Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 459–490

POTENTIAL THEORY OF QUASIMINIMIZERS

Juha Kinnunen and Olli Martio

Institute of Mathematics, P.O. Box 1100 FI-02015 Helsinki University of Technology, Finland; juha.kinnunen@hut.fi Department of Mathematics, P.O. Box 4 FI-00014 University of Helsinki, Finland; olli.martio@helsinki.fi

Abstract. We study nonlinear potential theory related to quasiminimizers on a metric measure space equipped with a doubling measure and supporting a Poincaré inequality. Our objective is to show that quasiminimizers create an interesting potential theory with new features although from the potential theoretic point of view they have several drawbacks: They do not provide a unique solution to the Dirichlet problem, they do not obey the comparison principle and they do not form a sheaf. However, many potential theoretic concepts such as harmonic functions, superharmonic functions and the Poisson modification have their counterparts in the theory of quasiminimizers and, in particular, we are interested in questions related to regularity, convergence and polar sets.

1. Introduction

Quasiminimizers minimize a variational integral only up to a multiplicative constant. More precisely, let Ω ⊂ R^{n} be an open set, K ≥ 1 and 1 < p < ∞. In the case of the p-Dirichlet integral, a function u belonging to the Sobolev space W_{loc}^{1,p}(Ω) is a K -quasiminimizer, if

(1.1)  ∫_{Ω′} |∇u|^{p} dx ≤ K ∫_{Ω′} |∇v|^{p} dx

for all functions v ∈ W^{1,p}(Ω′) with v −u ∈ W_{0}^{1,p}(Ω′) and for all open sets Ω′ with compact closure in Ω. A 1-quasiminimizer, called a minimizer, is a weak solution of the corresponding Euler equation

(1.2)  div(|∇u|^{p−2}∇u) = 0.

Clearly being a weak solution of (1.2) is a local property. However, being a K - quasiminimizer is not a local property as one-dimensional examples easily show. This indicates that the theory for quasiminimizers usually differs from the theory for minimizers and that there are some unexpected difficulties.

Quasiminimizers have been previously used as tools in studying regularity of minimizers of variational integrals, see [GG1–2]. The advantage of this approach is that it covers a wide range of applications and that it is based only

2000 Mathematics Subject Classification: Primary 31C45, 49N60.

on the minimization of the variational integral instead of the corresponding Euler equation. Hence regularity properties as Hölder continuity and L^{p} -estimates are consequences of the quasiminimizing property. For us an important fact is that nonnegative quasiminimizers satisfy the Harnack inequality, see [DT].

Instead of using quasiminimizers as tools, our objective is to show that quasiminimizers have a fascinating theory themselves. In particular, they form a basis for nonlinear potential theoretic model with interesting features. From the potential theoretic point of view quasiminimizers have several drawbacks: They do not provide unique solutions to the Dirichlet problem, they do not obey the comparison principle, they do not form a sheaf and they do not have a linear structure even when the corresponding Euler equation is linear. However, quasiminimizers form a wide and flexible class of functions in calculus of variations under very general circumstances. To emphasize this we study potential theory of quasiminimizers in metric measure spaces although most of the results are new even in the Euclidean setup. Observe that the quasiminimizing condition (1.1) applies not only to one particular variational integral but the whole class of variational integrals at the same time. For example, if a variational kernel F (x, ∇u) satisfies

(1.3)  α|h|^{p} ≤ F (x, h) ≤ β|h|^{p}

for some 0 < α ≤ β < ∞, then the minimizers of

∫ F (x, ∇u) dx

are quasiminimizers of the p-Dirichlet integral

(1.4)  ∫ |∇u|^{p} dx.

Hence the potential theory for quasiminimizers includes all minimizers of all variational integrals similar to (1.4). The essential feature of the theory is the control provided by the bounds in (1.3). We also mention that quasiminimizers are preserved under a bilipschitz change of coordinates, but the constant K may change.

This work is organized as follows. In Section 2 we recall the basic properties of the Sobolev spaces on metric measure spaces. Quasiminimizers and quasisuperminimizers are studied in Section 3. Quasisuperminimizers replace the class of superminimizers in the classical setup.

In Section 4 we introduce the Poisson modification of a quasisuperminimizer. The Poisson modification lies below the original quasisuperminimizer, but it need not be a minimizer inside the set of modification.

A minimizer satisfies a Harnack inequality and is locally Hölder continuous after a redefinition on a set of measure zero. In Section 5 we show, using the celebrated De Giorgi method adapted to metric spaces, that a quasisuperminimizer satisfies the weak Harnack inequality. From this it follows that quasisuperminimizers are lower semicontinuous.

Section 6 is devoted to convergence properties of quasisuperminimizers. We show that the class of quasisuperminimizers is closed under monotone convergence, provided the limit function is locally bounded. This will be a crucial fact for us in Section 7 where we define quasisuperharmonic functions. Quasisuperharmonic functions play a central role in the potential theory of quasiminimizers. However, it is not obvious how quasisuperharmonic functions should be defined. The standard definition based on the comparison principle is useless because the Dirichlet problem does not have a unique solution. Our definition has a global nature because of the lack of the sheaf property. In Section 8 we study equivalent definitions of superharmonicity. The Poisson modification of a quasisuperharmonic function is considered in Section 9. Finally, Section 10 deals with polar sets.

2. Newtonian spaces

Let X be a metric space and let μ be a Borel measure on X. Throughout the paper we assume that the measure of every nonempty open set is positive and that the measure of every bounded set is finite. Later we impose further requirements on the space and on the measure, see 2.9.

2.1. Upper gradients. Let u be a real-valued function on X. A nonnegative Borel measurable function g on X is said to be an upper gradient of u if for all rectifiable paths γ joining points x and y in X we have

(2.2)  |u(x) − u(y)| ≤ ∫_{γ} g ds.

See [C], [He] and [Sh1] for a discussion on upper gradients.

A property is said to hold for p-almost all paths, if the set of paths for which the property fails is of zero p-modulus. If (2.2) holds for p-almost all paths γ, then g is said to be a p-weak upper gradient of u.

2.3. Newtonian spaces. Let 1 ≤ p < ∞. We define the space Ñ^{1,p}(X) to be the collection of all p-integrable functions u on X that have a p-integrable p-weak upper gradient g on X. This space is equipped with the seminorm

∥u∥_{_{N^{1,p}}(X)} = ∥u∥_{L^{p}(X)} + inf ∥g∥_{L^{p}(X)},

where the infimum is taken over all p-weak upper gradients of u.

We define an equivalence relation in Ñ^{1,p}(X) by saying that u ∼ v if

∥u − v∥_{_{N^{1,p}}(X)} = 0.

The Newtonian space N^{1,p}(X) is defined to be the space Ñ^{1,p}(X)/ ∼ with the norm

∥u∥_{N^{1,p}(X)} = ∥u∥_{_{N^{1,p}}(X)}.

For basic properties of the Newtonian spaces we refer to [Sh1]. Cheeger [C] gives an alternative definition which leads to the same space when 1 < p < ∞, see [Sh1].

We recall that if 1 < p < ∞, every function u that has a p-integrable p-weak upper gradient has a minimal p-integrable p-weak upper gradient in X, denoted g_{u}, in the sense that if g is another p-weak upper gradient of u, then g_{u} ≤ g μ-almost everywhere in X.

2.4. Capacity. The p-capacity of a set E ⊂ X is defined by

C_{p}(E) = inf_{u} ∥u∥_{N^{1,p}(X)}^{p},

where the infimum is taken over all functions u ∈ N^{1,p}(X), with u = 1 on E. The discussion in [KM2] can easily be adapted to show that the capacity is an outer measure, see also [Sh1]. The p-capacity is the natural measure for exceptional sets of Sobolev functions.

Let Ω be an open subset of X. We say that a subset E of Ω is compactly contained in Ω, abbreviated E ⋐ Ω, if the closure of E is a compact subset of Ω. Let Ω ⊂ X be bounded and E ⋐ Ω. The relative p-capacity of E with respect to Ω is the number

C_{p}(E, Ω) = inf ∫_{X} g_{u}^{p} dμ,

where the infimum is taken over all functions u ∈ N^{1,p}(X) such that u = 1 on E and u = 0 on X \ Ω.

2.5. Newtonian spaces with zero boundary values. Let E be an arbitrary subset of X. We define N_{0}^{1,p}(E) to be the set of functions u ∈ N^{1,p}(X) for which

C_{p}({x ∈ X \ E : u(x) ≠ 0}) = 0.

The space N_{0}^{1,p}(E) equipped with the norm

∥u∥_{N_{0}^{1,p}(E)} = ∥u∥_{N^{1,p}(X)},

is the Newtonian space with zero boundary values. The norm is unambiguously defined by [Sh1] and the space N_{0}^{1,p}(E) with this norm is a Banach space.

2.6. Local Newtonian spaces. Let Ω be an open subset of X. We say that u belongs to the local Newtonian space N_{loc}^{1,p}(Ω) if u ∈ N^{1,p}(E) for every measurable set E ⋐ Ω. If u ∈ N_{loc}^{1,p}(Ω) with 1 < p < ∞, then u has a minimal p-weak upper gradient g_{u} in Ω in the following sense: If Ω′ ⋐ Ω is an open set and g is the minimal upper gradient of u in Ω′, then g_{u} = g μ-almost everywhere in Ω′.

2.7. Doubling property. The measure μ is doubling, if there is a constant c_{d} ≥ 1 so that

μ(B(z, 2r)) ≤ c_{d}μ(B(z, r))

for every open ball B(z, r) in X. The constant c_{d} is called the doubling constant of μ.

A metric space X is said to be doubling if there is a constant c < ∞ such that every ball B(x, r), x ∈ X, r > 0, can be covered by at most c balls with the radii r/2. If X is equipped with a doubling measure, then X is doubling.

2.8. Poincaré inequalities. Let 1 ≤ q < ∞. The space X is said to support a weak (1, q)-Poincaré inequality if there are constants c > 0 and τ ≥ 1 such that

\overline{∫} _{B(z,r)} |u − u_{B(z,r)}| dμ ≤ cr(\overline{∫} _{B(z,τr)} g^{q} dμ)^{1/q}

for all balls B(z, r) in X, for all integrable functions u in B(z, τr) and for all q - weak upper gradients g of u. In a doubling measure space a weak (1, q)-Poincaré inequality implies a weak (t, q)-Poincaré inequality for some t > q possibly with a different τ, see [BCLS] and [HaK].

2.9. Assumptions. Throughout the work we make the following rather standard assumptions:

From now on we assume, without further notice, that the complete metric measure space X is equipped with a doubling Borel measure for which the measure of every nonempty open set is positive and the measure of every bounded set is finite. Furthermore we assume that the space supports a weak (1, q)-Poincaré inequality for some q with 1 < q < p.

The assumption on the Poincaré inequality is needed in the regularity theory for quasiminimizers of variational integrals on metric spaces, see [KS1].

3. Quasiminimizers and quasisuperminimizers

Suppose that Ω ⊂ X is open and 1 < p < ∞. Let K ≥ 1. A function u ∈ N_{loc}^{1,p}(Ω) is called a K -quasiminimizer if for all open Ω′ ⋐ Ω and all functions v ∈ N_{loc}^{1,p}(Ω) with v − u ∈ N_{0}^{1,p}(Ω′) we have

(3.1)  ∫_{Ω′} g_{u}^{p} dμ ≤ K ∫_{Ω′} g_{v}^{p} dμ.

Here g_{u} and g_{v} are the minimal p-weak upper gradients of u and v in Ω, respectively.

A function u ∈ N_{loc}^{1,p}(Ω) is called a K -quasisuperminimizer if (3.1) holds for all open Ω′ ⋐ Ω and all functions v ∈ N_{loc}^{1,p}(Ω) such that v ≥ u μ-almost everywhere in Ω′ and v − u ∈ N_{0}^{1,p}(Ω′). A function u is called a K -quasisubminimizer if −u is a K -quasisuperminimizer. It is easy to see that a function is a K - quasiminimizer if and only if it is both K -quasisubminimizer and K -quasisuper-minimizer. In this work we concentrate on properties of K -quasisuperminimizers.

If K = 1, then 1-quasiminimizers and 1-quasisuperminimizers are called minimizers and superminimizers, respectively. If C_{p}(X \Ω) > 0 and f ∈ N^{1,p}(Ω), then there is a unique minimizer u ∈ N^{1,p}(Ω) such that u−f ∈ N_{0}^{1,p}(Ω). In other words, the Dirichlet problem has a unique solution with the given boundary values, see [C], [KM2] and [Sh2]. Observe, that for minimizers and superminimizers it is enough to test (3.1) with Ω′ = Ω.

The next lemma shows that open sets in (3.1) can be replaced by μ-measur-able sets.

3.2. Lemma. A function u ∈ N_{loc}^{1,p}(Ω) is a K -quasisuperminimizer if and only if for all μ-measurable sets E ⋐ Ω and all functions v ∈ N_{loc}^{1,p}(Ω) such that v ≥ u μ-almost everywhere in E and v − u ∈ N_{0}^{1,p}(E) we have

(3.3)  ∫_{E} g_{u}^{p} dμ ≤ K ∫_{E} g_{v}^{p} dμ.

Proof. Only the converse needs a proof. For this let E ⋐ Ω be μ-measurable, v ∈ N_{loc}^{1,p}(Ω) such that v ≥ u μ-almost everywhere in E and v − u ∈ N_{0}^{1,p}(E). Define ṽ: Ω → [−∞, ∞],

ṽ(x) = \{ \begin{matrix}v(x), & x ∈ E, \\ u(x), & x ∈ Ω \ E. \end{matrix}

Let ε > 0 and choose an open set Ω′ so that E ⊂ Ω′ ⋐ Ω and

∫_{Ω′\E} gp ; ṽ dμ < \frac{ε}{K} .

Then ṽ − u ∈ N_{0}^{1,p}(Ω′) and ṽ ≥ u μ-almost everywhere in Ω′. Therefore we can apply (3.1) and we obtain

\begin{aligned}∫_{E} g_{u}^{p} dμ & ≤ ∫_{Ω′} g_{u}^{p} dμ ≤ K ∫_{Ω′} gp ; ṽ dμ \\ & ≤ K ∫_{E} gp ; ṽ dμ + K ∫_{Ω′\E} gp ; ṽ dμ ≤ K ∫_{E} g_{v}^{p} dμ + ε. \end{aligned}

Since ε > 0 is arbitrary we obtain (3.3).

The condition (3.3) can be stated in a slightly different form. Observe that this is immediate for K = 1.

3.4. Lemma. A function u ∈ N_{loc}^{1,p}(Ω) is a K -quasisuperminimizer if and only if for all open Ω′ ⋐ Ω and all functions v ∈ N_{loc}^{1,p}(Ω) such that v ≥ u μ-almost everywhere in Ω′ and v − u ∈ N_{0}^{1,p}(Ω′) we have

(3.5)  ∫_{Ω′∩{v>u}} g_{u}^{p} dμ ≤ K ∫_{Ω′∩{v>u}} g_{v}^{p} dμ.

Proof. First suppose that (3.5) holds. Since g_{u} = g_{v} μ-almost everywhere on the set {x ∈ Ω′ : u(x) = v(x)} and v ≥ u μ-almost everywhere on Ω′, we have

\begin{aligned}∫_{Ω′} g_{u}^{p} dμ & = ∫_{Ω′∩\{v>u\}} g_{u}^{p} dμ + ∫_{Ω′∩\{v=u\}} g_{u}^{p} dμ \\ & ≤ K ∫_{Ω′∩\{v>u\}} g_{v}^{p} dμ + ∫_{Ω′∩\{v=u\}} g_{v}^{p} dμ ≤ K ∫_{Ω′} g_{v}^{p} dμ. \end{aligned}

Then suppose that u is a K -quasisuperminimizer. Let v ∈ N_{loc}^{1,p}(Ω) be such that v ≥ u μ-almost everywhere in Ω′ and v − u ∈ N_{0}^{1,p}(Ω′). Then the set

E = {x ∈ Ω′ : v(x) > u(x)}

is μ-measurable, v ≥ u μ-almost everywhere in E and v − u ∈ N_{0}^{1,p}(E). From Lemma 3.2 we conclude that

∫_{E} g_{u}^{p} dμ ≤ K ∫_{E} g_{v}^{p} dμ

and this is what we wanted to prove.

We observe that if u is a K -quasisuperminimizer, then αu and u + β are K -quasisuperminimizers when α ≥ 0 and β ∈ R. However, the sum of two K -quasisuperminimizers is not a K -quasisuperminimizer in general.

3.6. Lemma. Suppose that u_{i} is a K_{i} -quasisuperminimizer in Ω, i = 1, 2. Then min(u_{1}, u_{2}) is a (K_{1} + K_{2})-quasisuperminimizer in Ω.

Proof. Let u = min(u_{1}, u_{2}). Since N_{loc}^{1,p}(Ω) is a lattice we have u ∈ N_{loc}^{1,p}(Ω). Let Ω′ ⋐ Ω be an open set, and let v ∈ N_{loc}^{1,p}(Ω) be such that v − u ∈ N_{0}^{1,p}(Ω′) and v ≥ u μ-almost everywhere in Ω′. Set

E_{1} = {x ∈ Ω′ : v(x) ≥ u_{1}(x)} and E_{2} = {x ∈ Ω′ : v(x) > u_{2}(x)}.

Then

\begin{aligned}∫_{Ω′} g_{u}^{p} dμ & ≤ ∫_{Ω′∩\{u_{1}≤u_{2}\}} g_{u_{1}}^{p} dμ + ∫_{Ω′∩\{u_{1}>u_{2}\}} g_{u_{2}}^{p} dμ \\ & ≤ ∫_{E_{1}} g_{u_{1}}^{p} dμ + ∫_{E_{2}} g_{u_{2}}^{p} dμ. \end{aligned}

We observe that v −u_{1} ∈ N_{0}^{1,p}(E_{1}), v ≥ u_{1} μ-almost everywhere on E_{1} , v −u_{2} ∈ N_{0}^{1,p}(E_{2}), v ≥ u_{2} μ-almost everywhere on E_{2} . Then Lemma 3.2 implies

∫_{Ω′} g_{u}^{p} dμ ≤ K_{1} ∫_{E_{1}} g_{v}^{p} dμ + K_{2} ∫_{E_{2}} g_{v}^{p} dμ ≤ (K_{1} + K_{2}) ∫_{Ω′} g_{v}^{p} dμ.

This completes the proof.

3.7. Lemma. Suppose that u_{i} is a K_{i} -quasisuperminimizer in Ω, i = 1, 2. Then min(u_{1}, u_{2}) is a K_{1}K_{2} -quasisuperminimizer in Ω.

Proof. Let u and v be as in the proof of Lemma 3.6. Then

∫_{Ω′} g_{u}^{p} dμ ≤ ∫_{Ω′∩{u_{1}≤u_{2}}} g_{u_{1}}^{p} dμ + ∫_{Ω′∩{u_{1}>u_{2}}} g_{u_{2}}^{p} dμ.

Let w = max(min(u_{2}, v), u_{1}). Then w − u_{1} ∈ N_{0}^{1,p}(Ω′ ∩ {u_{1} ≤ u_{2}}), w ≥ u_{1} μ-almost everywhere in Ω′ and since u_{1} is a K_{1} -quasisuperminimizer we have

\begin{aligned}_{∫Ω′∩\{u_{1}≤u_{2}\}} & g_{u_{1}}^{p} dμ ≤ K_{1} ∫_{Ω′∩\{u_{1}≤u_{2}\}} g_{w}^{p} dμ \\ & ≤ K_{1} ∫_{Ω′∩\{u_{1}≤u_{2}\}∩\{v<u_{2}\}} g_{v}^{p} dμ + K_{1} ∫_{Ω′∩\{u_{1}≤u_{2}\}∩\{v≥u_{2}\}} g_{u_{2}}^{p} dμ. \end{aligned}

This implies that

∫_{Ω′} g_{u}^{p} dμ ≤K_{1} ∫_{Ω′∩{u_{1}≤u_{2}}∩{v<u_{2}}} g_{v}^{p} dμ + K_{1} ∫_{Ω′∩{u_{1}≤u_{2}}∩{v≥u_{2}}} g_{u_{2}}^{p} dμ
+ ∫_{Ω′∩{u_{1}>u_{2}}} g_{u_{2}}^{p} dμ
≤K_{1} ∫_{Ω′∩{u_{1}≤u_{2}}} g_{v}^{p} dμ + K_{1} ∫_{Ω′∩{v≥u_{2}}} g_{u_{2}}^{p} dμ.

Since max(u_{2}, v) − u_{2} ∈ N_{0}^{1,p}(Ω′ ∩ {v ≥ u_{2}}) and max(u_{2}, v) ≥ u_{2} μ-almost everywhere in Ω′, we obtain

∫_{Ω′∩{v≥u_{2}}} g_{u_{2}}^{p} dμ ≤ K_{2} ∫_{Ω′∩{v≥u_{2}}} g_{v}^{p} dμ

and we conclude that

\begin{aligned}∫_{Ω′} g_{u}^{p} dμ & ≤ K_{1} ∫_{Ω′∩\{u_{1}≤u_{2}\}∩\{v<u_{2}\}} g_{v}^{p} dμ + K_{1}K_{2} ∫_{Ω′∩\{v≥u_{2}\}} g_{v}^{p} dμ \\ & ≤ K_{1}K_{2} ∫_{Ω′} g_{v}^{p} dμ. \end{aligned}

From Lemmas 3.6 and 3.7 we obtain:

3.8. Corollary. Suppose that u_{i} is a K_{i} -quasisuperminimizer in Ω, i = 1, 2. Then min(u_{1}, u_{2}) is a min(K_{1} + K_{2}, K_{1}K_{2})-quasisuperminimizer in Ω.

The next corollary is important in our constructions. It provides a construction method which does not increase the constant K. Recall that 1-quasisuper-minimizers are called superminimizers.

3.9. Corollary. Suppose that u is a K -quasisuperminimizer and that h is a superminimizer in Ω. Then min(u, h) is a K -quasisuperminimizer in Ω.

Corollary 3.9 applied to constant functions h gives the following result.

3.10. Lemma. Suppose that u ∈ N_{loc}^{1,p}(Ω). Then u is a K -quasisupermini-mizer if and only if min(u, c) is a K -quasisuperminimizer for every c ∈ R.

Proof. If u is a K -quasisuperminimizer, then it follows from Corollary 3.9 that min(u, c) is a K -quasisuperminimizer for each c ∈ R, since constants are minimizers.

For the converse let Ω′ ⋐ Ω, v−u ∈ N_{0}^{1,p}(Ω′) and v ≥ u μ-almost everywhere in Ω′. Write v_{i} = min(v, i) and u_{i} = min(u, i), i = 1, 2, …. Then v_{i} − u_{i} ∈ N_{0}^{1,p}(Ω′) and v_{i} ≥ u_{i} μ-almost everywhere in Ω′. By assumption u_{i} is a K - quasisuperminimizer and hence

∫_{Ω′} g_{u_{i}}^{p} dμ ≤ K ∫_{Ω′} g_{v_{i}}^{p} dμ

for i = 1, 2, …. Since u, v ∈ N^{1,p}(Ω′), we obtain

∫_{Ω′} g_{u}^{p} dμ ≤ K ∫_{Ω′} g_{v}^{p} dμ

as i → ∞. This shows that u is a K -quasisuperminimizer in Ω.

4. Poisson modification of a quasisuperminimizer

Suppose that u is a superminimizer, i.e. 1-quasisuperminimizer, in Ω and that Ω′ ⋐ Ω is open. Define

P (u, Ω′)(x) = \{ \begin{matrix}h(x), & x ∈ Ω′, \\ u(x), & x ∈ Ω \ Ω′, \end{matrix}

where h is a minimizer in Ω′ with h − u ∈ N_{0}^{1,p}(Ω′). Then P (u, Ω′) ∈ N_{loc}^{1,p}(Ω) and the next lemma is well known in classical potential theory.

4.1. Lemma. Suppose that u is a superminimizer in Ω and that Ω′ ⋐ Ω is open. Then the function P (u, Ω′) is a superminimizer in Ω and u ≥ h μ-almost everywhere in Ω′.

Proof. We first show that u ≥ h μ-almost everywhere in Ω′. Let

A = {x ∈ Ω′ : u(x) < h(x)}.

Now u − h ∈ N_{0}^{1,p}(A) and by the superminimizing property of u

(4.2)  ∫_{A} g_{u}^{p} dμ ≤ ∫_{A} g_{h}^{p} dμ.

Now (4.2) holds as an equality because h is a minimizer in A with boundary values u and this means that u and h are both minimizers in A with the same boundary values. Since the minimizer with given boundary values is unique, see Theorem 7.14 in [C], this means that μ(A) = 0 as required.

To prove the superminimizing property of P (u, Ω′) write w = P (u, Ω′) and let Ω′′ ⋐ Ω be open and let v ∈ N_{loc}^{1,p}(Ω) such that v ≥ w μ-almost everywhere with w − v ∈ N_{0}^{1,p}(Ω′′). We show that

(4.3)  ∫_{Ω′′} g_{w}^{p} dμ ≤ ∫_{Ω′′} g_{v}^{p} dμ.

To this end let A_{1} = {x ∈ Ω′′ : v(x) ≥ u(x)} and A_{2} = Ω′′ \ A_{1} . Then A_{2} ⊂ Ω′ and define

(4.4)  v_{1}(x) = \{ \begin{matrix}v(x), & x ∈ A_{2}, \\ u(x), & x ∈ (Ω′ ∩ Ω′′) \ A_{2}. \end{matrix}

Since

v_{1} = min(v, u) = min(max(h, v), u)

in Ω′ ∩ Ω′′, it follows that v_{1} − h ∈ N_{0}^{1,p}(Ω′ ∩ Ω′′). The minimizing property of h in Ω′ ∩ Ω′′ implies

∫_{Ω′∩Ω′′} g_{h}^{p} dμ ≤ ∫_{Ω′∩Ω′′} g_{v_{1}}^{p} dμ = ∫_{A_{2}} g_{v}^{p} dμ + ∫_{(Ω′∩Ω′′)\A_{2}} g_{u}^{p} dμ

and the superminimizing property of u in A_{1} gives

∫_{Ω′′\Ω′} g_{u}^{p} dμ + ∫_{Ω′∩Ω′′∩A_{1}} g_{u}^{p} dμ = ∫_{A_{1}} g_{u}^{p} dμ ≤ ∫_{A_{1}} g_{v}^{p} dμ

because v ≥ u in A_{1} and v − u ∈ N_{0}^{1,p}(A_{1}). Combining these inequalities we obtain

∫_{Ω′′} g_{w}^{p} dμ = ∫_{Ω′′\Ω′} g_{u}^{p} dμ + ∫_{Ω′′∩Ω′} g_{h}^{p} dμ
≤ ∫_{A_{1}} g_{v}^{p} dμ − ∫_{Ω′∩Ω′′∩A_{1}} g_{u}^{p} dμ + ∫_{A_{2}} g_{v}^{p} dμ + ∫_{(Ω′∩Ω′′)\A_{2}} g_{u}^{p} dμ
= ∫_{Ω′′} g_{v}^{p} dμ,

since Ω′ ∩ Ω′′ ∩ A_{1} = (Ω′ ∩ Ω′′) \ A_{2} . This is (4.3) and the lemma follows.

The next theorem shows how the counterpart of the Poisson modification is constructed for K -quasisuperminimizers.

4.5. Theorem. Suppose that u is a K -quasisuperminimizer in Ω and that Ω′ ⋐ Ω is open. If h is a minimizer in Ω′ with h−u ∈ N_{0}^{1,p}(Ω′), then the function w: Ω → [−∞, ∞],

(4.6)  w(x) = \{ \begin{matrix}min(u(x), h(x)), & x ∈ Ω′, \\ u(x), & x ∈ Ω \ Ω′, \end{matrix}

is a K -quasisuperminimizer in Ω. Moreover, w ≤ u in Ω.

Proof. Clearly w ≤ u in Ω and it remains to show the K -quasisuperminimi-zing property of w. To this end, let Ω′′ ⋐ Ω be an open set and v ∈ N_{loc}^{1,p}(Ω) such that v ≥ w μ-almost everywhere in Ω′′ and v − w ∈ N_{0}^{1,p}(Ω′′). We show that

∫_{Ω′′} g_{w}^{p} dμ ≤ K ∫_{Ω′′} g_{v}^{p} dμ.

Let

U_{1} = {x ∈ Ω′′ : v(x) < u(x)} and U_{2} = {x ∈ Ω′′ : v(x) ≥ u(x)}.

Then v − u ∈ N_{0}^{1,p}(U_{2}). Since v ≥ u in U_{2} , the quasisuperminimizing property of u and Lemma 3.2 give

(4.7)  ∫_{U_{2}} g_{u}^{p} dμ ≤ K ∫_{U_{2}} g_{v}^{p} dμ.

We define h = u in Ω \ Ω′. Let D = {x ∈ Ω′′ : u(x) > h(x)}. Then U_{1} ⊂ D and we define

ψ(x) = \{ \begin{matrix}v(x), & x ∈ U_{1}, \\ u(x), & x ∈ D \ U_{1}. \end{matrix}

Then ψ−h ∈ N_{0}^{1,p}(D). Since D ⊂ Ω′ and h is a minimizer in Ω′, h is a minimizer in D as well and we obtain

(4.8)  ∫_{D} g_{h}^{p} dμ ≤ ∫_{D} g_{ψ}^{p} dμ.

The inequality (4.7) implies

K ∫_{Ω′′} g_{v}^{p} dμ = K ∫_{U_{2}} g_{v}^{p} dμ + K ∫_{U_{1}} g_{v}^{p} dμ
≥ K ∫_{U_{2}} g_{v}^{p} dμ + ∫_{U_{1}} g_{v}^{p} dμ
≥ ∫_{U_{2}} g_{u}^{p} dμ + ∫_{U_{1}} g_{v}^{p} dμ + ∫_{D\U_{1}} g_{u}^{p} dμ − ∫_{D\U_{1}} g_{u}^{p} dμ
= ∫_{U_{2}} g_{u}^{p} dμ + ∫_{D} g_{ψ}^{p} dμ − ∫_{D\U_{1}} g_{u}^{p} dμ.

Since U_{1} ⊂ D, the above inequality together with (4.8) imply

(4.9)  \begin{aligned}K ∫_{Ω′′} g_{v}^{p} dμ & ≥ ∫_{U_{2}} g_{u}^{p} dμ + ∫_{D} g_{h}^{p} dμ − ∫_{D\U_{1}} g_{u}^{p} dμ \\ & = ∫_{U_{2}\(D\U_{1})} g_{u}^{p} dμ + ∫_{D} g_{h}^{p} dμ. \end{aligned}

Finally we observe that Ω′′ \ D ⊂ U_{2} \ (D \ U_{1}), because if x ∈ Ω′′ \ D, then x belongs either to U_{2} \ D or to U_{1} \ D. However, we have U_{1} ⊂ D and hence x ∈ U_{2} \ D and x ∈ U_{2} \ (D \ U_{1}). This shows that

∫_{U_{2}\(D\U_{1})} g_{u}^{p} dμ ≥ ∫_{Ω′′\D} g_{u}^{p} dμ

and together with (4.9) this implies

K ∫_{Ω′′} g_{v}^{p} dμ ≥ ∫_{Ω′′\D} g_{u}^{p} dμ + ∫_{D} g_{h}^{p} dμ = ∫_{Ω′′} g_{w}^{p} dμ.

This is the required inequality.

As for the minimizers we write P (u, Ω′) = w. This is the Poisson modification of u in Ω′. Note that there is no ambiquity with the previous definition of P (u, Ω′) for a minimizer u since the definitions coincide for a minimizer u by Lemma 4.1.

The Poisson modification P (u, Ω′) of a K -quasisuperminimizer u in Ω′ ⋐ Ω has the following properties:

(1) P (u, Ω′) is a K -quasisuperminimizer in Ω,

(2) P (u, Ω′) = u in Ω \ Ω′,

(3) P (u, Ω′) ≤ u in Ω,

(4) P (u, Ω′) = h, if P (u, Ω′) < u, and where h is the minimizer with boundary values u in Ω′.

Note that in contrast to the classical potential theory there is no semicontinuity requirements for P (u, Ω′). Also Ω′ can be any open set such that Ω′ ⋐ Ω and it need not be regular for the Dirichlet problem.

5. Reqularity of quasisuperminimizers

Quasiminimizers satisfy the Harnack inequality and are locally Hölder continuous after redefinition on a set of measure zero, see [KS1]. Here we show that quasisuperminimizers satisfy the weak Harnack inequality and are lower semicontinuous as in the case of supersolutions and superharmonic functions. Our proof is based on the De Giorgi method. The basic work has been done in [KS1] and [KM2] since for this regularity result the cases K > 1 and K = 1 are similar.

We begin by recalling the definition of the De Giorgi class. Let k_{0} ∈ R and 0 < ϱ < R. A function u ∈ N_{loc}^{1,p}(Ω) belongs to the De Giorgi class DG_{p}(Ω, k_{0}), if there is a constant c < ∞ such that for all k ≥ k_{0} , z ∈ Ω such that B(z, R) ⋐ Ω the function u satisfies

∫_{A_{z}(k,ϱ)} g_{u}^{p} dμ ≤ c(R − ϱ)^{−p} ∫_{A_{z}(k,R)}(u − k)^{p} dμ

where

A_{z}(k, r) = {x ∈ B(z, r) : u(x) > k}

and g_{u} is the minimal p-weak upper gradient of u in Ω. If the inequality above holds for all k ∈ R, then we simply write u ∈ DG_{p}(Ω).

5.1. Lemma. Suppose that u is a K -quasisuperminimizer in Ω. Then −u belongs to DG_{p}(Ω).

The proof of Lemma 5.1 is similar to the proof of Lemma 4.1 in [KM2].

The next result shows that a nonnegative quasisuperminimizer satisfies a weak Harnack inequality.

5.2. Lemma. Suppose that u ≥ 0 is a K -quasisuperminimizer in an open set Ω ⊂ X. Then for every ball B(z, R) with B(z, 5R) ⊂ Ω we have

(\overline{∫} _{B(z,R)} u^{σ} dμ)^{1/σ} ≤ c ess inf_{B(z,3R)} u,

where c < ∞ and σ > 0 depend only on K and on the constants in the doubling condition and Poincaré inequality.

For the proof we refer to the proof of Lemma 4.7 in [KM2].

Next we observe that a quasisuperminimizer has a lower semicontinuous representative. We denote

ess lim inf_{y→x} u(y) = lim_{r→0} ess inf_{B(x,r)} u.

5.3. Lemma. Suppose that u is a K -quasisuperminimizer in Ω. Then the function u^{∗}: Ω → [−∞, ∞] defined by

u^{∗}(x) = ess lim inf_{y→x} u(y)

is a lower semicontinuous function in Ω and it belongs to the same equivalence class as u in N_{loc}^{1,p}(Ω).

The proof is similar to the proof of Theorem 5.1 in [KM2].

6. Convergence results for quasisuperminimizers

We show that the quasisuperminimizing property is preserved under increasing convergence if the limit function is locally bounded above or belongs to N_{loc}^{1,p}(Ω). The corresponding result for superminimizers has been studied in [KM2]. There are some unexpected difficulties for quasiminimizers and the argument is more involved in this case.

6.1. Theorem. Suppose that (u_{i}) is an increasing sequence of K -quasi-superminimizers in Ω and u = lim_{i→∞} u_{i} such that either

(i) u is locally bounded above or (ii) u ∈ N_{loc}^{1,p}(Ω). Then u is a K -quasisuperminimizer in Ω.

We first consider the case (i). In this case it follows from the De Giorgi type upper bound

∫_{B(x,ϱ)} g_{u_{i}}^{p} dμ ≤ c(R − ϱ)^{−p} ∫_{B(x,R)}(u_{i} − k)^{p} dμ,

where

k < − sup{ess sup_{B(x,R)} u_{i} : i = 1, 2, …},

0 < ϱ < R and B(x, R) ⋐ Ω, that the sequence (g_{u_{i}}) is uniformly bounded in L^{p}(Ω′) for every Ω′ ⋐ Ω. This implies that u ∈ N_{loc}^{1,p}(Ω) and we may assume that (g_{u_{i}}) converges weakly to g_{u} in L^{p}(Ω′), where g_{u} is an upper gradient of u.

We need a couple of lemmas. The first one is technical.

6.2. Lemma. Let u ∈ N_{loc}^{1,p}(Ω) and K ≥ 1. Suppose that for every open set Ω′ ⋐ Ω and for every v ∈ N_{loc}^{1,p}(Ω) such that v − u ∈ N_{0}^{1,p}(Ω′) and v ≥ u μ-almost everywhere in Ω′ we have

∫\overline{Ω′} g_{u}^{p} dμ ≤ K ∫\overline{Ω′} g_{v}^{p} dμ.

Then for every open set Ω′′ ⋐ Ω and for every v ∈ N_{loc}^{1,p}(Ω) such that v − u ∈ N_{0}^{1,p}(Ω′′) and v ≥ u μ-almost everywhere in Ω′′ we have

∫_{Ω′′} g_{u}^{p} dμ ≤ K ∫_{Ω′′} g_{v}^{p} dμ.

Proof. Let Ω′′ ⋐ Ω be open and v ∈ N_{loc}^{1,p}(Ω) such that v − u ∈ N_{0}^{1,p}(Ω′′) and v ≥ u μ-almost everywhere in Ω′′. Let ε > 0. By Theorem 5.2.5 of [Sh1] there is a Lipschitz function φ ≥ 0 such that spt φ ⋐ Ω′′ and

∥φ − (v − u)∥_{N^{1,p}(Ω′′)} < ε.

Choose an open set Ω′ such that spt φ ⋐ Ω′ ⋐ Ω′′.

By the assumption

(∫\overline{Ω′} g_{u}^{p} dμ)^{1/p} ≤ K^{1/p}(∫\overline{Ω′} g_{u+φ}^{p} dμ)^{1/p}

≤ K^{1/p}[(∫_{Ω′′} g_{v}^{p} dμ)^{1/p} + (∫_{Ω′} g_{φ−(v−u)}^{p} dμ)^{1/p}]

≤ K^{1/p}(∫_{Ω′′} g_{v}^{p} dμ)^{1/p} + K^{1/p}ε.

Since μ(Ω′′\\overline{Ω′}) can be made arbitrarily small, we see from the absolute continuity of the integral that

(∫_{Ω′′} g_{u}^{p} dμ)^{1/p} ≤ K^{1/p}(∫_{Ω′′} g_{v}^{p} dμ)^{1/p} + K^{1/p}ε.

We obtain the claim by letting ε → 0.

6.3. Lemma. Let u and u_{i} be as in Theorem 6.1. Then for every ball B(x, r) ⋐ Ω we have

lim sup_{i→∞} ∫_{B(x,r)} g_{u_{i}}^{p} dμ ≤ c ∫_{B(x,r)} g_{u}^{p} dμ,

where the constant c depends only on K and p.

Proof. Let B(x, ϱ) ⋐ B(x, r) ⋐ Ω and choose a Lipschitz cut-off function η such that 0 ≤ η ≤ 1, η = 0 in Ω \ B(x, r) and η = 1 in B(x, ϱ). Let

w_{i} = u_{i} + η(u − u_{i}), i = 1, 2, ….

Then w_{i} − u_{i} ∈ N_{0}^{1,p}(B(x, r)) and w_{i} ≥ u_{i} μ-almost everywhere in B(x, r). Hence the quasisuperminimizing property of u_{i} gives

\begin{aligned}_{∫B(x,ϱ)} & g_{u_{i}}^{p} dμ ≤ ∫_{B(x,r)} g_{u_{i}}^{p} dμ ≤ K ∫_{B(x,r)} g_{w_{i}}^{p} dμ \\ & ≤ αK(∫_{B(x,r)}(1 − η)^{p}g_{u_{i}}^{p} dμ + ∫_{B(x,r)} g_{η}^{p}(u − u_{i})^{p} dμ + ∫_{B(x,r)} η^{p}g_{u}^{p} dμ), \end{aligned}

where α = 2^{p−1} and we used the fact that

g_{w_{i}} ≤ (1 − η)g_{u_{i}} + g_{η}(u − u_{i}) + ηg_{u},

see Lemma 2.4 in [KM2]. Adding the term

αK ∫_{B(x,ϱ)} g_{u_{i}}^{p} dμ

to both sides and taking into account that η = 1 in B(x, ϱ) we obtain

(6.4) \begin{aligned}(1 + αK) ∫_{B(x,ϱ)} g_{u_{i}}^{p} dμ & ≤ αK ∫_{B(x,r)} g_{u_{i}}^{p} dμ + αK ∫_{B(x,r)} g_{η}^{p}(u − u_{i})^{p} dμ \\ & + αK ∫_{B(x,r)} g_{u}^{p} dμ. \end{aligned}

Set Ψ: (0, dist(x, ∂Ω)) → R,

Ψ(r) = lim sup_{i→∞} ∫_{B(x,r)} g_{u_{i}}^{p} dμ.

Since −u_{i} belongs to the De Giorgi class (see Lemma 5.1), we observe that Ψ is a finite-valued and increasing function of r. Hence the points of discontinuities form a countable set. Let r, 0 < r < dist(x, ∂Ω), be a point of continuity of Ψ. Letting i → ∞, we obtain from (6.4) the estimate

(6.5)  (1 + αK)Ψ(ϱ) ≤ αKΨ(r) + αK ∫_{B(x,r)} g_{u}^{p} dμ,

because

∫_{B(x,r)} g_{η}^{p}(u − u_{i})^{p} dμ → 0

as i → ∞. Since r is a point of continuity of Ψ, we conclude from (6.5) that

(1 + αK)Ψ(r) ≤ αKΨ(r) + αK ∫_{B(x,r)} g_{u}^{p} dμ,

or in other words

(6.6)  Ψ(r) ≤ αK ∫_{B(x,r)} g_{u}^{p} dμ.

This holds at each point of continuity of Ψ. Since Ψ is increasing and

r ↦ ∫_{B(x,r)} g_{u}^{p} dμ

is a continuous function of r, it is easy to see that (6.6) holds for every r with 0 < r < dist(x, ∂Ω). This is the required estimate.

Proof of Theorem 6.1. Case (i). As noted before u ∈ N_{loc}^{1,p}(Ω) and hence it suffices to prove inequality (3.1). To this end let Ω′ ⋐ Ω be open and v ∈ N_{loc}^{1,p}(Ω′), v ≥ u μ-almost everywhere and v − u ∈ N_{0}^{1,p}(Ω′). By Lemma 6.2 it suffices to show that

∫\overline{Ω′} g_{u}^{p} dμ ≤ K ∫\overline{Ω′} g_{v}^{p} dμ.

Choose an open set Ω′′ such that Ω′ ⋐ Ω′′ ⋐ Ω and a Lipschitz cut-off function η with the properties η = 1 on Ω′, 0 ≤ η ≤ 1 and η = 0 on Ω \ Ω′′. Set

w_{i} = u_{i} + η(v − u_{i}), i = 1, 2, ….

Then w_{i} − u_{i} ∈ N_{0}^{1,p}(Ω′′) and w_{i} ≥ u_{i}. By Lemma 2.4 in [KM2] we have

g_{w_{i}} ≤ (1 − η)g_{u_{i}} + ηg_{v} + g_{η}(v − u_{i}),

and we obtain

\begin{aligned}(∫_{Ω′′} g_{w_{i}}^{p} dμ)^{1/p} & ≤ (∫_{Ω′′}((1 − η)g_{u_{i}} + ηg_{v})^{p} dμ)^{1/p} + (∫_{Ω′′} g_{η}^{p}(v − u_{i})^{p} dμ)^{1/p} \\ & = α_{i} + β_{i}. \end{aligned}

Now (α_{i} + β_{i})^{p} ≤ α_{i}^{p} + pβ_{i}(α_{i} + β_{i})^{p−1} and hence

(6.7)  ∫_{Ω′′} g_{w_{i}}^{p} dμ ≤ ∫_{Ω′′}(1 − η)g_{u_{i}}^{p} dμ + ∫_{Ω′′} ηg_{v}^{p} dμ + pβ_{i}(α_{i} + β_{i})^{p−1},

where we also used the convexity of the function t ↦ t^{p}. We estimate the terms on the right-hand side separately.

First, since g_{η} = 0 μ-almost everywhere in Ω′, by the Lebesgue monotone convergence theorem

(6.8)  β_{i}^{p} = ∫_{Ω′′\Ω′} g_{η}^{p}(v − u_{i})^{p} dμ → 0

as i → ∞, because v = u in Ω′′ \ Ω′ and u_{i} → u in Ω.

Next we consider

∫_{Ω′′}(1 − η)g_{u_{i}}^{p} dμ.

Since η = 1 on \overline{Ω′}, we obtain

(6.9)  ∫_{Ω′′}(1 − η)g_{u_{i}}^{p} dμ ≤ ∫_{Ω′′\\overline{Ω′}} g_{u_{i}}^{p} dμ.

Choose balls B(x_{j}, r_{j}), j = 1, 2, …, such that B(x_{j}, r_{j}) ⊂ Ω′′ \ \overline{Ω′},

Ω′′ \ \overline{Ω′} ⊂ ⋃_{j=1}^{∞} B(x_{j}, r_{j})  and  ∑_{j=1}^{∞} χ_{B(x_{j},r_{j})} ≤ N < ∞,

where N depends only on the doubling constant, see [He].

Define

Ψ_{x_{j}}(r) = lim sup_{i→∞} ∫_{B(x_{j},r)} g_{u_{i}}^{p} dμ

and let ε > 0. By Lemma 6.3

Ψ_{x_{j}}(r) ≤ c ∫_{B(x_{j},r_{j})} g_{u}^{p} dμ

and letting i → ∞ we obtain

(6.10)

lim sup_{i→∞} ∫_{Ω′′\\overline{Ω′}} g_{u_{i}}^{p} dμ ≤ lim sup_{i→∞} ∑_{j=1}^{∞} ∫_{B(x_{j},r_{j})} g_{u_{i}}^{p} dμ

≤ ∑_{j=1}^{∞} Ψ_{x_{j}}(r_{j}) ≤ c ∑_{j=1}^{∞} ∫_{B(x_{j},r_{j})} g_{u}^{p} dμ

≤ c ∑_{j=1}^{∞} ∫_{Ω′′\\overline{Ω′}} χ_{B(x_{j},r_{j})}g_{u}^{p} dμ ≤ cN ∫_{Ω′′\\overline{Ω′}} g_{u}^{p} dμ.

Next we choose μ(Ω′′ \ \overline{Ω′}) so small that

∫_{Ω′′\\overline{Ω′}} g_{u}^{p} dμ < ε

and from (6.9) and (6.10) we obtain

(6.11)  lim sup_{i→∞} ∫_{Ω′′}(1 − η)g_{u_{i}}^{p} dμ ≤ cNε.

Note that c and N are independent of ε. Since α_{i} remains bounded as i → ∞, we obtain from (6.7), (6.8) and (6.11) that

(6.12)  lim sup_{i→∞} ∫_{Ω′′} g_{w_{i}}^{p} dμ ≤ cNε + ∫_{Ω′′} ηg_{v}^{p} dμ.

Now u_{i} is a K -quasisuperminimizer and hence

∫\overline{Ω′} g_{u_{i}}^{p} dμ ≤ ∫_{Ω′′} g_{u_{i}}^{p} dμ ≤ K ∫_{Ω′′} g_{w_{i}}^{p} dμ.

Together with the lower semicontinuity of the L^{p} -norm, p > 1, and (6.12) we have

∫\overline{Ω′} g_{u}^{p} dμ ≤ lim sup_{i→∞} ∫\overline{Ω′} g_{u_{i}}^{p} dμ ≤ KcNε + K ∫_{Ω′′} ηg_{v}^{p} dμ
≤ KcNε + K ∫\overline{Ω′} g_{v}^{p} dμ + K ∫_{Ω′′\\overline{Ω′}} g_{v}^{p} dμ.

We can still choose μ(Ω′′ \ \overline{Ω′}) so small that

∫_{\overline{Ω′}\Ω′′} g_{v}^{p} dμ < ε

and hence

∫\overline{Ω′} g_{u}^{p} dμ ≤ KcNε + Kε + K ∫\overline{Ω′} g_{v}^{p} dμ.

Finally letting ε → 0 we obtain the claim. This completes the proof of the case (i).

Proof of Theorem 6.1. Case (ii). In Case (ii) we consider functions

u_{i,c} = min(u_{i}, c), i = 1, 2, …, c ∈ R.

Since u_{i,c} is a K -quasisuperminimizer, it follows from Case (i) that u_{c} = min(u, c) is a K -quasisuperminimizer. Finally, since u ∈ N_{loc}^{1,p}(Ω), Lemma 3.10 implies that u is a K -quasisuperminimizer.

7. Quasisuperharmonic functions

In the classical potential theory a superharmonic function can be defined as a limit of an increasing sequence of supersolutions of the Laplace equation, provided the limit is not identically ∞. If u is a superharmonic function, then one of the aforementioned sequences is min(u, i), i = 1, 2, …. Our defininition for a quasisuperharmonic function has a global nature. We show later that the cutoff method leads to an equivalent definition for quasisuperharmonic functions.

7.1. Definition. Let Ω ⊂ X be open and K ≥ 1. We say that a function u: Ω → (−∞, ∞] is K -quasisuperharmonic in Ω, if u is not identically ∞ in any component of Ω and there is a sequence of open sets Ω_{i} and K -quasiminimizers v_{i}: Ω_{i} → (−∞, ∞] such that

(i) Ω_{i} ⋐ Ω_{i+1} , (ii) ⋃_{i=1}^{∞} Ω_{i} = Ω, (iii) v_{i} ≤ v_{i+1} in Ω_{i}, (iv) lim_{i→∞} v_{i}^{∗} = u in Ω. Here v_{i}^{∗} is the lower semicontinuous representative of v_{i} given by Lemma 5.3.

Since v_{i}^{∗} is lower semicontinuous, a K -quasisuperharmonic function is lower semicontinuous.

7.2. Proposition. If u is a K -quasisuperminimizer in Ω such that

u(x) = ess lim inf_{y→x} u(y)

for every x ∈ Ω, then u is K -quasisuperharmonic in Ω.

Proof. Let Ω_{i} be any sequence of open sets satisfying (i) and (ii) in Definition 7.1. Choose v_{i} = u^{∗} for i = 1, 2, …. Then v_{i} satisfy (iii) and (iv) in Definition 7.1.

7.3. Theorem. Suppose that u is a K -quasisuperharmonic function in Ω and u is locally bounded above. Then u ∈ N_{loc}^{1,p}(Ω) and u is a K -quasisuper-minimizer in Ω.

Proof. It suffices to show that u is a K -quasisuperminimizer in each open Ω′ ⋐ Ω. Let Ω′ ⋐ Ω. Then there is an increasing sequence of K -quasisupermini-mizers v_{i} in Ω′ such that u = lim_{i→∞} v_{i}^{∗} in Ω′. Since u is locally bounded above, Theorem 6.1 implies that u is a K -quasisuperminimizer in Ω′.

7.4. Theorem. Suppose that u is a K -quasisuperharmonic function in Ω and h is a continuous minimizer in Ω. Then min(u, h) is a K -quasisuperminimizer and K -quasisuperharmonic in Ω.

Proof. First we show that min(u, h) is K -quasisuperharmonic. Since min(u, h) is lower semicontinuous and locally bounded, it suffices to check the conditions (i)– (iv) in Definition 7.1. Let Ω_{i} and v_{i} be as in Definition 7.1. Then each min(v_{i}^{∗}, h) is a K -quasisuperminimizer in Ω_{i} by Corollary 3.9 and

min(v_{i}, h) = min(v_{i}^{∗}, h) → min(u, h)

in Ω as i → ∞. Hence min(u, h) is K -quasisuperharmonic and by Theorem 6.1 it is also a K -quasisuperminimizer in Ω.

7.5. Corollary. Suppose that u is a K -quasisuperharmonic function in Ω and c ∈ R. Then min(u, c) is a K -quasisuperminimizer and K -quasisuperhar-monic in Ω.

If u is K -quasisuperharmonic, then αu and u+β are K -quasisuperharmonic when α ≥ 0 and β ∈ R. However, the sum of two quasisuperharmonic functions is not quasisuperharmonic in general.

7.6. Theorem. Suppose that u_{j} is K_{j} -quasisuperharmonic in Ω, j = 1, 2. Then min(u_{1}, u_{2}) is min(K_{1} + K_{2}, K_{1}K_{2})-quasisuperharmonic.

Proof. Clearly min(u_{1}, u_{2}) is lower semicontinuous. Let Ω_{j,i} and v_{j,i} be as in Definition 7.1 for u_{j}, j = 1, 2. Then Ω_{i} = Ω_{1,i} ∩ Ω_{2,i}, i = 1, 2, …, is an increasing sequence of open sets and min(v_{1,i}, v_{2,i}) is a min(K_{1} + K_{2}, K_{1}K_{2})- quasisuperminimizer in Ω_{i} by Lemmas 3.6 and 3.7. Moreover

min(v_{1,i}, v_{2,i})^{∗} = min(v_{1,i}^{∗}, v_{2,i}^{∗}) → min(v, v)

in Ω as i → ∞. This proves the claim.

7.7. Theorem. Suppose that u is K -quasisuperharmonic in Ω. Then u is locally integrable to a power σ > 0 and, in particular, |u| < ∞ μ-almost everywhere in Ω.

Proof. Let z ∈ Ω be such that u(z) < ∞. Let Ω′ ⋐ Ω such that B(z, 5R) ⋐ Ω′. Let Ω_{i} and v_{i} be as in Definition 7.1. Then inf_{B(z,R)} u < ∞ whenever R > 0 is such that B(z, R) ⊂ Ω′. By Lemma 5.2 for every ball B(z, R) with B(z, 5R) ⊂ Ω′ we have

(\overline{∫} _{B(z,R)} v_{i}^{σ} dμ)^{1/σ} ≤ c inf_{B(z,3R)} v_{i} ≤ c inf_{B(z,3R)} v,

where c < ∞ and σ > 0 are as in Lemma 5.2. In particular, they are independent of i. Letting i → ∞ we conclude that

(\overline{∫} _{B(z,R)} u^{σ} dμ)^{1/σ} ≤ c inf_{B(z,3R)} u < ∞.

We can use the same reasoning as in [KM2] and obtain the following two results.

7.8. Theorem. If u is K -quasisuperharmonic in Ω, then

u(x) = ess lim inf_{y→x} u(y)

for every x ∈ Ω.

7.9. Theorem. If u and v are K -quasisuperharmonic in Ω and u = v μ-almost everywhere in Ω, then u = v in Ω.

We close this section by a characterization of K -quasisuperharmonic functions and by a description of K -quasisuperminimizers among K -quasisuperharmonic functions.

7.10. Theorem. Suppose that u: Ω → (−∞, ∞] is not identically ∞ in any component of Ω. Then u is K -quasisuperharmonic if and only if min(u^{∗}, k) is a K -quasisuperminimizer for every k ∈ R.

Proof. First suppose that u is K -quasisuperharmonic in Ω. It follows from Theorem 7.8 and Corollary 7.5 that min(u^{∗}, k) is a K -quasisuperminimizer in Ω for every k ∈ R.

Then suppose that min(u^{∗}, k) is a K -quasisuperminimizer for every k ∈ R. If Ω_{i} is a sequence of open sets satisfying (i) and (ii) in Definition 7.1, then the sequence v_{i} = min(u^{∗}, i), i = 1, 2, …, satisfies (iii) and (iv) in Definition 7.1.

7.11. Theorem. Suppose that u is K -quasisuperharmonic in Ω. Then u is a K -quasisuperminimizer in Ω if and only if there is v ∈ N_{loc}^{1,p}(Ω) such that u ≤ v μ-almost everywhere.

Proof. If u is a K -quasisuperminimizer, then we can choose v = u. For the converse let v ∈ N_{loc}^{1,p}(Ω) with u ≤ v μ-almost everywhere. Fix Ω′ ⋐ Ω and choose a Lipschitz cutoff function φ such that 0 ≤ φ ≤ 1, φ = 1 in Ω′ and spt φ is compact in Ω. By the lower semicontinuity of u we may assume that u > 0 on spt φ. Fix c > 0 and write

A = {x ∈ spt φ : u(x) ≤ φ(x)v(x)}.

Now A is μ-measurable and A ⋐ Ω. The function u_{c} = min(u, c) is a K - quasisuperminimizer and u_{c} − (φv)_{c} ∈ N_{0}^{1,p}(A). The quasiminimizing property of u_{c} yields

∫_{A} g_{u_{c}}^{p} dμ ≤ K ∫_{A} g_{(φv)_{c}}^{p} dμ ≤ K ∫_{A} g_{φv}^{p} dμ ≤ K ∫_{spt φ} g_{φv}^{p} dμ = M < ∞.

Since A ∩ {u < c} ⊃ Ω′ ∩ {u < c}, we obtain

∫_{Ω′∩{u<c}} g_{u_{c}}^{p} dμ ≤ M.

Letting c → ∞ and noting that μ({u = ∞}) = 0 by Theorem 7.7 we see that g_{u} = lim_{c→∞} g_{u_{c}} defines an L^{p}(Ω′)-function. Since u ≤ v μ-almost everywhere, v ∈ L^{p}(Ω′) and u is locally bounded below, u belongs to L^{p}(Ω′) and since it is easy to see that g_{u} is a weak upper gradient of u in Ω′, it follows that u ∈ N^{1,p}(Ω′). The definition of quasisuperharmonicity together with Theorem 6.1 implies that u is a K -quasisuperminimizer.

8. Definitions of superharmonicity

There are alternative definitions for 1-quasisuperminimizers. One possible choice has been studied in [KM2]. In this section we study how these definitions are related to Definition 7.1. We begin with a definition, which is slightly different from that of [KM2].

8.1. Definition. A function u: Ω → (−∞, ∞] is called superharmonic in Ω, if

(i) u is lower semicontinuous in Ω, (ii) u is not identically ∞ in any component of Ω, (iii) for every open Ω′ ⋐ Ω the comparison principle holds: if v ∈ C(\overline{Ω′})∩N^{1,p}(Ω′)

and v ≤ u on \overline{Ω′}, then 𝒪(v, Ω′) ≤ u in Ω′. Here 𝒪(v, Ω′) denotes the unique solution to the 𝒦_{v,v}(Ω′)-obstacle problem in Ω′. This means that the function 𝒪(v, Ω′) minimizes the p-Dirichlet integral among all functions w such that w − v ∈ N_{0}^{1,p}(Ω′) and w ≥ v μ-almost everywhere in Ω′, see [KM2].

Observe that 𝒪(v, Ω′) is a superminimizer and continuous in Ω′ by [KM2, Theorem 5.5].

8.2. Lemma. Suppose that u is a superharmonic function in Ω and let Ω′ be an open set such that Ω′ ⋐ Ω. Then there is an increasing sequence of continuous superminimizers u_{i}, i = 1, 2, …, in Ω′ such that u = lim_{i→∞} u_{i} everywhere in Ω′.

The proof of this lemma is similar to the proof of [KM2, Theorem 7.7].

8.3. Lemma. If u is a superminimizer in Ω such that

u(x) = ess lim inf_{y→x} u(y)

for every x ∈ Ω, then u is superharmonic.

Proof. It suffices to prove the comparison principle (iii) in Definition 8.1. Let Ω′ be an open set such that Ω′ ⋐ Ω and v ∈ C(\overline{Ω′}) ∩ N^{1,p}(Ω′) and v ≤ u on \overline{Ω′}. Let

A = {x ∈ Ω′ : 𝒪(v, Ω′)(x) > u(x)}.

Since v − 𝒪(v, Ω′) ∈ N_{0}^{1,p}(Ω′) and u ≥ v, we have 𝒪(v, Ω′) − u ∈ N_{0}^{1,p}(A). Now 𝒪(v, Ω′) ≥ u in A and the minimizing property of u yields

(8.4)  ∫_{A} g_{𝒪(v,Ω′)}^{p} dμ ≥ ∫_{A} g_{u}^{p} dμ.

On the other hand 𝒪(v, Ω′) is the unique solution to the obstacle problem 𝒦_{v,v}(Ω′) and hence (8.4) holds as an equality. This implies that 𝒪(v, Ω′) = u μ-almost everywhere in A and thus μ(A) = 0. This means that 𝒪(v, Ω′) ≤ u μ-almost everywhere in Ω′. Since u satisfies

u(x) = ess lim inf_{y→x} u(y)

for every x ∈ Ω and since 𝒪(v, Ω′) is continuous in Ω′, the condition 𝒪(v, Ω′) ≤ u μ-almost everywhere in Ω′ implies 𝒪(v, Ω′) ≤ u everywhere in Ω′.

Next we show that the condition (iii) is stronger than the comparison condition: (iii′) for every open Ω′ ⋐ Ω the comparison principle holds: if v ∈ C(\overline{Ω′})∩N^{1,p}(Ω′)

and v ≤ u on ∂Ω′, then ℋ (v, Ω′) ≤ u in Ω′. Here ℋ (v, Ω′) denotes the unique harmonic function in Ω′ with v − ℋ (v, Ω′) ∈ N_{0}^{1,p}(Ω′).

8.5. Lemma. Suppose that u is superharmonic in the sense of Definition 8.1. Then u satisfies (iii′).

Proof. Let v ∈ C(\overline{Ω′}) ∩ N^{1,p}(Ω′) with v ≤ u on ∂Ω′. For every ε > 0 we construct a function v_{1} such that v − ε − v_{1} ∈ N_{0}^{1,p}(Ω′), v_{1} ∈ C(\overline{Ω′}) and v_{1} ≤ u in \overline{Ω′}.

Since the function u is lower semicontinuous, there is m > −∞ such that u(x) ≥ m in \overline{Ω′}. Let ε > 0. The set

U_{1} = \{x ∈ \overline{Ω′} : v(x) − ε < u(x)\}

is open in \overline{Ω′} and contains ∂Ω′. Choose another open set U_{2} in \overline{Ω′} such that ∂Ω′ ⋐ U_{2} ⋐ U_{1} . Let

λ(x) = min(1, dist(x, (X \ U_{1}) ∩ Ω′)/ dist(U_{2}, (X \ U_{1}) ∩ Ω′)).

Then λ is a Lipschitz continuous function in Ω′, λ = 1 in U_{2} and λ = 0 in (X \ U_{1}) ∩ Ω′. Let

v_{1}(x) = (1 − λ(x))m + λ(x)(v(x) − ε).

Then v_{1} ∈ C(\overline{Ω′}) and v−ε−v_{1} ∈ N_{0}^{1,p}(Ω′) because v−ε = v_{1} in a neighbourhood of ∂Ω′ in Ω′. Moreover, it easily follows that v_{1} ≤ u in \overline{Ω′}.

Now (iii) in Definition 8.1 implies that 𝒪(v_{1}, Ω′) ≤ u in Ω′. On the other hand 𝒪(v_{1}, Ω′) ≥ ℋ (v_{1}, Ω′) in Ω′. Since v −ε−v_{1} ∈ N_{0}^{1,p}(Ω′) and the minimizer is unique, we obtain

ℋ (v_{1}, Ω′) = ℋ (v − ε, Ω′) = ℋ (v, Ω′) − ε.

Hence

ℋ (v, Ω′) − ε ≤ 𝒪(v_{1}, Ω′) ≤ u

in Ω′. Since ε > 0 was arbitrary, we conclude that ℋ (v, Ω′) ≤ u in Ω′ as required.

8.6. Remark. Lemma 8.5 shows that all results in [KM2] which deal with superharmonic functions apply to superharmonic functions in the sense of Definition 8.1.

8.7. Lemma Suppose that u: Ω → (−∞, ∞] is superharmonic in Ω and Ω_{i} is a sequence of open sets such that Ω_{1} ⋐ Ω_{2} ⋐ ⋯ and ⋃_{i=1}^{∞} Ω_{i} = Ω. Then there is a sequence (v_{i}) of functions such that

(i) v_{i} ∈ C(\overline{Ω_{i}}) ∩ N^{1,p}(Ω_{i}), (ii) v_{i} is a superminimizer in Ω_{i}, (iii) v_{i} ≤ v_{j} in \overline{Ω_{i}} if j ≥ i, (iv) lim_{i→∞} v_{i} = u in Ω.

Proof. Let Ω_{i} be a sequence of open sets satisfying the assumptions of the lemma. By the lower semicontinuity of u for every i there is a sequence of functions φ_{i,j}: \overline{Ω_{i}} → R such that φ_{i,j} < φ_{i,j+1} in \overline{Ω_{i}}, φ_{i,j} is Lipschitz and lim_{j→∞} φ_{i,j} = u in \overline{Ω_{i}}. Set v_{1} = 𝒪(φ_{1,1}, Ω_{1}) and if v_{1}, …, v_{i} have been chosen, then choose j such that φ_{i+1,j} ≥ v_{i} in \overline{Ω_{i}} and take v_{i+1} = 𝒪(φ_{i+1,j}, Ω_{i+1}). The sequence (v_{i}) has the required properties.

In particular, Lemma 8.7 implies that if u is superharmonic in the sense of Definition 8.1, then it is 1-quasisuperharmonic in the sense of Definition 7.1. The converse is also true:

8.8. Theorem. A function is superharmonic in the sense of Definition 8.1 if and only if it is 1-quasisuperharmonic in the sense of Definition 7.1.

Proof. Let u be 1-quasisuperharmonic. It suffices to prove (iii) in Definition 8.1. To this end, fix an open set Ω′ ⋐ Ω and let v ∈ C(\overline{Ω′}) ∩ N^{1,p}(Ω′) with v ≤ u on \overline{Ω′}. By Definition 7.1 there is Ω′′ with Ω′ ⋐ Ω′′ ⋐ Ω and an increasing sequence of 1-quasisuperminimizers (that is, superminimizers) v_{i}^{∗} in Ω′′ such that v_{i}^{∗} → u in Ω′′.

Now for each ε > 0 there is i such that v_{i}^{∗} ≥ v − ε in \overline{Ω′}. To prove this, let ε > 0. If there are points x_{i}, i = 1, 2, …, in \overline{Ω′} such that v_{i}^{∗}(x_{i}) < v(x_{i}) − ε, then possibly passing to a subsequence we may assume that x_{i} → x_{0} ∈ \overline{Ω′}. For j ≤ i we have

v_{j}^{∗}(x_{i}) ≤ v_{i}^{∗}(x_{i}) < v(x_{i}) − ε

and since v is continuous in \overline{Ω′}, this yields

lim inf_{i→∞} v_{j}^{∗}(x_{i}) ≤ v(x_{0}) − ε ≤ u(x_{0}) − ε

for each j = 1, 2, …. Now each v_{j}^{∗} is lower semicontinuous and thus

v_{j}^{∗}(x_{0}) ≤ v(x_{0}) − ε ≤ u(x_{0}) − ε

for each j. If u(x_{0}) < ∞, then this is impossible because

lim_{j→∞} v_{j}^{∗}(x_{0}) = u(x_{0}) > u(x_{0}) − ε,

and if u(x_{0}) = ∞, then this is also impossible because v(x_{0}) < ∞. Hence there is i such that v_{i}^{∗} ≥ v − ε in \overline{Ω′}.

Consider the function 𝒪(v, Ω′). Now 𝒪(v − ε, Ω′) = 𝒪(v, Ω′) − ε and the function min(𝒪(v, Ω′) − ε, v_{i}^{∗}) is a superminimizer (the minimum of two superminimizers is a superminimizer by Lemma 3.7). If 𝒪(v, Ω′) − ε > v_{i}^{∗} in some open subset of Ω′, then this leads to a contradiction with the uniqueness of the solution to an obstacle problem. Consequently 𝒪(v, Ω′) − ε ≤ v_{i}^{∗} in Ω′. We thus have

𝒪(v, Ω′) − ε ≤ v_{i}^{∗} ≤ u

in Ω′. Since this holds for all ε > 0, we obtain 𝒪(v, Ω′) ≤ u in Ω′ as required for (iii) in Definition 8.1. This completes the proof.

9. Poisson modification of a quasisuperharmonic function

Suppose that u: Ω → (−∞, ∞] is a K -quasisuperharmonic function and that Ω′ ⋐ Ω is open. Then there are sequences Ω_{i} and v_{i} as in Definition 7.1. Fix i_{0} such that Ω′ ⋐ Ω_{i} for i ≥ i_{0} . Let P (v_{i}, Ω′) denote the Poisson modification of v_{i} in Ω′, see Section 4. Then P (v_{i}, Ω′) is a K -quasisuperminimizer in Ω_{i}.

Let P (v_{i}, Ω′)^{∗} denote the lower semicontinuous representative of P (v_{i}, Ω′) in Ω_{i} as in Lemma 5.3. Then

P (v_{i+1}, Ω′)^{∗} ≥ P (v_{i}, Ω′)^{∗}

in Ω_{i} and

P (u, Ω′) = lim_{i→∞} P (v_{i}, Ω′)^{∗}

defines a K -quasisuperharmonic function in Ω. Note that P (u, Ω′) ≤ u in Ω and hence P (u, Ω′) cannot be ∞ in any component of Ω. The function P (u, Ω′) is called the Poisson modification of u in Ω′. The following theorem summarizes the properties of P (u, Ω′).

9.1. Theorem. Suppose that u: Ω → (−∞, ∞] is a K -quasisuperharmonic function and that Ω′ ⋐ Ω is open. The function P (u, Ω′): Ω → (−∞, ∞] has the properties:

(1) P (u, Ω′) is K -quasisuperharmonic in Ω,

(2) P (u, Ω′) = u in Ω \ Ω′,

(3) P (u, Ω′) ≤ u in Ω, and

(4) P (u, Ω′) is a minimizer in the open set A = {x ∈ Ω′ : P (u, Ω′)(x) < u(x)}.

Proof. Only the property (4) needs a proof. Let v_{i} and Ω_{i}, i = 1, 2, …, be as above. We may assume that Ω′ ⋐ Ω_{i} for all i and that Ω′ is connected, otherwise we consider a component of Ω′. Let h_{i} be the minimizer with boundary values v_{i} in Ω′.

Suppose that x_{0} ∈ A. Then P (u, Ω′)(x_{0}) < u(x_{0}) and, in particular, we have P (u, Ω′)(x_{0}) < ∞. The sequence (h_{i}) is an increasing sequence of minimizers in Ω′ and let h = lim_{i→∞} h_{i}. Now

(9.2)  h(x_{0}) = P (u, Ω′)(x_{0}) < ∞

since

P (u, Ω′)(x_{0}) = lim_{i→∞} P (v_{i}, Ω′)^{∗}(x_{0}) = lim_{i→∞} min(v_{i}, h_{i})(x_{0}) = lim_{i→∞} h_{i}(x_{0}) = h(x_{0}),

because x_{0} ∈ Ω′ and

lim_{i→∞} v_{i}(x_{0}) = u(x_{0}) > P (u, Ω′)(x_{0}).

The Harnack convergence theorem, see [KM2], and (9.2) imply that h is a minimizer in Ω′.

Suppose that u(x_{0}) < ∞ and write

ε = u(x_{0}) − P (u, Ω′)(x_{0}) > 0.

Since u is lower semicontinuous and h is continuous, there is a neighbourhood U of x_{0} such that

u(x) > u(x_{0}) − \frac{1}{2}ε  and  h(x) < h(x_{0}) + \frac{1}{2}ε

for x ∈ U. Because P (v_{i}, Ω′)^{∗} ≤ h_{i} in Ω′, we have P (u, Ω′) ≤ h in Ω′ and thus for all x ∈ U we have

u(x) > u(x_{0}) − \frac{1}{2}ε = P (u, Ω′)(x_{0}) + \frac{1}{2}ε = h(x_{0}) + \frac{1}{2}ε > h(x) ≥ P (u, Ω′)(x).

Hence U ⊂ A and thus A is open. An obvious modification takes care of the case u(x_{0}) = ∞. Now (9.2) holds for each x ∈ A and (4) follows.

10. Polar sets

Here we show that the set where a quasisuperharmonic function is equal to ∞ is of zero p-capacity. For superharmonic functions on metric spaces this question has been studied in [KS2]. We start with a lemma which gives a characterization of compact sets of zero p-capacity. See 2.4 for the definition of the relative pcapacity.

10.1. Lemma. Suppose that C ⊂ X is a compact set. Then C_{p}(C) > 0 if and only if

(10.2)  lim_{t→0} C_{p}(C, C_{t}) = ∞.

Here C_{t} = {x ∈ X : dist(x, C) < t}.

Proof. If (10.2) holds, then clearly C_{p}(C) > 0. Suppose then that C_{p}(C) > 0. For t > 0, let E(t) denote the condenser (C, C_{t}) and for 0 < t < r, let E(t, r) be the the condenser (\overline{C_{t}}, C_{r}). We claim that

(10.3)  lim_{t→0} C_{p}(E(t, r)) = C_{p}(E(r)).

Indeed, let u be an admissible Lipschitz function for the condenser E(r). For ε > 0 the function (1 + ε)u is admissible for E(t, r) provided t is small and hence

C_{p}(E(t, r)) ≤ (1 + ε)^{p} ∫_{X} g_{u}^{p} dμ.

Now X is proper and thus admissible Lipschitz functions are dense in the class of admissible functions for E(r), see [KaS]. Hence we obtain

C_{p}(E(t, r)) ≤ (1 + ε)^{p} C_{p}(E(r))

for small t. This implies

lim_{t→0} C_{p}(E(t, r)) ≤ C_{p}(E(r))

and since C_{p}(E(t, r)) ≥ C_{p}(E(r)) for every t, 0 < t < r, we obtain (10.3). Then we claim that

(10.4)  C_{p}(E(r))^{−1/(p−1)} ≥ C_{p}(E(t))^{−1/(p−1)} + C_{p}(E(t, r))^{−1/(p−1)}.

Indeed, let u_{E(t)} and u_{E(t,r)} be the capacitary potentials of E(t) and E(t, r), respectively. This means that u_{E(t)} is the unique minimizer in C_{t}\C with boundary values 0 in X \ C_{t} and 1 in C. Let

u = αu_{E(t)} + βu_{E(t,r)},

where α, β ≥ 0, α + β = 1. Then u is admissible for E(r) and hence

C_{p}(E(r)) ≤ α^{p} C_{p}(E(t)) + β^{p} C_{p}(E(t, r)).

Note that g_{u_{E(t)}} = 0 μ-almost everywhere in X \ C(t) and g_{u_{E(t,r)}} = 0 μ-almost everywhere in C(t). If C_{p}(E(t)) > 0 and C_{p}(E(t, r)) > 0, we set

α = C_{p}(E(t))^{−1/(p−1)}(C_{p}(E(t))^{−1/(p−1)} + C_{p}(E(t, r))^{−1/(p−1)})^{−1}

and

β = C_{p}(E(t, r))^{−1/(p−1)}(C_{p}(E(t))^{−1/(p−1)} + C_{p}(E(t, r))^{−1/(p−1)})^{−1}

and the claim follows. If either C_{p}(E(t)) = 0 or C_{p}(E(t, r)) = 0, then (10.4) is obvious.

Letting t → 0 we obtain from (10.3) and (10.4) that lim_{t→0} C_{p}(E(t)) = ∞ as required.

We make the following assumptions:

(1) C is a compact set in X,

(2) v is a K -quasisuperminimizer in X such that v ≥ 1 on C, and

(3) v ≥ 0 on C_{r}, r > 0. We let u_{E(r)} be the capacitary potential of the condenser E(r).

10.5. Lemma. If C_{p}(C) > 0, then there exist t, 0 < t < r, r > 0, and a point x_{0} such that d(x_{0}, C) = t with the property that any neighbourhood of x_{0} contains a set A with μ(A) > 0 where

v(x) ≥ \frac{1}{4}u_{E(r)}(x).

The number t depends on C and K and on the other data but not on the function v otherwise. The point x_{0} and the set A depend on v.

Proof. Let 0 < t < r, 0 < ε < 1 and suppose that

εu_{E(r)}(x) ≥ v(x), x ∈ ∂C_{t}.

Since u_{E(r)} ≤ 1, this implies that ε ≥ v(x), x ∈ ∂C_{t}. Let w = min(v, u_{E(r)}), then the function (1 − ε)^{−1}(w − ε) is admissible for the condenser E(t). Thus

\begin{aligned}C_{p}(E(t)) & ≤ (1 − ε)^{−p} ∫_{C_{t}\C} g_{w}^{p} dμ \\ & = (1 − ε)^{−p}(∫_{(C_{t}\C)∩\{u_{E(r)}<v\}} g_{u_{E(r)}}^{p} dμ + ∫_{(C_{t}\C)∩\{u_{E(r)}≥v\}} g_{v}^{p} dμ). \end{aligned}

The K -quasisuperminimizing property of v yields

∫_{(C_{t}\C)∩{u_{E(r)}≥v}} g_{v}^{p} dμ ≤ K ∫_{(C_{t}\C)∩{u_{E(r)}≥v}} g_{u_{E(r)}}^{p} dμ.

Note that u_{E(r)} − v ∈ N_{0}^{1,p}((C_{r} \ C) ∩ {u_{E(r)} ≥ v}) because v ≥ 1 ≥ u_{E(r)} on C, v ≥ 0 and u_{E(r)} = 0 in X \ C_{r}. From this we conclude that

\begin{aligned}C_{p}(E(t)) & ≤ (1 − ε)^{−p}(∫_{C_{r}\C} g_{u_{E(r)}}^{p} dμ + K ∫_{C_{r}\C} g_{u_{E(r)}}^{p} dμ) \\ & ≤ \frac{1 + K}{(1 − ε)^{p}} ∫_{C_{r}\C} g_{u_{E(r)}}^{p} dμ = \frac{1 + K}{(1 − ε)^{p}} C_{p}(E(r)). \end{aligned}

This implies that

(1 − ε)^{p} ≤ (1 + K)\frac{C_{p}(E(r))}{C_{p}(E(t))},

or in other words,

ε ≥ 1 − [(1 + K)\frac{C_{p}(E(r))}{C_{p}(E(t))}]^{1/p}.

Since C_{p}(C) > 0, by Lemma 10.1 we have

lim_{t→0} C_{p}(E(t)) = ∞

and we can choose t, independent of v, such that

[(1 + K)\frac{C_{p}(E(r))}{C_{p}(E(t))}]^{1/p} ≤ \frac{1}{2}.

Then ε ≥ \frac{1}{2} and this means that there is a point x_{0} ∈ ∂C_{t} such that v(x_{0}) > \frac{1}{4} and hence

\frac{1}{4}u_{E(r)}(x_{0}) ≤ \frac{1}{4} < v(x_{0}).

Since v is a K -quasisuperminimizer, by Lemma 5.3 we may assume that

ess lim inf_{x→x_{0}} v(x) = v(x_{0}).

From this we conclude that there exists in any neighbourhood of x_{0} a set A of positive measure such that

\frac{1}{4}u_{E(r)}(x) < v(x), x ∈ A.

This completes the proof.

We say that a set C_{0} ⊂ X is K -polar, if there is an open neighbourhood Ω of C_{0} and a K -quasisuperharmonic function u in Ω such that u(x) = ∞ for every x ∈ C_{0} .

The following theorem is the main result of this section.

10.6. Theorem. If C_{0} is a K -polar set, then C_{p}(C_{0}) = 0.

Proof. Let C_{0} be a K -polar set in X. Then there is an open set Ω such that C_{0} ⊂ Ω and a K -quasisuperharmonic function u_{0} in Ω such that u_{0} = ∞ on C_{0} . By Definition 7.1 there are open sets Ω_{i} and K -quasisuperminimizers v_{i}, i = 1, 2, …, satisfying the properties (i)–(iv) in Definition 7.1. Because the statement of the theorem is local and the functions v_{i} are lower semicontinuous we may assume that Ω_{i} = Ω and v_{i} > 0 for every i = 1, 2, ….

We observe that

C_{0} = ⋂_{j=j_{0}}^{∞} ⋃_{i=i_{0}}^{∞} C_{i,j},

where C_{i,j} = {x ∈ Ω : v_{i}(x) > j} for any integers j_{0} and i_{0} . To see this let x_{0} ∈ C_{0} . Then v_{i}(x_{0}) → ∞ as i → ∞ and hence for any j there are arbitrary large i such that v_{i}(x_{0}) > j. Consequently x_{0} ∈ ⋂_{j=j_{0}}^{∞} ⋃_{i=i_{0}}^{∞} C_{i,j}. For the converse, let x_{0} ∈ ⋂_{j=j_{0}}^{∞} ⋃_{i=i_{0}}^{∞} C_{i,j}. Then for every j ≥ j_{0} there is i such that v_{i}(x_{0}) > j. Since v_{i} is an increasing sequence, v_{i′}(x_{0}) > j for i′ ≥ i. This means that lim_{i→∞} v_{i}(x_{0}) = ∞ and hence x_{0} ∈ C_{0} .

Note that the set C_{i,j} is open by the lower semicontinuity of v_{i} and hence C_{0} is a 𝒢_{δ} -set.

Suppose that C_{p}(C_{0}) > 0. Then there is a compact set C ⊂ C_{0} such that C_{p}(C) > 0. This is due to the fact that the p-capacity is a Choquet capacity, see [KM1]. This means that if we choose r > 0 such that r < dist(C, X \ Ω), then C_{p}(C, C_{t}) > 0. For every j = 1, 2, …, we have

C ⊂ ⋃_{i=i_{0}}^{∞} C_{i,j}

for arbitrarily large i_{0} . Now the sets C_{i,j} are open and C is compact. Hence for any fixed j and i_{0} there is i′ = i′(j, i_{0}) such that

C ⊂ ⋃_{i=i_{0}}^{i′} C_{i,j}.

On the other hand, we have C_{i,j} ⊂ C_{i+1,j} and hence C ⊂ C_{i′,j}.

Let E(r), r > 0, denote the condenser (C, C_{r}) and u_{E(r)} the capacitary function of E(r). We can now apply Lemma 10.5 to the K -quasiminimizer v_{i′}/j. By Lemma 10.5 there are t, 0 < t < r, independent of i′ and j, and a point x_{0} such that dist(x_{0}, C) = t and any neighbourhood of x_{0} contains a set A with μ(A) > 0 and

\frac{v_{j′}(x)}{j} ≥ \frac{1}{4}u_{E(r)}(x), x ∈ A.

Since v_{i} is an increasing sequence, this means that

(10.7)  \frac{v_{i}(x)}{j} ≥ \frac{1}{4}u_{E(r)}(x), x ∈ A,

for all i ≥ i′.

By Theorem 7.7 we have u_{0}(x) < ∞ for μ-almost every x ∈ Ω. Since v_{i}(x) ≤ u_{0}(x) for all x and since (10.7) holds, we can pick x̃_{0} ∈ A such that u(x̃_{0}) < ∞, (10.7) holds at x̃_{0} for i ≥ i′ and

t − ε_{1} < dist(x̃_{0}, C) < t + ε_{1},

where ε_{1} = min(t, r − t)/10.

By (10.7) this implies

\frac{u_{0}(x̃_{0})}{j} ≥ \frac{1}{4}u_{E(r)}(x̃_{0})

for each j. Letting j → ∞ we obtain that u_{E(r)}(x̃_{0}) = 0 and by the Harnack inequality u_{E(r)} = 0 in a component U of C_{t} \ C containing x̃_{0} . Since the above reasoning can be applied to each such component, this means that u_{E(r)} = 0 in C_{t} \ C, but this implies that C_{p}(C) = 0, which is a contradiction. This completes the proof.

490 Juha Kinnunen and Olli Martio

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Received 26 February 2003