Mixed boundary value problem for $p$-harmonic functions in an infinite cylinder

We study a mixed boundary value problem for the $p$-Laplace equation $\Delta_p u=0$ in an open infinite circular half-cylinder with prescribed Dirichlet boundary data on a part of the boundary and zero Neumann boundary data on the rest. Existence of weak solutions to the mixed problem is proved both for Sobolev and for continuous data on the Dirichlet part of the boundary. We also obtain a boundary regularity result for the point at infinity in terms of a variational capacity adapted to the cylinder.


Introduction
When solving the Dirichlet problem for a given partial differential equation, in a nonempty open set Ω ⊂ R n , one primarily seeks a solution u which is constructed from the boundary data f ∈ C(∂Ω) so that lim Ω∋x→x0 u(x) = f (x 0 ) for x 0 ∈ ∂Ω. (1.1) This may or may not be possible for all boundary points. Therefore, the solution u is often found in a suitable Sobolev space associated with the studied equation and the boundary data are only attained in a weak sense. We say that x 0 ∈ ∂Ω is regular for the considered equation if (1.1) holds for all continuous boundary data f . If all the boundary points are regular, the solution attains its continuous boundary data in the classical sense. At irregular boundary points, equality (1.1) may fail even for continuous boundary data. The first examples of this phenomenon were given for the Laplace equation ∆u = 0 in 1911 by Zaremba [20] in the punctured ball and in 1912 by Lebesgue [12] in the complement of the so-called Lebesgue spine.
Regularity of a boundary point x 0 ∈ ∂Ω for the Laplace equation ∆u = 0 can be characterized by the celebrated Wiener criterion which was established in 1924 by Wiener [18]. With this criterion, one measures the thickness of the complement of Ω near x 0 in terms of capacities. Roughly speaking, x 0 is regular if the complement is thick enough at x 0 .
Boundary regularity has been later studied for more general elliptic equations, mainly in bounded open sets. These studies include linear uniformly elliptic equations with bounded measurable coefficients in Littman-Stampacchia-Weinberger [14], degenerate linear elliptic equations in Fabes-Jerison-Kenig [3], as well as many nonlinear elliptic equations. In particular, Maz ′ ya [15] obtained pointwise estimates near a boundary point for weak solutions of elliptic quasilinear equations, including the p-Laplace equation (1.2). These estimates lead to a sufficient condition for boundary regularity for such equations. Gariepy-Ziemer [4] generalized Maz ′ ya's result to a larger class of elliptic quasilinear equations.
The necessity part of the Wiener criterion for elliptic quasilinear equations was for p > n − 1 proved by Lindqvist-Martio [13] and for all p > 1 by Kilpeläinen-Malý [10]. For weighted elliptic quasilinear equations, the sufficiency part was obtained in Heinonen-Kilpeläinen-Martio [6], while the necessity condition was established by Mikkonen [17].
In this paper, we consider a mixed boundary value problem for the p-Laplace equation ∆ p u := div(|∇u| p−2 ∇u) = 0, 1 < p < ∞, (1.2) in an open infinite circular half-cylinder with zero Neumann boundary data on a part of the boundary and prescribed Dirichlet data on the rest of the boundary. In Theorem 6.3, we prove the existence of weak solutions to the mixed boundary value problem for (1.2) with Sobolev type Dirichlet data. For continuous Dirichlet data, we obtain the following result. Then there exists a bounded continuous weak solution u ∈ W 1,p loc (G \ F ) of the p-Laplace equation (1.2) in G \ F , with zero Neumann boundary data on ∂G \ F , attained in the weak sense of (2.1), and Dirichlet boundary data f on F 0 , attained as the limit lim G\F ∋x→x0 for all x 0 ∈ F 0 , except possibly for a set of Sobolev C p -capacity zero. Moreover, the limit lim G\F ∋x→x0 u(x) exists and is finite for all x 0 ∈ ∂G \ F .
Note that the set F need not be a part of the boundary ∂G and thus, equation (1.2) can be considered on a more general subset of the cylinder G. The zero Neumann condition is, however, prescribed only on a part of the lateral boundary ∂G.
We also study boundary regularity of the point at infinity for these solutions. More precisely, in Theorem 8. Here the capacity cap p,Gt−1 is for compact sets K ⊂ G t−1 := B ′ × (t − 1, ∞) defined by with the infimum taken over all v ∈ C ∞ 0 (R n ) satisfying v ≥ 1 on K and v = 0 on G \ G t−1 . We also relate cap p,Gt−1 to the standard Sobolev p-capacity in R n . In particular, Lemmas 7.7 and 7.8 show that for K ⊂ G t \ G t+1 , the two capacities are comparable, but this is not true for general K ⊂ G t−1 .
To obtain these results, we use the change of variables introduced in Björn [1] to transform the infinite half-cylinder G and the p-Laplace equation (1.2) into a unit half-ball and a weighted elliptic quasilinear equation div A(ξ, ∇u(ξ)) = 0, (1.6) respectively. In order to use the theory of Dirichlet problems, developed in Heinonen-Kilpeläinen-Martio [6] for the equation (1.6), the Neumann data are removed by reflecting the unit half-ball and the equation (1.6) to the whole unit ball. We then use the Wiener criterion for such equations, together with tools from [6], to determine the regularity of the point at infinity and to prove the existence of continuous weak solutions to the mixed boundary value problem for (1.2). Compared to the Dirichlet problem in bounded domains, there are relatively few studies of boundary value problems with respect to unbounded domains and with mixed boundary data. Early work on mixed boundary value problems was due to Zaremba [19] and such problems are therefore sometimes called Zaremba problems. Kerimov-Maz ′ ya-Novruzov [8] characterized regularity of the point at infinity for the Zaremba problem for the Laplace equation ∆u = 0 in an infinite half-cylinder. Björn [1] studied a similar problem for certain linear weighted elliptic equations. Our results partially extend the ones in [1] and [8] to the p-Laplace equation (1.2), even though the necessary and sufficient conditions obtained therein are formulated differently.
The organization of the paper is as follows. In Section 2 we introduce the notation and give the definition of weak solutions. Section 3 is devoted to transforming the infinite half-cylinder together with the p-Laplace equation (1.2) into a unit half-ball with the weighted elliptic quasilinear equation (1.6). In Section 4, we state and prove some properties of the obtained operator div A(ξ, ∇u(ξ)), such as ellipticity and monotonicity, needed to apply the results from Heinonen-Kilpeläinen-Martio [6].
In Section 5, the Neumann boundary data are removed by means of a reflection and the mixed boundary value problem is turned into a Dirichlet problem. This makes it is possible to use the tools developed for weighted elliptic quasilinear equations in [6]. Sections 4 and 5 also contain comparisons of appropriate function spaces on the half-cylinder and those on the ball. In Section 6, we prove the existence of continuous weak solutions to the mixed boundary value problem for (1.2). Section 7 is devoted to comparing two variational capacities: one associated with the weighted Sobolev spaces on the unit ball and the other defined on the half-cylinder. These are crucial for studying the boundary regularity at infinity in Section 8.

Notation and formulation of the mixed problem
Throughout the paper, we represent points in the n-dimensional Euclidean space R n = R n−1 × R, n ≥ 2, as x = (x ′ , x n ) = (x 1 , ... , x n−1 , x n ). We shall consider the open infinite circular half-cylinder Let F be a closed subset of G. Assume also that F contains the base B ′ × {0} of G. Let 1 < p < ∞ be fixed. We shall consider a mixed boundary value problem for the p-Laplace equation ∆ p u = 0 in G \ F with Dirichlet boundary data where n is the outer normal of G. Note that F is not necessarily a subset of ∂G, which makes it possible to consider more general domains contained in G. If ∂G ⊂ F , then the mixed boundary value problem reduces to a purely Dirichlet problem on such domains contained in G.
The p-Laplace equation (1.2) and the Neumann condition will be considered in the weak sense as follows: where · denotes the scalar product in R n and Recall that for an open set Ω ⊂ R n , the space C ∞ 0 (Ω) consists of all infinitely many times continuously differentiable functions with compact support in Ω, extended by zero outside Ω if needed.
In (2.1) it is implicitly assumed that the integral exists for all test functions ϕ ∈ C ∞ 0 (G \ F ). This need not be the case for a general u ∈ W 1,p loc (G \ F ).

Transforming the half-cylinder into a half-ball
In this section, we shall see that the p-Laplace operator in the open infinite circular half-cylinder G corresponds to a weighted quasilinear elliptic operator on the unit half-ball. The following change of variables was introduced in Björn [1, Section 3]. Let κ > 0 be a fixed constant and define where we adopt the notation ξ = (ξ ′ , ξ n ) = (ξ 1 , ... , ξ n−1 , ξ n ) ∈ R n , similar to x = (x ′ , x n ) ∈ R n . The mapping x → ξ = T (x) is defined on R n with values in T (R n ) = R n \ {(ξ ′ , ξ n ) ∈ R n : ξ ′ = 0 and ξ n ≤ 0}.
We will mainly consider T on G and its closure G. It is easily verified that T (G) = {ξ ∈ R n : |ξ| < 1 and ξ n > 0}, are the open and the closed upper unit half-balls, respectively, with the origin ξ = 0 removed. Note that so the point at infinity for the half-cylinder G corresponds to the origin ξ = 0. Throughout the paper we will use x for points in G, while ξ will be used for points in the target space of T . A direct calculation shows that the inverse mapping T −1 of T is given by: The following lemma is then easily proved by induction.
Note that is positive and bounded away from 0 as long as x stays within a bounded set in R n , or equivalently, as long as ξ stays away from 0 and from the negative ξ n -axis. In particular, |ξ|(|ξ| + ξ n ) > 0 in T (R n ) and hence T is a smooth diffeomorphism between R n and T (R n ). Our next step is to see how the p-Laplace equation (1.2) transforms under the diffeomorphism T . For notational purposes, we regard the differential of T as the left-multiplication of the column vector h ∈ R n by the Jacobian matrix of partial derivatives With this matrix convention, the chain rule for u andũ = u • T −1 can be written as ∇u(x) = dT * (x)∇ũ(ξ), where ξ = T (x), (3.4) dT * (x) is the transpose of the matrix dT (x) and the distributional gradients ∇u(x) and ∇ũ(ξ) are seen as column vectors in R n . Formula (3.4) clearly holds when u andũ are smooth, while for functions in L 1 loc with distributional gradients in L 1 loc it is obtained by mollification and holds a.e., see for example Ziemer [21, Theorem 2.2.2 and Section 1.6] or Hörmander [7,Section 6.1].
We shall substitute the chain rule (3.4) into equation (2.1) to obtain the corresponding integral identity on the unit half-ball T (G). A(ξ, ∇ũ) · ∇ ϕ dξ, (3.5) where A is for ξ = T (x) ∈ T (G) and q ∈ R n defined by Here, J T (x) = det(dT (x)) denotes the Jacobian of T at x.
Note that by the assumptions on u and ϕ, all the integrals are finite.
In view of the integral identity (2.1), Lemma 3.2 indicates that the p-Laplace equation (1.2) on G \ F will be transformed by T into the equation with a proper interpretation of the function spaces and the zero Neumann condition, which will be made precise later, see Proposition 4.7, Theorem 5.6 and Section 6.
In the next section, we will study the operator (3.7) in more detail. For this, we will use the following geometric lemma. Its proof is rather straightforward, but requires good control of all the involved expressions. We provide it for the reader's convenience.
Throughout the paper, unless otherwise stated, C will denote any positive constant whose real value is not important and need not be the same at each point of use. It can even vary within a line. By a b we mean that there exists a nonnegative constant C, independent of a and b, such that a ≤ Cb. We also write a ≃ b if a b a.

Lemma 3.3.
For all x, y ∈ R n , it holds that In particular, if |x ′ | ≤ M and q ∈ R n then where the comparison constants in ≃ depend on κ and M , but are independent of x and q.
Proof. Let ξ = T (x) and η = T (y). We can assume that |y ′ | ≤ |x ′ |. By (3.1) and the triangle inequality, In the above two estimates, we have 1 − |y Since the mean-value theorem shows that |e −κxn − e −κyn | ≤ κe −κ min{xn,yn} |x n − y n |, we conclude that which gives the second inequality in (3.8). Conversely, (3.2) and the triangle inequality yield because of (3.2) and (3.3). Since also where the first inequality follows from the mean-value theorem, we conclude that which proves the first inequality in (3.8).

Properties of the operator div A(ξ, ∇ũ)
We shall now study some properties of the operator div A(ξ, ∇ũ) on T (G). It will turn out to be degenerate elliptic with a degeneracy given by the weight function and w(0) = 0. This will make it possible to treat equation (3.7) using methods from Heinonen-Kilpeläinen-Martio [6].
where the comparison constants are independent of ξ and q.
Proof of Theorem 4.1. Using the above matrix notation and (3.6), we have for all q ∈ R n and ξ = T x ∈ T (G),

Lemma 3.3 then gives
which concludes the proof of the first statement. For the second statement, note that by Lemma 3.3, we have Thus by Lemma 3.3 together with (3.6), we get Theorem 4.2. The mapping A : T (G) × R n → R n , defined by (3.6), satisfies for all ξ ∈ T (G) and q 1 , q 2 ∈ R n the monotonicity condition with equality if and only if q 1 = q 2 .
Proof. Expand the left-hand side of (4.2) as where Using (4.1), we have Estimating the first term of A 2 using (3.6), together with the Cauchy-Schwarz and Young inequalities, yields Similarly, with the roles of q 1 and q 2 interchanged, the second term in A 2 is estimated as Substituting the last three estimates back into (4.3) reveals that We notice that the left-hand side is zero if and only if equality holds both in the Cauchy-Schwarz and Young inequalities, from which it is easily concluded that this requires dT * (x)q 1 = dT * (x)q 2 and thus q 1 = q 2 , since dT * (x) is invertible.
The following lemma is well-known, cf. Heinonen-Kilpeläinen-Martio [6, p. 298]. For the reader's convenience, we include the short proof. Here and in the rest of the paper, we let denote the open ball centred at the origin and with radius r > 0.
where C α,p,n = nω n n + α(p − n) and ω n is the Lebesgue measure of the unit ball in R n .
Moreover, w belongs to the Muckenhoupt class A p , that is, for all balls B ⊂ R n ,

5)
where |B| stands for the Lebesgue measure of B.
Proof. Estimate (4.4) is easily obtained by direct calculation using spherical coordinates. To prove (4.5), we let B = B(ζ, r) be a ball and consider two cases: If r < 1 2 |ζ|, then w(ξ) ≃ w(ζ) for all ξ ∈ B and hence the left-hand side in (4.5) is comparable to On the other hand, if r ≥ 1 2 |ζ|, then B ⊂ B 3r and hence, by the first part of the lemma with α = 1 and Note that α(n − p) < n for both choices of α.
Weights from the Muckenhoupt class A p are known to be p-admissible, i.e. the measure dµ(ξ) = w(ξ) dξ is doubling and supports a p-Poincaré inequality on R n , see Heinonen-Kilpeläinen-Martio [6,Chapters 15 and 20]. Such measures are suitable for the theory of Sobolev spaces and partial differential equations, as developed in [6].
Similarly, H 1,p (Ω, w) is the completion of the set In other words, a function u belongs to H 1,p (Ω, w) if and only if u ∈ L p (Ω, w) and there is a vector-valued function v such that for some sequence of smooth , we know from [6, Section 1.9] that v = ∇u is the distributional gradient of u. Moreover, by Kilpeläinen [9], u ∈ H 1,p (Ω, w) if and only if both u and its distributional gradient ∇u belong to L p (Ω, w). For the unweighted Sobolev space with w ≡ 1, we use the notation W 1,p (Ω).
We can now make more precise the statement that the p-Laplace equation on G \ F transforms into the equation (3.7). First, we formulate the following simple consequence of the estimates in Lemma 3.3, which will also be useful later when dealing with function spaces on G and T (G), and when comparing capacities.
with comparison constants depending on R but independent of A and u.
Note that, in general, the above integrals can be infinite, but then they are infinite simultaneously.
Proof. As in Lemma 3.2, we use the change of variables ξ = T (x). The chain rule (3.4), together with Lemma 3.3 and e −κxn = |ξ|, implies that Proof. Using Lemma 4.6, we conclude that u ∈ W 1,p loc (G \ F ) if and only ifũ ∈ H 1,p loc (T (G \ F ), w), since e −pκxn ≃ 1 for every compact subset of G \ F . We need to show that u satisfies the integral identity in (2.1) for all test functions ϕ ∈ C ∞ 0 (G\F ) if and only ifũ satisfies , then implies that the integral identity in (2.1) becomes (4.6). In order to be able to use the theory of Dirichlet problems, developed for weighted elliptic equations in Heinonen-Kilpeläinen-Martio [6], the part of the boundary, where the Neumann data are prescribed, will be eliminated by reflection in the hyperplane {ξ ∈ R n : ξ n = 0}.

Removing the Neumann data
More precisely, consider the reflection mapping and let the open set D consist of T (G \ F ), together with its reflection P T (G \ F ) and the "Neumann" part of the boundary T (∂G \ F ) added, that is, Clearly, F is closed and hence D is open. We recall that T maps the base B ′ × {0} of G onto the upper unit half-sphere {ξ ∈ ∂B 1 : ξ n > 0} and that the point at infinity in G corresponds to the origin ξ = 0. In particular, since we assume that B ′ × {0} ⊂ F and F is closed, we have ∂D ⊂ F . Hence, the whole boundary ∂D will carry a Dirichlet condition. Now, let T = P • T represent the map from the open circular half-cylinder G to the lower unit half-ball {ξ ∈ B 1 : ξ n < 0}. We extend A(ξ, q) from T (G) to the whole unit ball B 1 as follows: Clearly, Lemma 3.3 holds with T replaced by T as well. It then immediately follows from Theorems 4.1 and 4.2 that A satisfies the ellipticity and monotonicity assumptions (3.3)-(3.7) from Heinonen-Kilpeläinen-Martio [6] in the whole unit ball B 1 . The above reflection makes it possible to remove the Neumann boundary data on T (∂G \ F ) and obtain an equivalence with a Dirichlet problem on D. First, we make a suitable identification of the function spaces. Note that Lemmas 3.2 and 4.6 clearly hold also with T replaced by T .
We alert the reader that the L 1,p κ (G \ F )-norm also includes the function v, not only its gradient, and because of the weight e −pκxn it differs from the standard Sobolev norm. Also note that functions in L 1,p κ,0 (G \ F ) are required to vanish on F (in the Sobolev sense), but not on the rest of the lateral boundary ∂G \ F . For t ≥ 0, we let Note that the truncated cylinder G t is open at its base B ′ × {t}, but contains the lateral boundary ∂B ′ × (t, ∞). Proof.
Since v can be approximated in the L 1,p κ (G \ F ) norm by its truncations v k := min{k, max{v, −k}} at levels ±k, we can without loss of generality assume that v is bounded and |v| ≤ 1.
The following result relates the space L 1,p κ (G \ F ) to the weighted Sobolev space on D. We shall write Moreover, the functionũ Remark 5.4. In Lemma 7.6, we shall see that the origin 0 has zero (p, w)-capacity in H 1,p (B(0, 1), w) and hence for bounded F , we also have In particular, this applies when F = B ′ × {0} is the base of G, and thus F = ∂B 1 ∪ {0}.
We also need to compare the spaces of test functions. Clearly, if ϕ ∈ C ∞ 0 (D) then ϕ • T ∈ C ∞ 0 (G \ F ), by Lemma 3.1. For Sobolev functions with zero boundary values, we have the following statement.
Proof. To prove the first statement, choose a sequence ϕ j ∈ C ∞ 0 (D) such that For solutions in L 1,p κ (G \ F ), we are now able to remove the zero Neumann condition and transfer the mixed boundary value problem for (1.2) in G \ F to a Dirichlet problem in D.
Theorem 5.6. Assume that u ∈ L 1,p κ (G \ F ) is a weak solution of ∆ p u = 0 in G \ F with zero Neumann boundary data on ∂G \ F , i.e. (2.1) holds. Letũ be as in (5.4).
Hence, Theorem 5.6 also implies that lim x→x0 u(x) exists and is finite for every x 0 belonging to the Neumann boundary ∂G \ F .

Existence of solutions
In this section, we shall prove the existence of weak solutions to equation (1.2) in G \ F with zero Neumann boundary data on ∂G \ F and prescribed continuous Dirichlet boundary data u = f on This will be done using uniform approximations by Lipschitz boundary data from the space L 1,p κ (G \ F ). Let therefore f ∈ L 1,p κ (G \ F ) and letf be defined as in (5.4). By Theorems 3.17 and 3.70 in Heinonen-Kilpeläinen-Martio [6], there is a unique continuous weak solutionũ ∈ H 1,p (D, w) of div(A(ξ, ∇ũ)) = 0 (6.1) with boundary dataf in the sense thatũ −f ∈ H 1,p 0 (D, w). We shall show that u =ũ • T satisfies (2.1) and that u − f ∈ L 1,p κ,0 (G \ F ). To do this, we use the integral formulation D A(ξ, ∇ũ) · ∇ ϕ dξ = 0 for all test functions ϕ ∈ C ∞ 0 (D) (6.2) and split the left-hand side into integrals over D+ and D−. We shall see that the corresponding integrals are the same and that each is zero. For this, we prove that u =ũ • P is also a solution of (6.1) withū −f ∈ H 1,p 0 (D, w), and thusũ =ũ • P , by uniqueness. The following identity obtained from (5.1) will be useful, namely A(ξ, P q) = P A(P ξ, q) (6.3) for all q ∈ R n and all ξ ∈ B 1 . First, we prove the following symmetry result.
We shall now use uniform approximations to treat continuous boundary data on F 0 . Suppose that f ∈ C(F 0 ) and, if F 0 is unbounded, also that the limit exists and is finite. Replacing f by f − f (∞), we can assume without loss of generality that f (∞) = 0. We then find a sequence of compactly supported Lipschitz functionsf k : F 0 → R such that for k = 1, 2, ... , By the McShane-Whitney extension theorem (see Heinonen [5, Theorem 2.3]), there exist Lipschitz functions f k : G → R such that f k | F0 =f k . The Lipschitz constant of f k is preserved whenf k is extended to G. Multiplying f k by a cut-off function, if necessary, we may assume that f k has compact support.
Theorem 6.4. Let f ∈ C(F 0 ) and, if F 0 is unbounded, assume also that the limit in (6.6) is zero. Let {f k } ∞ k=1 be a sequence of compactly supported Lipschitz functions on G such that for all k = 1, 2, ... , Let u k ∈ L 1,p k (G \ F ) be the unique continuous weak solution of (2.1) with u k − f k ∈ L 1,p κ,0 (G \ F ), provided by Theorem 6.3. Then u k converge uniformly in G \ F and the function u := lim k→∞ u k is a bounded continuous weak solution of the p-Laplace equation (1.2) in G \ F with zero Neumann boundary data on ∂G \ F , in the sense of (2.1).
By the comparison principle [6,Lemma 3.18], the sequenceũ k + 3 · 2 −k is decreasing to a functionũ in D, while the sequenceũ k − 3 · 2 −k is increasing toũ. Clearly, the convergence is uniform. Sincef k are bounded, so areũ k by the maximum principle. Hence also the functions u k converge uniformly to the bounded continuous function u =ũ • T in G \ F . The Harnack convergence theorem [6, Theorem 6.13] implies thatũ is a solution of div(A(ξ, ∇ũ)) = 0 in D. In particular, u ∈ H 1,p loc (D, w) and (6.2) holds for all ϕ ∈ C ∞ 0 (D) and, by a density argument, also for all ϕ ∈ H 1,p (D, w) which have compact support in D. Sinceũ ∈ H 1,p loc (D, w), it follows from Lemma 4.6 that u ∈ W 1,p (U ) for every open set U ⋐ G \ F and hence u ∈ W 1,p loc (G \ F ).
Finally, we show that u satisfies (2.1). Let ϕ ∈ C ∞ 0 (G \ F ) and define ϕ as in (6.4). By Proposition 5.5, the function ϕ belongs to H 1,p 0 (D, w) and has compact support in D. As in the proof of Theorem 6.3, we can therefore conclude from Lemmas 3.2 and 6.1 that (6.5), and thus (2.1), holds for all ϕ ∈ C ∞ 0 (G \ F ), i.e. that u is a weak solution of the p-Laplace equation (1.2) in G\F with zero Neumann boundary data on ∂G \ F .
We shall now see that the function u obtained in Theorem 6.4 attains its continuous boundary data on F 0 , except for a set of zero p-capacity. The definition below follows Chapter 2 in Heinonen-Kilpeläinen-Martio [6]. Definition 6.5. Suppose that K is a compact subset of an open set Ω ⊂ R n . The variational (p, w)-capacity of K in Ω, is where the infimum is taken over all v ∈ C ∞ 0 (Ω) satisfying v ≥ 1 on K.
By a density argument, the infimum in (6.9) can equivalently be taken over all v ∈ H 1,p 0 (Ω, w) ∩ C(Ω) such that v ≥ 1 on K, see [6, pp. 27-28]. The capacity cap p, w is extended using a standard procedure to open and then to arbitrary sets, see [6, p. 27]. By Theorem 2.5 in [6], it is a Choquet capacity and for all Borel (even Suslin) sets E ⊂ Ω, Recall that the Sobolev C p -capacity is the capacity associated with the usual Sobolev space W 1,p (R n ) and is for compact sets defined as where the infimum is taken over all v ∈ C ∞ 0 (R n ) such that v ≥ 1 on K ⊂ R n , see [6, Section 2.35 and Lemma 2.36]. Similarly to cap p, w , it extends to general sets as a Choquet capacity and (6.13) Lemma 6.6. Let Z ⊂ T (G) be a set of (p, w)-capacity zero. Then C p (T −1 (Z)) = 0.
We will not need it, but it is not difficult to show that the converse of Lemma 6.6 is also true. For Z ⊂ T (G 0 ) it also follows from Lemmas 7.5 and 7.7.
Proof. Because of (6.12) and (6.13), it suffices to show that for every compact set K ⊂ T −1 (Z) there are v j ∈ C ∞ 0 (R n ) such that v j ≥ 1 on K and v j W 1,p (R n ) → 0 as j → ∞. We therefore choose a bounded open set Ω ⊃ K. Then T (K) is also compact and T (K) ⊂ Z. Moreover, T (Ω) ⊃ T (K) is a bounded open set.
Since Z is of (p, w)-capacity zero, we have cap p, w (T (K), T (Ω)) = 0 and hence we can find functions 0 ≤v j ∈ C ∞ 0 (T (Ω)) satisfyinĝ v j ≥ 1 on T (K) and The Poincaré inequality [6, (1.5)] implies that also where the constant C depends on T (Ω). Now, because T is a smooth diffeomorphism by Lemma 3.1, letting v j =v j • T provides us with functions 0 ≤ v j ∈ C ∞ 0 (Ω) such that v j ≥ 1 on K.
Lemma 4.6, together with the fact that e −pκxn ≃ 1 on the bounded set Ω, implies that with comparison constants depending on Ω. Thus, K (and consequently T −1 (Z)) has zero Sobolev C p -capacity. Let Z ⊂ ∂D be the set of irregular boundary points for the equation (6.1). The Kellogg property [6, Theorem 8.10] and Lemma 6.6 imply that the set Z 0 := T −1 (Z ∩ T (G)) has zero Sobolev C p -capacity.
It remains to show that (1.4) holds for all x 0 ∈ F 0 \ Z 0 . Let therefore ε > 0 be arbitrary. Then ξ 0 := T (x 0 ) = 0 is a regular boundary point of D for the equation (6.1). Recall from Theorem 6.4 and its proof that u =ũ • T , whereũ is the uniform limit of solutionsũ k to (6.1) in D with Lipschitz boundary dataf k such thatf k →f uniformly on ∂D, wheref is defined in terms of f as in (6.8). Thus, we can find k so that Since ξ 0 is a regular boundary point for (6.1), there is a neighbourhood V ⊂ T (R n ) of ξ 0 such that |ũ k −f k (ξ 0 )| < ε in V ∩ D. The triangle inequality then implies that for all Since ε > 0 was arbitrary, this shows that (1.4) holds. Finally, the continuity ofũ in D shows that the limit lim x→x0 u(x) exists and is finite for every x 0 ∈ ∂G \ F . The proof of Theorem 6.4, together with (6.14), also leads to the following comparison principle. Corollary 6.7. If f, h ∈ C(F 0 ) and f ≤ h, then the corresponding continuous weak solutions u and v, provided by Theorem 1.1, satisfy u ≤ v in G \ F .
Proof. By (6.7), the functions f k and h k , uniformly approximating f − f (∞) and h − h(∞) in Theorem 6.4, satisfy for all k = 1, 2, ... , The comparison principle in Theorem 6.3 then shows that also the continuous so- Letting k → ∞, together with (6.14), concludes the proof.

Capacity estimates
In this section we compare the variational capacity cap p, w from Definition 6.5 with a new variational capacity defined on the cylinder G and adapted to the mixed boundary value problem. These capacities will play an essential role for the boundary regularity at infinity. Recall from (5.2) that for t ≥ 0, G t := {x ∈ G : x n > t} = B ′ × (t, ∞). Note that G t contains the lateral boundary, but not the base B ′ × {t}, of the truncated cylinder B ′ × (t, ∞). It can also be written as The results from the previous sections concerning function spaces on G \ F are therefore available for G t by replacing F with is the upper half of the ball B r , with the origin removed, where r = e −κt . Inspired by (6.9), we define the following variational p-capacity on G t .
where the infimum is taken over all functions v ∈ L 1,p It follows directly from the definition that cap p,Gt is an outer capacity, i.e. for every E ⊂ G t , It is also clearly a monotone set function, i.e. cap p,Gt (E 1 ) ≤ cap p,Gt (E 2 ) whenever also follows directly by considering the function max{v 1 , v 2 }, with v j admissible for cap p,Gt (E j ), j = 1, 2. By truncation, the admissible functions v in (7.1) can be assumed to satisfy 0 ≤ v ≤ 1. As in [6, pp. 27-28], the following approximation argument allows us to test the capacity of compact sets with smooth admissible functions. Recall the definition of C ∞ 0 (G \ Q t ) and L 1,p κ,0 (G \ Q t ) in (2.2) and Definition 5.1. That is, in (7.1) we have v(x) = 0 when x n ≤ t and when x n is sufficiently large, but there is no such requirement on the lateral boundary of G t . Lemma 7.2. If K ⊂ G t is compact, then the infimum in (7.1) can equivalently be taken over all Proof. Denote the latter infimum by I.
Then it is easily verified that the functions belong to C ∞ 0 (G \ Q t ) and satisfy u j = 1 on K. We therefore have , where C depends on U and η. Letting j → ∞ and then taking infimum over all v admissible in the definition of cap p,Gt (K) shows one inequality. The opposite inequality is straightforward.
For monotone sequences of sets, the capacity cap p,Gt has the following continuity properties, which show that it is a Choquet capacity. Proof. This follows immediately from the monotonicity of cap p,Gt and from (7.2) since for every open U ⊃ K, there is some j such that K j ⊂ G ∩ U and hence Proof. The proof follows the arguments from Kinnunen-Martio [11]. The inequality lim j→∞ cap p,Gt (E j ) ≤ cap p,Gt (E) follows immediately from the monotonicity of cap p,Gt . For the opposite inequality, let u j ∈ L 1,p We can assume that lim j→∞ cap p,Gt (E j ) < ∞ and hence the sequence u j is bounded in L 1,p κ,0 (G \ Q t ). Since the space L p (G t , e −pκxn dx) × L p (G t ) is reflexive, there is a subsequence (also denoted u j ) such that (u j , ∇u j ) → (u, ∇u) weakly in L p (G t , e −pκxn dx) × L p (G t ).
Mazur's lemma (see e.g. [6,Lemma 1.29]) applied to each of the subsequences It follows that w j : So it is admissible in the definition of cap p,Gt (E) and hence Letting j → ∞ concludes the proof. We shall now compare the two variational capacities cap p, w and cap p,Gt .
Lemma 7.5. There exist constants C ′ , C ′′ > 0, independent of t ≥ 0, such that for all Borel sets E ⊂ G t , where E = T (E) ∪ T (E) and r = e −κt .
Proof. To prove the first inequality, letv ∈ H 1,p 0 (B r , w) ∩ C(B r ) be such thatv ≥ 1 on E. By considering the open sets {ξ ∈ B r :v(ξ) > 1 − ε} and letting ε → 0, we can assume thatv ≥ 1 in an open neighbourhood of E. Letting v =v • T on G t , we have from Proposition 5.5 and Lemma 4.6 that v ∈ L 1,p κ,0 (G \ Q t ) and Gt Since v is admissible for cap p,Gt (E), taking infimum over allv admissible in the definition of cap p, w ( E, B r ) proves the first inequality in the lemma. For the second inequality, we need continuous test functions in (6.9). Let therefore ε > 0 and using (6.10) choose a compact set K ⊂ E such that and cap p, w ( K, B r ) > 1/ε otherwise. Replacing K by its symmetrization K ∪ P ( K) and noting that 0 / ∈ E, we can assume that K = T (K) ∪ T (K) for some compact set K ⊂ E. Now, use Lemma 7.2 to find v ∈ C ∞ 0 (G \ Q t ) satisfying v ≥ 1 on K and Gt |∇v| p dx ≤ cap p,Gt (K) + ε. (7.4) For ξ ∈ B r , defineṽ as in (6.8) with f k replaced by v. Asṽ =ṽ • P , applying Lemma 4.6 together with (7.4) then gives is admissible for cap p, w ( K, B r ) as in (6.9), we have if cap p, w ( E, B r ) < ∞, and 1/ε < cap p, w ( K, B r ) cap p,Gt (E) + ε otherwise. Letting ε → 0 completes the proof.
The following simple capacity estimates for spherical condensers will be useful when dealing with the Wiener criterion.
Lemma 7.6. For all 0 < r < R < ∞, with comparison constants depending only on n and p.
In particular, the origin 0 has zero (p, w)-capacity.
The equality for B r follows from [6, (6.40)]. For (7.5) note that the function is admissible for the condenser (B r , B R ). The change of variables ρ = |ξ| yields The last statement follows by letting r → 0.
We end this section with the following two lemmas which give a comparison between the capacities C p from (6.12) and cap p,Gt−1 from (7.1).
Lemma 7.7. Let 0 ≤ s < t and E ⊂ G t . Then cap p,Gs (E) min{1, C p (E)} with the comparison constant depending on t − s.

Boundary regularity at ∞
The solution u of the mixed boundary value problem (2.1), obtained in Section 6, is continuous in G \ F and at the Neumann boundary ∂G \ F . If f ∈ C(F 0 ), then u is also continuous at the Dirichlet boundary F 0 , except for a set of zero C p -capacity. We shall now study its continuity at ∞.
Recall that F is a closed subset of G, containing the base B ′ × {0}, and that If F is bounded, then the solutionũ ∈ H 1,p loc (D, w) of (6.1) in D, constructed in the proofs of Theorems 6.3 and 6.4, belongs to H 1,p (D, w) (when f ∈ L 1,p κ (G \ F )) or is bounded (when f ∈ C(F 0 )). Since the origin 0 has zero (p, w)-capacity by Lemma 7.6, the removability results [6, Lemma 7.33 and Theorem 7.36] imply that u is a solution of (6.1) in Ω = D ∪ {0}. Consequently,ũ is continuous at ξ = 0 and it follows that the limit lim G\F ∋x→∞ u(x) always exists when F is bounded.
Definition 8.1. Assume that F is unbounded. We say that the point at ∞ is regular for the mixed boundary value problem (2.1) in G \ F with zero Neumann data on ∂G \ F if for all Dirichlet boundary data f ∈ C(F 0 ) with a finite limit  where A is as in (3.6).
Proof. First, assume that 0 is regular with respect to the equation (8.3) and let f ∈ C(F 0 ) be such that the limit in (8.1) is finite. Also, assume without loss of generality that f (∞) = 0.
Conversely, assume that ∞ is regular for the mixed boundary value problem (2.1) and letf ∈ H 1,p (D, w) ∩ C(D) be arbitrary. We shall show that the solution u of (8.3) in D with boundary dataf satisfies lim ξ→0û (ξ) =f (0).
Finally, sinceû j =û j • P by Corollary 6.2, it follows that Note thatf 1 ≤f ≤f 2 and henceû 1 ≤û ≤û 2 , from which we conclude that where the last inequality follows as in the proof of [6,Lemma 2.16]. Inserting this into (8.5) shows that (8.5) is equivalent to We will now further rewrite this condition to better match the transformation T −1 back to the cylinder G. Proof. One implication is clear since the integral in (8.5) majorizes the one in (8.7). Conversely, use the subadditivity to majorize the integral in (8.5) by a sum of two integrals, one for each of the sets in the right-hand side of (8.8) as follows. For all 0 < δ < 1, we have The first integral on the right-hand side of (8.9) is for all δ > 0 majorized by the integral in (8.7). For the second integral, we use the change of variables ρ = r α , together with the fact that and estimate it as For α > 2 1/(p−1) , the last integral in (8.10) can be subtracted from the integral on the left-hand side of (8.9). The remaining integral on the right-hand side in (8.10) is estimated using (7.5) as follows, Inserting this into (8.10) and (8.9), together with letting δ → 0, shows that with C independent of α. We therefore conclude that the integral in (8.5) is finite whenever the one in (8.7) is finite. Thus (8.5) and (8.7) are equivalent. Proof. We already know that (8.7) is equivalent to (8.5) and thus to (8.6). The integral in (8.6) clearly majorizes the integral in (8.11) and so (8.11) implies (8.7). Conversely, we can without loss of generality assume that α = 2 m for some integer m ≥ 1, and hence r α ≤ r 2 . By the subadditivity and monotonicity of cap p, w , see [6, Theorem 2.2], we have for all 0 < r < 1, Finally, we prove the following concrete Wiener criterion for the boundary regularity at ∞ for the mixed boundary value problem (2.1) in G \ F , stated as (1.5) in the introduction. . By Lemma 7.6, the Wiener criterion at 0 ∈ ∂D reduces to (8.5). Lemma 8.3 shows that (8.5) is for α > 2 1/(p−1) equivalent to (8.7), which is in turn equivalent to (8.11), by Lemma 8.4. What now remains is to show that the integral in (8.11) diverges if and only if the one in (8.13) does.
As in [6,Lemma 2.16], it can be shown that replacing the ball B 2r in (8.11) by B λr , for any λ > 1, results in an integral that is comparable to the one in (8.11). The convergence of the integral is thus not influenced by the change to B λr . In particular, we can take λ = e κ . Moreover, since by Lemma 7.6, To finish the proof, apply the change of variables r = e −κt and the fact that the ball B r , 0 < r ≤ e −κ , corresponds to the truncated cylinder G t := {x ∈ G : x n > t}, with t = − 1 κ log r ≥ 1.
Also note that with this notation, B r 2 and B e κ r correspond to G 2t and G t−1 , respectively. Comparing cap p, w and cap p,Gt−1 using Lemma 7.5, it therefore follows that the regularity condition (8.14) is equivalent to ∞ 1 cap p,Gt−1 (F ∩ (G t \ G 2t )) 1/(p−1) dt = ∞.
Theorem 8.5 and Remark 8.6 therefore imply that ∞ is irregular for the mixed boundary value problem (2.1) in G \ F with F defined as in (8.15).
Theorem 8.5 and Remark 8.6 show that ∞ is regular for the mixed boundary value problem (2.1) in G \ F with F defined as in (8.17).
A particular example of this situation is K j = E × [j, j + 1], where E ⊂ B ′ is any nonempty closed set of Hausdorff dimension dim H (E) > n − 1 − p.
Then dim H (E × [0, 1]) > n − p, which implies that C p (K j ) = C p (E × [0, 1]) > 0, by e.g. Ziemer [21, Remark 2.6.15 and Theorem 2.6.16]. If p > n then singletons have positive capacity and it is thus sufficient for regularity at infinity that F is unbounded.