Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 20, 1995, 315–326

TANGENTIAL LIMITS OF MONOTONE SOBOLEV FUNCTIONS

Yoshihiro Mizuta

Hiroshima University, Faculty of Integrated Arts and Sciences The Division of Mathematical and Information Sciences Higashi-Hiroshima 724, Japan; mizuta@mis.hiroshima-u.ac.jp

Abstract. Tangential limits have been discussed by several authors for harmonic functions with finite Dirichlet integral. This paper deals mostly with tangential limits for monotone functions in the half space of R^{n}, which are extensions of monotone functions on the one dimensional space R^{1} . Harmonic functions together with solutions in a wider class of nonlinear elliptic equations are monotone in our sense; of course, the coordinate functions of quasiregular mappings are monotone. We first give the fine limit result for Sobolev functions, and then apply the estimate of the oscillations over balls by the p-th means of partial derivatives over balls.

1. Introduction

Our aim in this paper is to study tangential boundary limits of monotone functions u with finite Dirichlet integral in the half space R_{+}^{n} = {x = (x_{1}, …, x_{n−1}, x_{n}) : x_{n} > 0}, n ≧ 2. We say that u has finite Dirichlet integral if

(1)  ∫_{R_{+}^{n}} |grad u(x)|^{n} dx < ∞.

Further we say that a continuous function u on R_{+}^{n} is monotone (in the sense of Lebesgue) if

max_{\overline{G}} u = max_{∂G} u  and  min_{\overline{G}} u = min_{∂G} u

hold for any relatively compact open set G in R_{+}^{n}, where \overline{G} = G ∪ ∂G (see Vuorinen [22], [23]). Harmonic functions, (weak) solutions in a wider class of (non)linear elliptic partial differential equations and the coordinate functions of quasiregular mappings are monotone (see e.g. Gilbarg–Trudinger [5], Heinonen– Kilpeläinen–Martio [6], Reshetnyak [19], Serrin [20] and Vuorinen [23]). For γ ≧ 1, ξ ∈ ∂R_{+}^{n} and a > 0, consider the set

T_{γ}(ξ; a) = {x = (x_{1}, …, x_{n}) ∈ R_{+}^{n} : |x − ξ|^{γ} < ax_{n}}.

1991 Mathematics Subject Classification: Primary 31B25.

If lim_{x→ξ,x∈T_{γ}(ξ,a)} u(x) = ℓ for every a > 0, then u is said to have a T_{γ} -limit ℓ at ξ; u is said to have a nontangential limit at ξ if it has a T_{1} -limit at ξ. We say further that u has a T_{∞} -limit ℓ at ξ ∈ ∂R_{+}^{n} if

lim_{x→ξ,x∈T_{γ}(ξ,a)} u(x) = ℓ

for every γ > 1 and a > 0 (cf. [14]). To evaluate the size of exceptional sets, we use the capacity

C_{1,p}(E; G) = inf ∥f∥_{p}^{p},

where G is an open set in R^{n} and the infimum is taken over all nonnegative measurable functions f such that f = 0 outside G and

∫ |x − y|^{1−n}f(y) dy ≧ 1 for every x ∈ E;

see [8] for the basic properties of capacity. Since C_{1,p}(E; R^{n}) = 0 for any set E when p ≧ n, we write C_{1,p}(E) = 0 simply if

C_{1,p}(E ∩ G; G) = 0 for every bounded open set G.

In case 1 < p < n, C_{1,p}(E) = 0 if and only if C_{1,p}(E; R^{n}) = 0.

Our main aim in this paper is to establish the following theorem.

Theorem 1. If u is a monotone function on R_{+}^{n} satisfying (1), then u has a finite T_{∞} -limit at every boundary point except for a set E ⊂ ∂R_{+}^{n} such that C_{1,n}(E) = 0.

The nontangential case for harmonic functions has been dealt by many mathematicians (cf. Beurling [1], Carleson [2], Gavrilov [4], Wallin [24] and the author [11]). Miklyukov [10] discussed the nontangential limits for quasiregular mappings with finite Dirichlet integral. Recently, Manfredi and Villamor [7] have proved the existence of nontangential limits for monotone functions on the unit ball. The present tangential limit result for harmonic functions was obtained by Cruzeiro [3].

It is well-known (through an application of change of variables) that the coordinate functions of bounded quasiconformal mappings defined on R_{+}^{n} have finite Dirichlet integral, so that Theorem 1 gives the following result (see Väisälä [21] and Vuorinen [23] for the definition of quasiconformal mappings).

Corollary 1. Every coordinate function of bounded quasiconformal mappings on R_{+}^{n} has a finite T_{∞} -limit at every boundary point except for a set E ⊂ ∂R_{+}^{n} such that C_{1,n}(E) = 0.

This corollary gives an affirmative answer to the open problem given by Vuorinen [23, 15.16].

The key of proving our theorem mentioned above is the fact that if u is monotone on the ball B(x, r) centered at x with radius r and n − 1 < p ≦ n, then

|u(x) − u(y)|^{p} ≦ Mr^{p−n} ∫_{B(x,r)} |grad u(z)|^{p} dz

whenever y ∈ B(x, r/2), with a positive constant M independent of r (see [7, Remark, p. 9] and [23, Section 16]). In the author’s paper [12], we used this inequality to study the existence of nontangential limits of weak solutions for nonlinear Laplace equations.

We also give an improvement of the result by Manfredi and Villamor [7] concerning the Lindelöf-type theorem.

Theorem 2. Let u be a monotone function on R_{+}^{n} satisfying

(2)  ∫_{R_{+}^{n}} |grad u(x)|^{p} dx < ∞

for n − 1 < p ≦ n. Consider the set

E_{p} = {ξ ∈ ∂R_{+}^{n} : lim sup_{r→0} r^{p−n} ∫_{R_{+}^{n}∩B(ξ,r)} |grad u(y)|^{p} dy > 0}.

If ξ ∈ ∂R_{+}^{n} − E_{p} and u has a finite limit along a rectifiable curve in R_{+}^{n} ending to ξ, then u has a nontangential limit at ξ.

It is easy to see that E_{p} has (n − p)-dimensional Hausdorff measure zero; in case p = n, the exceptional set in this discussion is empty as was also pointed out in [7, Theorem 2], because E_{n} = ∅. Note further that C_{1,p}(E_{p}) = 0.

The author wishes to express his deep gratitude to the referee for his kind and valuable suggestions.

2. Preliminary lemmas

Throughout this paper, let M denote various constants independent of the variables in question.

To represent Sobolev functions in the integral form, we use the kernel functions

k_{j}(x, y) = \{ \begin{matrix}(x_{j} − y_{j})|x − y|^{−n}, & y ∈ B(0, 1), \\ (x_{j} − y_{j})|x − y|^{−n} − (−y_{j})|y|^{−n}, & y ∈ R^{n} − B(0, 1). \end{matrix}

Lemma 1 (cf. [16, Lemma 3]). Let u be a continuous function on R_{+}^{n} satisfying (2). Then there exist functions u_{j} ∈ L^{n}(R^{n}) and a constant C for which

u(x) = ∑_{j=1}^{n} ∫_{R^{n}} k_{j}(x, y)u_{j}(y) dy + C

for every x ∈ R_{+}^{n}.

In view of Lemma 1, we have for x ∈ B(0, N) ∩ R_{+}^{n}, N > 0,

(3)  u(x) = ∑_{j=1}^{n} ∫_{B(0,2N)}(x_{j} − y_{j})|x − y|^{−n}u_{j}(y) dy + v_{N}(x)

with a continuous function v_{N} on B(0, N). For a function u satisfying (1), set

A_{1} = {ξ ∈ ∂R_{+}^{n} : ∫_{B(ξ,1)} |ξ − y|^{1−n}f(y) dy = ∞}

and

A_{2} = {ξ ∈ ∂R_{+}^{n} : lim sup_{r→0}[log(1/r)]^{n−1} ∫_{B(ξ,r)} f(y)^{n} dy > 0},

where f = \sqrt{|u_{1}|^{2} + ⋯ + |u_{n}|^{2}} .

The following is easy.

Lemma 2. Let u be a continuous function on R_{+}^{n} satisfying (1). Then

C_{1,n}(A_{1}) = 0.

If h(r) = [log(2 + 1/r)]^{1−n}, then A_{2} has zero Hausdorff measure with respect to the measure function h, that is,

H_{h}(A_{2}) = 0.

In view of [9], we find

Lemma 3. Let u be a continuous function on R_{+}^{n} satisfying (1). Then

C_{1,n}(A_{2}) = 0.

Lemma 4 (cf. [14, Theorem 2′ and Remark 1]). Let u be a continuous function on R_{+}^{n} satisfying (1). If ξ ∈ ∂R_{+}^{n} − (A_{1} ∪ A_{2}), then there exists a set E ⊂ R_{+}^{n} such that

(i) lim_{x→ξ,x∈T_{γ}(ξ,a)−E} u(x) exists and is finite for any γ > 1 and any a > 0; (ii) lim_{j→∞} j^{n−1}C_{1,n}(E_{j}; B_{j}) = 0, where E_{j} = {x ∈ E : 2^{−j} ≦ |x| < 2^{−j+1}} and B_{j} = {x ∈ R^{n} : 2^{−j−1} < |x| < 2^{−j+2}}.

Proof. Let ξ ∈ B(0, N) ∩ ∂R_{+}^{n}. If we note (3), then we may assume that

u(x) = ∑_{j=1}^{n} ∫_{B(0,2N)}(x_{j} − y_{j})|x − y|^{−n}u_{j}(y) dy

for every x ∈ R_{+}^{n}. The right-hand side is denoted by \overline{u}(x) for x ∈ R^{n}, at which the integrals are convergent in absolute value. For x ∈ R_{+}^{n}, write

\begin{aligned}U_{1}(x) & = ∑_{j=1}^{n} ∫_{B(0,2N)−B(ξ,2|x−ξ|)} (x_{j} − y_{j})|x − y|^{−n}u_{j}(y) dy, \\ U_{2}(x) & = ∑_{j=1}^{n} ∫_{B(ξ,2|x−ξ|)−B(x,x_{n}/2)}(x_{j} − y_{j})|x − y|^{−n}u_{j}(y) dy, \\ U_{3}(x) & = ∑_{j=1}^{n} ∫_{B(x,x_{n}/2)}(x_{j} − y_{j})|x − y|^{−n}u_{j}(y) dy. \end{aligned}

Since ξ ∉ A_{1} , \overline{u}(ξ) is finite, so that we apply Lebesgue’s dominated convergence theorem to see that

lim_{x→ξ} U_{1}(x) = \overline{u}(ξ).

As before, set

f = \sqrt{|u_{1}|^{2} + ⋯ + |u_{n}|^{2}}.

By Hölder’s inequality we have for x ∈ B(ξ, 1/2) ∩ R_{+}^{n}

|U_{2}(x)| ≦ M([log(4|x − ξ|/x_{n})]^{n−1} ∫_{B(ξ,2|x−ξ|)} f(y)^{n} dy)^{1/n},

so that, since ξ ∉ A_{2} ,

lim_{x→ξ,x∈T_{γ}(ξ,a)} U_{2}(x) = 0.

For a sequence {b_{j}} of positive numbers, we consider the sets

E_{j} = {x : 2^{−j} ≦ |x| < 2^{−j+1}, |U_{3}(x)| ≧ b_{j}^{−1}}, j = 1, 2, …,

and

E = ⋃_{j=1}^{∞} E_{j}.

Note that for x ∈ E_{j},

|U_{3}(x)| ≦ ∫_{B(x,x_{n}/2)} |x − y|^{1−n}f(y) dy ≦ ∫_{B_{j}} |x − y|^{1−n}f(y) dy.

Consequently we have by the definition of capacity

C_{1,n}(E_{j}; B_{j}) ≦ b_{j}^{n} ∫_{B_{j}} f(y)^{n} dy.

Since ξ ∉ A_{2} , we can find a sequence {b_{j}} of positive numbers such that lim_{j→∞} b_{j} = ∞ but

lim_{j→∞} b_{j}^{n}j^{n−1} ∫_{B_{j}} f(y)^{n} dy = 0.

Now it follows that

lim_{j→∞} j^{n−1}C_{1,n}(E_{j}; B_{j}) = 0.

On the other hand, we see that

lim sup_{x→0,x∈R_{+}^{n}−E} |U_{3}(x)| ≦ lim sup_{j→∞} b_{j}^{−1} = 0.

Thus Lemma 4 is established.

Lemma 5 (cf. [18, Lemma 7.3]. If x ∈ T_{γ}(ξ, a) and x_{n} < 1/2, then

C_{1,n}(B(x, x_{n}/2); B(ξ, 2|x − ξ|)) ∼ [log(4|x − ξ|/x_{n})]^{1−n}.

Proof. For our later use, we shall show only that

C_{1,n}(B(x, x_{n}/2); B(ξ, 2|x − ξ|)) ≧ M[log(4|x − ξ|/x_{n})]^{1−n}.

For this purpose, take a nonnegative function f such that f = 0 outside B(ξ, 2|x− ξ|) and

∫ |z − y|^{1−n}f(y) dy ≧ 1 for every z ∈ B(x, x_{n}/2).

Then we have by Fubini’s theorem and Hölder’s inequality

∫_{B(x,x_{n}/2)} dz ≦ ∫_{B(x,x_{n}/2)}(∫ |z − y|^{1−n}f(y) dy) dz
= ∫ (∫_{B(x,x_{n}/2)} |z − y|^{1−n} dz)f(y) dy
≦ Mx_{n}^{n} ∫_{B(x,3|x−ξ|)}(x_{n} + |x − y|)^{1−n}f(y) dy
≦ Mx_{n}^{n}[log(3|x − ξ|/x_{n})]^{(n−1)/n}∥f∥_{n},

which yields the required inequality.

3. Proof of Theorem 1

For ξ ∈ ∂R_{+}^{n} − (A_{1} ∪ A_{2}), take a set E as in Lemma 4. Since u is monotone on R_{+}^{n},

(4)  |u(x) − u(y)|^{n} ≦ M ∫_{B(x,x_{n})} |grad u(z)|^{n} dz

whenever y ∈ B(x, x_{n}/2), where x = (x_{1}, …, x_{n}) ∈ R_{+}^{n} (see [7] and [23]). If x ∈ T_{γ}(ξ, a), then Lemma 5 implies that B(x, x_{n}/2) − E is not empty, so that there exists y(x) ∈ B(x, x_{n}/2) − E (when x_{n} is small enough). Then we see from (4) that

lim_{x→ξ,x∈T_{γ}(ξ,a)}|u(x) − u(y(x))| = 0.

Hence it follows that

lim_{x→ξ,x∈T_{γ}(ξ,a)} u(x) = lim_{x→ξ,x∈T_{γ}(ξ,a)} u(y(x)),

so that the limit on the left exists and is finite. Thus E = A_{1} ∪ A_{2} has all the required properties, with the aid of Lemmas 2 and 3.

4. Proof of Theorem 2

Manfredi and Villamor [7] proved the following result concerning the existence of nontangential limits.

Theorem 3 ([7]). Let u be a monotone function on R_{+}^{n} satisfying (2) for n − 1 < p < n. Then u has a nontangential limit at every ξ ∈ ∂R_{+}^{n} − E, where C_{1,p}(E) = 0.

As will be shown soon, we may take the above E as

E = A_{1} ∪ A_{2,p},

where A_{1} is defined as after Lemma 1, that is,

A_{1} = {ξ ∈ ∂R_{+}^{n} : ∫_{B(ξ,1)} |ξ − y|^{1−n}f(y) dy = ∞}

and

A_{2,p} = {ξ ∈ ∂R_{+}^{n} : lim sup_{r→0} r^{p−n} ∫_{B(ξ,r)} f(y)^{p} dy > 0},

where f(y) = |grad u(y)|.

Now we give a proof of Theorem 2. Our end in this direction is to show that the condition that ξ ∉ A_{1} may be replaced by the existence of asymptotic values.

Proof of Theorem 2. Without loss of generality we may assume that ξ = 0. Let C be a rectifiable curve in R_{+}^{n} tending to ξ = 0, and assume that u has a finite limit along C. For 2^{−j} ≦ r < 2^{−j+1} , take a point C(r) on C ∩ ∂B(0, r). Letting e(r) = (0, …, 0, r), we see that

(5)  |u(e(r)) − u(y)|^{p} ≦ Mr^{p−n} ∫_{B(e(r),r/2)} f(z)^{p} dz

whenever y ∈ B(e(r), r/2), where f(z) = |grad u(z)|. Moreover, letting θ denote the angle between the x_{n} -axis and the vector 0x, we find

|u(y) − u(C(r))| ≦ ∫_{0}^{π} f(rΘ)r dθ

along the circular arc {rΘ} through y and C(r), for y ∈ B(e(r), r/2) ∩ ∂B(0, r). By Hölder’s inequality we have

|u(y) − u(C(r))| ≦ (∫_{0}^{π}[r^{1−(n−1)/p}(sin θ)^{−(n−2)/p}]^{p′} dθ)^{1/p′}

× (∫_{0}^{π} f(rΘ)^{p}r^{n−1} sin^{n−2} θ dθ)^{1/p}

≦ Mr^{1−(n−1)/p}(∫_{0}^{π} f(rΘ)^{p}r^{n−1} sin^{n−2} θ dθ)^{1/p},

where 1/p + 1/p′ = 1. Hence it follows that

inf_{y∈B(e(r),r/2)}|u(y) − u(C(r))|^{p} ≦ Mr^{p−n+1} ∫_{0}^{π} f(rΘ)^{p}r^{n−1} sin^{n−2} θ dθ,

so that, by considering polar coordinates with the north pole C(r)/|C(r)|, we have

inf_{y∈B(e(r),r/2)}|u(y) − u(C(r))|^{p} ≦ Mr^{p−n+1} ∫_{∂B(0,r)} f(z)^{p} dS(z).

By integrating both sides with respect to r, we obtain

∫_{2^{−j}}^{2^{−j+1}}[ inf_{y∈B(e(r)),r/2)}|u(y) − u(C(r))|^{p}]r^{n−p−1}dr ≦ M ∫_{B_{j}} f(z)^{p} dz.

Now we can find r_{j} such that 2^{−j} ≦ r_{j} < 2^{−j+1} and

inf_{y∈B(e(r_{j}),r_{j}/2)}|u(y) − u(C(r_{j}))|^{p} ≦ M2^{j(n−p)} ∫_{B_{j}} f(z)^{p} dz.

In view of (5),

|u(e(r_{j})) − u(C(r_{j}))|^{p} ≦ M2^{j(n−p)} ∫_{B_{j}} f(z)^{p} dz.

If the origin does not belong to A_{2,p}, then this implies that {u(e(r_{j}))} has a finite limit as j → ∞. Applying (5) again, we see that

|u(x) − u(e(r_{j}))|^{p} ≦ M2^{j(n−p)} ∫_{B_{j}∩T_{1}(0,2a)} |grad u(z)|^{p} dz

for all x ∈ T_{1}(0, a) with 2^{−j} ≦ |x| < 2^{−j+1} . Therefore u has a nontangential limit at the origin. Thus we have proved that u has a nontangential limit at every ξ ∈ ∂R_{+}^{n} − A_{2,p}, and the proof of Theorem 2 is completed.

Theorem 4. Let u be a monotone function on R_{+}^{n} satisfying (2) for n−1 < p < n, and let γ > 1. Then there exists E_{γ} ⊂ ∂R_{+}^{n} such that H_{γ(n−p)}(E_{γ}) = 0 and u has a finite T_{γ} -limit at every ξ ∈ ∂R_{+}^{n} − E_{γ}.

In fact we may take E_{γ} = A_{1} ∪ A_{2,p,γ}, where

A_{2,p,γ} = {ξ ∈ ∂R_{+}^{n} : lim sup_{r→0} r^{γ(p−n)} ∫_{B(ξ,r)} |grad u(y)|^{p} dy > 0}.

In the harmonic case, we refer the reader to [15].

Theorem 5. Let u be a harmonic function on R_{+}^{n} satisfying (1) and ξ ∈ ∂R_{+}^{n} − A_{2} . If u has a finite limit along a rectifiable curve in R_{+}^{n} ending to ξ, then u has a T_{∞} -limit at ξ.

To prove this theorem, let ξ = 0. It suffices to note that

|u(x) − u(e(|x|))|^{n} ≦ M[log(2|x|/x_{n})]^{n−1} ∫_{B_{j}∩R_{+}^{n}} |grad u(z)|^{n} dz

for all x ∈ T_{γ}(0, a) with 2^{−j} ≦ |x| < 2^{−j+1} (see [17, Theorem 1 and its proof]), because Theorem 2 implies that u(e(|x|)) has a limit at the origin.

5. Remarks

Remark 1. According to [15, Remark 5], for given γ > 1 and 1 < p ≦ n, we can find a harmonic function u on R_{+}^{n} satisfying (2) such that

(i) u has a nontangential limit at the origin. (ii) lim sup_{x→0,x∈T_{γ′} (0,a′)−T_{γ′} (0,a)} u(x) = ∞ for every γ′ > γ and a′ > a.

This shows that the assumption ξ ∈ ∂R_{+}^{n} − A_{2} is needed in Theorem 5.

Remark 2. We do not know whether Theorem 5 remains valid for general monotone functions or not.

Remark 3. For x^{(j)} = (2^{−j}, 0, …, 0) ∈ ∂R_{+}^{n} and 0 < r_{j} < 2^{−j−1} , consider the sets

B_{j} = [B(x^{(j)}, 2^{−j−2}s_{j}) − B(x^{(j)}, r_{j}s_{j})] − R_{+}^{n}, where s_{j} = (log \frac{1}{2^{j}r_{j}} )^{(2−n)/n}.

Suppose {r_{j}} is chosen so small that

∑_{j}(log \frac{1}{2^{j}r_{j}} )^{1−n} < ∞;

if this is the case, B = ⋃_{j} R_{+}^{n} ∩ B(x^{(j)}, r_{j}) is called C_{1,n} -thin at the origin in the sense of [13]. Taking a sequence {a_{j}} of positive numbers such that

lim_{j→∞} a_{j} = ∞

and

(6)  ∑_{j} a_{j}^{n}(log \frac{1}{2^{j}r_{j}} )^{1−n} < ∞,

we now define

f(y) = \{ \begin{matrix}a_{j}(log \frac{1}{2^{j}r_{j}} )^{−1}|x^{(j)} − y|^{−1} & when y ∈ B_{j}, \\ 0 & elsewhere, \end{matrix}

and

u(x) = ∫_{R^{n}} \frac{x_{n} − y_{n}}{|x − y|^{n}} f(y) dy, x = (x_{1}, …, x_{n}), y = (y_{1}, …, y_{n}).

Then, as in [13, Proposition], we can prove:

(i) u is a harmonic function on R_{+}^{n} with finite Dirichlet integral. (ii) u has a nontangential limit at the origin. (iii) lim_{j→∞} u(x^{(j)} + (0, …, 0, r_{j})) = ∞. To show (i) and (ii), we note by (6) that

∫ f(y)^{n} dy ≦ M ∑_{j} a_{j}^{n}(log \frac{1}{2^{j}r_{j}} )^{−n+1} < ∞

Tangential limits of monotone Sobolev functions 325

and

u(0) = ∫ (−y_{n})|y|^{−n}f(y) dy

≦ M ∑_{j} a_{j}(log \frac{1}{2^{j}r_{j}} )^{−1}2^{jn} ∫_{B_{j}}(−y_{n})|x^{(j)} − y|^{−1} dy

≦ M ∑_{j} a_{j}(log \frac{1}{2^{j}r_{j}} )^{−n+1} < ∞.

Finally we see that for x ∈ R_{+}^{n} ∩ B(x^{(j)}, r_{j}),

u(x) ≧ Ma_{j}(log \frac{1}{2^{j}r_{j}} )^{−1} ∫_{r_{j}s_{j}}^{2^{−j−2}s_{j}}(|x − x^{(j)}| + r)^{1−n}r^{−1}r^{n−1} dr ≧ Ma_{j},

which implies that

lim_{x→0,x∈B} u(x) = ∞.

Remark 4. The examples in Remarks 1 and 3 show that the existence of nontangential limits may not always imply that of tangential limits.

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326 Yoshihiro Mizuta

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Received 18 January 1994