑖 defined on 𝒪_𝑖 with above properties. We denote by 𝑑𝒪_𝑘=inf_{𝑚∈𝒪𝑘}ℎ(𝑚),𝑑=max_{1≤𝑘≤𝑁}𝑑𝒪_𝑘. (5.51)
Fix 𝜏₀>𝑑. Using the inequality (5.4) from Theorem 5.1 for an arbitrary 𝑘=1,2,…,𝑁 and a.e. 0<𝜏₀<𝜏^′<ℎ₀ we have 𝐼_𝑘𝜏₀exp𝜈₁^𝜏′_𝜏0𝜀_𝑘(𝑡)𝑑𝑡≤𝐼_𝑘𝜏^′, (5.52) where 𝐼_𝑘𝜏^′=_{𝒪𝑘∩𝐵ℎ(𝜏′)}𝑑𝑍_𝑘^𝑝∗𝟙,𝜀_𝑘𝜏^′=𝜀𝜏^′;𝒪_𝑘. (5.53) Adding these inequalities, we get min_{1≤𝑘≤𝑁}𝐼_𝑘𝜏₀^𝑁_𝑘=1exp𝜈₁^𝜏′_𝜏0𝜀_𝑘(𝑡)𝑑𝑡≤𝐼𝜏^′, (5.54) where 𝐼𝜏^′=_{𝐵ℎ(𝜏′)}|𝑑𝑍|^𝑝∗𝟙. (5.55) Applying the arithmetic-geometric mean inequality 1𝑁^𝑁_𝑘=1exp𝜈₁^𝜏′_𝜏0𝜀_𝑘(𝑡)𝑑𝑡≥^𝑁_𝑘=1𝜈exp¹𝑁^𝜏′_𝜏0𝜀_𝑘(𝑡)𝑑𝑡 (5.56) we get min_{1≤𝑘≤𝑁}𝐼_𝑘𝜏₀𝜈𝑁exp¹𝑁^𝜏′_𝜏01𝑁^𝑁_𝑘=1𝜀_𝑘(𝑡)𝑑𝑡≤𝐼𝜏^′. (5.57)
The domains 𝒪₁,𝒪₂,…,𝒪_𝑁 are nonintersecting. Therefore for all 𝜏₀<𝑡<ℎ₀ we have 1𝑁^𝑁_𝑘=1𝜀_𝑘(𝑡)≥𝐸(𝑡;𝑁). (5.58) The preceding inequality gives now min_{1≤𝑘≤𝑁}𝐼_𝑘𝜏₀𝑁exp𝜈₁^𝜏′_𝜏0𝐸(𝑡;𝑁)𝑑𝑡≤𝐼𝜏^′. (5.59)
For the estimation of the integral 𝐼(𝜏^′) we use the inequalities (5.16) and (5.28) and obtain min_{1≤𝑘≤𝑁}𝐼_𝑘𝜏₀≤𝐶_{1𝜏′<ℎ(𝑚)<𝜏′′}|∇ℎ||𝑍∧𝜃|∗𝟙exp−𝜈₁^𝜏′_𝜏0𝐸(𝑡;𝑁)𝑑𝑡 (5.60) or min_{1≤𝑘≤𝑁}𝐼_𝑘𝜏₀≤𝐶_{2𝜏′<ℎ(𝑚)<𝜏′′}|∇ℎ|^𝑝|𝑍|^𝑝∗𝟙exp−𝜈₁^𝜏′_𝜏0𝐸(𝑡;𝑁)𝑑𝑡, (5.61) where 𝐶₁=𝑝𝜏^′′−𝜏^′,𝐶₂=𝑝𝜈₂𝜏^′′−𝜏^′𝜈₁. (5.62)
On the basis of the conditions (5.47)–(5.50) imposed on the form 𝑍, for some 𝑘, 1≤𝑘≤𝑁, we have 𝐼_𝑘(𝜏₀)=0. Thus 𝑑𝑍_𝑘(𝑚)≡0 on 𝒪_𝑘∩𝐵_ℎ(𝜏₀). Because we have chosen 𝜏₀>𝑑 arbitrarily, we can conclude that at least on one of the components 𝒪_𝑘, 𝑑𝑍_𝑘≡0. A contradiction.