Stagnation zones for $\mathcal{A}$-harmonic functions on canonical domains

We study stagnation zones of $\mathcal{A}$-harmonic functions on canonical domains in the Euclidean $n$-dimensional space. Phragmen-Lindel\"of type theorems are proved.


Introduction
In this article we investigate solutions of the A-Laplace equation on canonical domains in the n-dimensional Euclidean space.
Suppose that D is a domain in R n , and let f : D → R be a function. For s > 0, a subset ∆ ⊂ D is called s-zone (stagnation zone with the deviation s) of f , if there exists a constant C such that the difference between C and the function f is smaller than s on ∆. We may, for example, consider difference in the sense of the sup norm, For discussion about the history of the question, recent results and applications, see [SS06,SS07].
Some estimates of stagnation zone sizes for solutions of the A-Laplace equation on locally Lipschitz surfaces and behaviour of solutions in stagnation zones, were given in [Mik07]. In this paper we consider solutions of the A-Laplace equation in subdomains of R n of a special form, canonical domains. In two-dimensional case, such domains are sectors and strips. In higher dimensions, they are conical and cylindrical regions. The special form of domains allows us to obtain more precise results.
Below we study stagnation zones of generalized solutions of the A-Laplace equation div A(x, ∇f ) = 0 (see [HKM93]) with boundary conditions of types (see definitions 1.6 and 1.10 below): A(x, ∇f ), n = 0 , x ∈ ∂D \ G and f A(x, ∇f ), n = 0 , x ∈ ∂D \ G on canonical domains in the Euclidean n-dimensional space, where G is a closed subset of ∂D. We will prove Phragmén-Lindelöf type theorems for solutions of the A-Laplace equation with such boundary conditions.
For every 0 < k < n we set Let 0 < α < β < ∞ be fixed, and let D k α,β = {x ∈ R n : α < p k (x) < β}. For k = n − 1 we also assume that x n > 0. Then for k = n − 1 the D n−1 α,β is the a layer between two parallel hyperplanes, and for 1 ≤ k < n − 1 the boundary of the domain D k α,β consists of two coaxial cylindrical surfaces. The intersections Σ k (t) ∩ D k α,β are precompact for all t > 0. Thus, the functions d k (x) are exhaustion functions for D k α,β . Structure conditions. Let D be a subdomain of R n and let be a vector function such that for a.e. x ∈ D the function A(x, ξ) : R n → R n is defined and continuous with respect to ξ. We assume that the function is measurable in the Lebesgue sense for all ξ ∈ R n and Suppose that for a.e. x ∈ D and for all ξ ∈ R n the following properties hold: with p ≥ 1 and some constants ν 1 , ν 2 > 0. We consider the equation An important special case of (1.3) is the Laplace equation Frequencies. Fix t ≥ 0 and p ≥ 1. Let O be an open subset of Σ * k (t) (with respect to the relative topology of Σ * k (t)), and let P be a nonempty closed subset of ∂O. We set The second frequency is the following quantity: where the supremum is taken over all constants C and u ∈ Lip loc (O) ∩ C 0 (O) . See also Pólya and Szegö [PS51], Lax [Lax57].
Generalized boundary conditions. Suppose that D is a proper subdomain of R n . Let ϕ : D → R be a locally Lipschitz function. We denote by D b (ϕ) the set of all points x ∈ D at which ϕ does not have the differential. Let U ⊂ D be a subset and let ∂ ′ U = ∂U \ ∂D be its boundary with respect to D. If ∂ ′ U is (H n−1 , n − 1)rectifiable, then it has locally finite perimeter in the sense of De Giorgi, and therefore a unit normal vector n exists H n−1 -almost everywhere on ∂U [Fed69, Sections 3.2.14, 3.2.15]. Let D ⊂ R n be a domain and let G ⊂ ∂D be a subset of the boundary of D. Define the concept of a generalized solution of (1.3) with zero boundary conditions on ∂D \ G. A subset U ⊂ D is called admissible, if U ∩ G = ∅ and U has a (H n−1 , n − 1)-rectifiable boundary with respect to D.
Suppose that D is unbounded. Let G ⊂ ∂D be a set closed in R n ∪ {∞}. We denote by (G, D) the collection of all subdomains U ⊂ D with ∂U ⊂ (D ∪ (∂D \ G)) and (H n−1 , n − 1)-rectifiable boundaries ∂ ′ U = ∂U \ ∂D.
1.6. Definition. We say that a locally Lipschitz function f : D → R is a generalized solution of (1.3) with the boundary condition (1.7) A(x, ∇f ), n = 0 , x ∈ ∂D \ G , if for every subdomain U ∈ (G, D), and for every locally Lipschitz function ϕ : U \ G → R the following property holds: (1.9) Here n is the unit normal vector of ∂ ′ U and dH n is the volume element on R n .
1.10. Definition. We say that a locally Lipschitz function f : D → R is a generalized solution of (1.3) with the boundary condition if for every subdomain U ∈ (G, D) with (1.8), and for every locally Lipschitz function ϕ : U \ G → R the following property holds: (1.12) In the case of a smooth boundary ∂D, and f ∈ C 2 (D), the relation (1.9) implies (1.3) with (1.7) everywhere on ∂D \ G. This requirement (1.12) implies (1.3) with (1.11) on ∂D \ G. See [Mik08, Section 9.2.1].
The surface integrals exist by (1.8). Indeed, this assumption guarantees that ∇f (x) exists H n−1 -a.e. on ∂ ′ U. The assumption that U ∈ (G, D) implies existence of a normal vector n for H n−1 -a.e. points on ∂ ′ U [Fed69, Chapter 2 Section 3.2]. Thus, the scalar product A(x, ∇f ) , n is defined and finite a.e. on ∂ ′ U.

Saint-Venant principle
In this section, we will prove the Saint-Venant principle for solutions of the A-Laplace equation. The Saint-Venant principle states that strains in a body produced by application of a force onto a small part of its surface are of negligible magnitude at distances that are large compared to the diameter of the part where the force is applied. This well known result in elasticity theory is often stated and used in a loose form. For mathematical investigation of the results of this type, see e.g. [BT08].
In this paper the inequalities of the form (2.3), (2.2) are called the Saint-Venant principle (see also, [OY77,BT08]). Here we consider only the case of canonical domains. We plan to consider the general case in another article.
Let 0 < k < n. Fix a domain D 0 in R k with compact and smooth boundary, and write 2.1. Theorem. Let α < τ ′ < τ ′′ < β, and let 0 < k < n. If f : D → R is a generalized solution of (1.3) with the generalized boundary condition (1.7) on ∂D \ G, then the inequality holds for all t ∈ (α, τ ′ ]. If f : D → R is a generalized solution of (1.3) with the generalized boundary condition (1.11) then Here Proof. The case A. At first we consider the case in which f is a generalized solution of (1.3) with the generalized boundary condition (1.11) on ∂D \ G. It is easy to see that a.e. on D k α,β , |∇p k (x)| = 1 .
The domain ∆ k (t, τ ) belongs to (G, D). Let ϕ : U \ G → R be a locally Lipschitz function. By (1.12) we have For ϕ ≡ 1, we have by (1.2) and (1.12) Note that we may also choose to obtain an inequality similar to (2.5). Next we will estimate the right side of (2.5). By (1.2) and the Hölder inequality By using (1.4) we may write By (2.5) and the Fubini theorem and By integrating this differential inequality we have The case B. Now we assume that f is a generalized solution of (1.3) with the boundary condition (1.7) on ∂D \ G. Fix t < τ . By choosing ϕ ≡ 1 in (1.9) we see that For an arbitrary constant C, we get from this and (1.9) As above, we obtain (2.11) By using (1.5) we get (2.12) is the constant from (1.5). Then by (2.11) and (2.12), and by (2.10) we have By integrating this inequality we have shown that

Stagnation zones
Next we apply the Saint-Venant principle to obtain information about stagnation zones of generalized solutions of the equation (1.3). We first consider zones with respect to the Sobolev norm. Other results of this type follow immediately from well-known imbedding theorems.
Stagnation zones with respect to the W 1 p -norm. We rewrite (2.2) and (2.3) in another form. Let 0 < k < n and let 0 < α < β. Fix a domain D 0 in R k with compact and smooth boundary, and write For x ∈ D k α,β and Let 0 < τ ′ < τ ′′ < β * . By (2.3) we have for t ∈ (−τ, τ ) By choosing the estimate as in (2.6) we also have By adding these inequalities and noting that C 4 (t) +C 4 (t) = 0 we obtain Thus we have the estimate Similarly, from (2.2) we obtain From this we obtain the following theorem on stagnation W 1 p -zones: 3.7. Theorem. Let 0 < k < n, β > α > 0, and let −β * < τ ′ ≤ τ ′′ < β * where β * is as in (3.1). If f is a solution of (1.3) on D with the generalized boundary condition or a solution of (1.3) on D with the generalized boundary condition (1.11) on ∂D \ G and where ∆ * ,k is as in (3.2).
Stagnation zones with respect to the L p -norm. Let β > α > 0, and let −β * < τ ′ ≤ τ ′′ < β * where β * is as in (3.1). Denote by C 5 the best constant of the imbedding theorem from W 1 p (D * ,k β * ) to L p (D * ,k β * ), i.e. in the inequality g − C L p (D * ,k β * ) ≤ C 5 g W 1 p (D * ,k β * ) , if such constant exist (see Maz'ya [Maz85] or [AF03]). Then we obtain from (3.5), (3.6) These relations can be used to obtain information about stagnation zones with respect to the L p -norm. Namely, we have: 3.10. Theorem. Let 0 < k < n, and let where D 0 is a domain in R k with compact and smooth boundary. If f is a solution of (1.3) on D, with the generalized boundary condition (1.7) (or (1.11)) on ∂D \ G, where G = {x ∈ ∂D : p * k (x) = ±β * }, and the right side of (3.8) (or (3.9)) is smaller than s > 0, then the domain ∆ * ,k (−τ ′ , τ ′ ) is a stagnation zone with the deviation s p in the sense of the L p -norm on D.
In this case from (3.5), (3.6) we obtain These relations can be used to obtain theorems about stagnation zones for bounded uniformly continuous functions.

Other applications
Next we prove Phragmén-Lindelöf type theorems for the solutions of the A-Laplace equation with boundary conditions (1.7) and (1.11).
As in (2.9) we may by (1.9) write By the construction of φ, (4.1) and (4.2), the surface integral is equal to zero, and we have Thus by (1.2) Now we note that |∇φ(p * k (x))| = |φ ′ (p * k (x))| and by the Hölder inequality From this inequality and (4.3) we obtain Because φ(p * k (x)) ≡ 1 on ∆ * ,k (−τ ′ , τ ′ ) we have the following inequality: Next we will find where the minimum is taken over all ψ in (4.2). We have |f (x) − C| p dH n−1 and (4.5) min Because by the Hölder inequality and hence It is easy to see that here the equality holds for a special choice of ψ. Thus Similarly, From (4.5) we obtain By using (4.4) we obtain the inequality: where C is an arbitrary constant. From this we obtain (4.6) Similarly, for the solutions of the A-Laplace equation with the boundary condition (1.11) we may prove that It follows that (4.7) Phragmén-Lindelöf type theorems I. We prove Phragmén-Lindelöf type theorems for cylindrical domains. Let k = n − 1. Fix a domain D 0 in R n−1 with compact and smooth boundary. Consider the domain Let f 0 : D → R be a generalized solution of (1.3) with (1.1) and (1.2) satisfying the boundary condition (1.7) on ∂D. Fix β > α > 0, and let β * be as in (3.1). Let f (x) = f 0 (x − β * e n ), where e n is the n:th unit coordinate vector, and let 0 < τ ′ < τ ′′ < β * < ∞ . By (4.6) Similarly for a solution f of (1.3) with (1.1) and (1.2), satisfying the boundary condition (1.11) we may write However here we do not have any identity similar to (4.8). We have: 4.12. Theorem. Fix a domain D 0 in R n−1 with compact and smooth boundary. Let and let f : D → R be a generalized solution of (1.3) with (1.1) and (1.2) satisfying the boundary condition (1.11) on ∂D. If the right side of (4.11) tends to 0 as τ ′′ → ∞, then f ≡ 0 on ∂D.
Phragmén-Lindelöf type theorems II. We prove Phragmén-Lindelöf type theorems for canonical domains of an arbitrary form. Let 1 ≤ k < n − 1. We consider a domain where D 0 is a domain in R k with compact and smooth boundary. Let f be a generalized solution of (1.3) with (1.1) and (1.2) satisfying the boundary condition (1.7) on ∂D.