Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 20, 1995, 433–436

INVARIANT SETS FOR A-HARMONIC MEASURE

Jukka Kurki

University of Helsinki, Department of Mathematics P.O. Box 4, FIN-00014 Helsinki, Finland

Abstract. We prove that the zero capacity is a sufficient condition for invariant sets for the 𝒜 -harmonic measure, i.e., if cap_{p} F = 0 then ω(E ∪ F, Ω; 𝒜) = ω(E, Ω; 𝒜) for any closed E ⊂ ∂Ω.

1. Introduction

The 𝒜 -harmonic measure ω is a function similar to the classical harmonic measure. However, it is associated with a more general, possibly non-linear, elliptic partial differential equation ∇ · 𝒜 (x, ∇u) = 0 than the Laplace equation. An invariant set is a set F ⊂ ∂Ω such that F does not change the 𝒜 -harmonic measure of the original set E, i.e., ω(E ∪ F, Ω; 𝒜) = ω(E, Ω; 𝒜). If 𝒜 (x, ∇u) = ∇u, then invariant sets are, of course, nothing else but sets of harmonic measure zero. The p-harmonic case, i.e., 𝒜 (x, ∇u) = |∇u|^{p−2}∇u, is studied by P. Aviles and J. Manfredi [AM]. They proved that if F is a closed set such that the Hausdorff dimension of F is small enough, then ω(E ∪ F, Ω; p) = ω(E, Ω; p). The linearization method employed by Aviles and Manfredi does not work for arbitrary 𝒜 . In this paper we derive the following sufficient condition for invariant sets:

Theorem 1.1. Let E, F ⊂ ∂Ω and let E be closed. If cap_{p} F = 0, then

ω(E ∪ F, Ω; 𝒜) = ω(E, Ω; 𝒜).

For p < n it is known that dim_{H} F < n − p implies cap_{p} F = 0 where dim_{H} F refers to the Hausdorff dimension of F. Hence dim_{H} F < n − p for a set F yields that F is invariant in the sense of Theorem 1.1. Bounds of this type have been obtained in [AM]. These, however, depend on the set Ω. By the paper of Tukia [T] it is easy to see that the result is the best possible involving a general class of equations and Hausdorff dimensions. In particular, for each γ < p = 2, there are compact sets K, on the boundary of unit disks B ⊂ R^{2} , such that dim_{H} K < 2 − γ and ω(K, B, 𝒜) > 0 for some operator 𝒜 .

For p > n no non-empty set is of p-capacity zero and Theorem 1.1 gives nothing in this case.

I would like to thank Professor Olli Martio and Dr Tero Kilpeläinen for helpful discussions. Thanks are also due to the referee, whose suggestion shortened the proof of Theorem 1.1.

1991 Mathematics Subject Classification: Primary 31C99.

2. Definitions for 𝒜 -harmonic measure

Throughout this paper we assume that Ω is an open, bounded and connected set in R^{n}. We also assume that the operator 𝒜 satisfies assumptions 2.1–2.5 below for some 1 < p < ∞ and 0 < α ≤ β < ∞:

(2.1)  x → 𝒜 (x, h) is measurable for all h ∈ R^{n} and
h → 𝒜 (x, h) is continuous for a.e. x ∈ Ω,

and for all h ∈ R^{n} and a.e. x ∈ Ω

(2.2)  𝒜 (x, h) · h ≥ α|h|^{p}
(2.3)  |𝒜 (x, h)| ≤ β|h|^{p−1}
(2.4)  (𝒜 (x, h_{1}) − 𝒜 (x, h_{2})) · (h_{1} − h_{2}) > 0, whenever h_{1} ≠ h_{2}, and
(2.5)  𝒜 (x, λh) = |λ|^{p−2}λ𝒜 (x, h) for all λ ∈ R \ {0}.

A function u ∈ W_{loc}^{1,p}(Ω) is a solution of the equation

(2.6)  ∇ · 𝒜 (x, ∇u) = 0

if

∫_{Ω} 𝒜 (x, ∇u) · ∇φ dx = 0

for all φ ∈ C_{o}^{∞}(Ω). Any solution of (2.6) can be redefined in a set of measure zero so that it becomes continuous in Ω. This redefined continuous solution of (2.6) is said to be 𝒜 -harmonic in Ω. We denote by ℋ (Ω) the set of all 𝒜 -harmonic functions in Ω. If v: Ω → R ∪ {∞} is lower semicontinuous and if v is not identically infinite in Ω, then v is 𝒜 -superharmonic if for each domain D ⊂⊂ Ω and for each u ∈ ℋ (D) ∩ C(D̄) the condition u ≤ v in ∂D implies that u ≤ v in D. We let 𝒮 (Ω) denote the family of all 𝒜 -superharmonic functions.

Definition 2.1. Let f: ∂Ω → R ∪ {±∞} be any function and

H̄_{f}(x) = inf{v(x) |v ∈ 𝒮 (Ω), bounded below and
lim inf_{z→y} v(z) ≥ f(y) for all y ∈ ∂Ω}.

The function H̄_{f} is called the upper Perron solution of f.

Now H̄_{f} ∈ ℋ (Ω) if it is bounded in Ω. Let E ⊂ ∂Ω and let χ_{E} be the characteristic function of E. The function ω(E, Ω; 𝒜) = H̄_{χ_{E}} is called the 𝒜 - harmonic measure of set E with respect to Ω. For these constructions see [HKM].

The next lemma is employed in the proof of Theorem 1.1. The lemma is proved in [HKM, Theorem 9.3].

Invariant sets for 𝒜 -harmonic measure 435

Lemma 2.2 Let f_{j}: ∂Ω → R be a decreasing sequence of continuous functions and let f = lim f_{j}. Then

H̄_{f} = lim_{j→∞} H̄_{f_{j}}.

Let θ ∈ W^{1,p}(Ω). We write
𝒦_{θ} = {v ∈ W^{1,p}(Ω) : v ≥ θ a.e., v − θ ∈ W_{o}^{1,p}(Ω)}.

We call a function v a solution to the obstacle problem with obstacle and boundary values θ if v ∈ 𝒦_{θ} and if

∫_{Ω} 𝒜 (x, ∇v) · ∇(φ − v) dx ≥ 0

whenever φ ∈ 𝒦_{θ}.

Lemma 2.3. Let ϕ_{j} ∈ W^{1,p}(Ω) be a decreasing sequence such that ϕ_{j} → ϕ in W^{1,p}(Ω). Let u_{j} ∈ W^{1,p}(Ω) be a solution to the obstacle problem with obstacle and boundary values ϕ_{j}. Then the sequence u_{j} is decreasing and u = lim u_{j} is a solution to the obstacle problem with ϕ as an obstacle and boundary value.

Lemma 2.3 is proved in [HKM, Theorem 3.79]. For further details see [HKM, Chapter 3].

3. Proof of Theorem 1.1

Let I be the set of all irregular points, for the p-Dirichlet problem, in the boundary of Ω. We may assume that I ⊂ F because I is also a set of p-capacity zero [HKM, Theorem 9.11].

Let φ_{i} ∈ C_{o}^{∞}(R^{n}) be a decreasing sequence of non-negative functions such that φ_{i} ↘ χ_{E}. Let the function H̄_{φ_{i}} be as in Definition 2.1.

Let B ⊂ R^{n} be a ball such that Ω ⊂⊂ \frac{1}{2}B. Because cap_{p} F = 0, there exists a sequence of open sets U_{j} such that F ⊂ U_{j} and cap_{p}(U_{j}, B) < 1/j. Let ψ_{j} = R̂_{U_{j}}^{1}(B), where R̂_{U_{j}}^{1}(B) is the 𝒜 -potential of U_{j} in B (see [HKM, Chapter 8]). By using the estimates in [HKM] we get that ψ_{j} = 1 in U_{j}, ψ_{j} ∈ W_{o}^{1,p}(B) and ∫_{B} |∇ψ_{j}|^{p} dx < c/j where the constant c depends only on α, β and p. Let v_{ij} ∈ 𝒮 (Ω) be the solution to the obstacle problem with the function H̄_{φ_{i}} + ψ_{j} as an obstacle and boundary value. Now the continuity of H̄_{φ_{i}} yields v_{ij} ≥ H̄_{φ_{i}} in Ω and ψ_{j} ≡ 1 in U_{j} gives v_{ij} ≥ 1 in U_{j} ∩ Ω. It follows that

lim inf_{x→y} v_{ij}(x) ≥ χ_{E∪F} (y)

for all y ∈ ∂Ω and for all i and j. Thus v_{ij} ≥ ω(E ∪F, Ω; 𝒜) for all i and j. The solution to the obstacle problem with obstacle and boundary values φ_{i} is clearly H̄_{φ_{i}}. By Lemma 2.3 the limit function of the sequence v_{ij} is H̄_{φ_{i}} as j → ∞. Hence H̄_{φ_{i}} ≥ ω(E ∪ F, Ω; 𝒜) for all i. Lemma 2.2 says that H̄_{φ_{i}} ↘ ω(E, Ω; 𝒜) as i → ∞. So ω(E, Ω; 𝒜) ≥ ω(E ∪ F, Ω; 𝒜) and the theorem follows since the opposite inequality is obvious.

436 Jukka Kurki

References

[AM] Aviles, P., and J.J. Manfredi: On null sets of p-harmonic measures. - In: Partial Differential Equations with Minimal Smoothness and Applications, Springer-Verlag, Berlin, 1992.

[HK] Heinonen, J., and T. Kilpeläinen: 𝒜 -superharmonic functions and supersolutions of degenerate elliptic equations. - Ark. Mat. 26, 1988, 87–105.

[HKM] Heinonen, J., T. Kilpeläinen, and O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. - Clarendon Press, 1993.

[M] Martio, O.: Sets of zero elliptic harmonic measures. - Ann. Acad. Sci. Fenn. Ser. A I Math. 14, 1989, 47–55.

[T] Tukia, P.: Hausdorff dimension and quasisymmetric mappings. - Math. Scand. 65, 1989, 152–160.

Received 18 August 1994