Existence and almost uniqueness for p-harmonic Green functions on bounded domains in metric spaces
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This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Existence and almost uniqueness for p-harmonic Green functions on bounded domains in metric spaces © 2020 The Authors. Published by Elsevier Inc Published version Björn, Anders; Björn, Jana; Lehrbäck, Juha Björn, A., Björn, J., & Lehrbäck, J. (2020). Existence and almost uniqueness for p-harmonic Green functions on bounded domains in metric spaces. Journal of Differential Equations, 269(9), 6602- 6640. https://doi.org/10.1016/j.jde.2020.04.044 2020
Available online at www.sciencedirect.com ScienceDirect J. Differential Equations 269 (2020) 6602–6640 www.elsevier.com/locate/jde Existence and almost uniqueness for p-harmonic Green functions on bounded domains in metric spaces Anders Björn a,∗, Jana Björn a, Juha Lehrbäck b aDepartment of Mathematics, Linköping University, SE-581 83 Linköping, Sweden bDepartment of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland Received 4 February 2020; accepted 24 April 2020 Abstract We study (p-harmonic) singular functions, defined by means of upper gradients, in bounded domains in metric measure spaces. It is shown that singular functions exist if and only if the complement of the domain has positive capacity, and that they satisfy very precise capacitary identities for superlevel sets. Suitably normalized singular functions are called Green functions. Uniqueness of Green functions is largely an open problem beyond unweighted Rn, but we show that all Green functions (in a given domain and with the same singularity) are comparable. As a consequence, for p-harmonic functions with a given pole we obtain a similar comparison result near the pole. Various characterizations of singular functions are also given. Our results hold in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, or under similar local assumptions. ©2020 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). MSC: primary 31C45; secondary 30L99, 31C12, 31C15, 31E05, 35J08, 35J92, 46E36, 49Q20 Keywords: Capacitary potential; Doubling measure; Metric space; p-harmonic Green function; Poincaré inequality; Singular function *Corresponding author. E-mail addresses: [email protected] (A. Björn), [email protected] (J. Björn), [email protected] (J. Lehrbäck). https://doi.org/10.1016/j.jde.2020.04.044 0022-0396/©2020 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 6603 1. Introduction Let ⊂Rnbe a bounded domain, and let x0∈. Then uis a p-harmonic Green function in with singularity at x0if pu:= div(|∇u|p−2∇u) =−δx0in (1.1) with zero boundary values on ∂(in Sobolev sense), where δx0is the Dirac measure at x0. Such a Green function is in particular p-harmonic in \{x0}and p-superharmonic in the whole domain . If 1 <p≤n, it is also unbounded. For example, the p-harmonic Green function in the unit ball in unweighted Rnis given by u(x) =ω1/(1−p) n−1⎧ ⎨ ⎩ p−1 |n−p||x|(p−n)/(n−1)−1,if p= n, −log |x|,if p=n, where ωn−1is the surface area of Sn−1. In metric measure spaces, Holopainen–Shanmugalingam [32]gave a definition of singular functions, which behave similarly to the Green functions in Rn. In this paper we introduce a simpler definition of singular functions, and then define Green functions as suitably normalized singular functions. See Section 12 for the definition from [32] and for a discussion on the relation between these different definitions. In a metric measure space X=(X, d, μ) there is (a priori) no equation available for defining p-harmonic functions, and they are instead defined as local minimizers of the p-energy integral ˆgp udμ, where guis the minimal p-weak upper gradient of u, see Definition 2.1. This definition of pharmonic functions is in, e.g., Rnequivalent to the definition using the p-Laplace operator pu. Definition 1.1. Let ⊂Xbe a bounded domain. A positive function u: →(0, ∞] is a singular function in with singularity at x0∈if it satisfies the following properties: (S1) uis p-superharmonic in ; (S2) uis p-harmonic in \{x0}; (S3) u(x0) =supu; (S4) infu =0; (S5) ˜u∈N1,p loc (X \{x0}), where ˜u=uin , 0onX\. There is actually some redundancy in this definition under very mild assumptions, see Theorem 1.6 and Remark 6.3. Singular functions are sometimes called Green functions in the
6604 A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 literature, and vice versa. Moreover they can be normalized, or pseudonormalized, in different ways. For Green functions, we require the following precise normalization in terms of the variational capacity of superlevel sets. Definition 1.2. Let ⊂Xbe a bounded domain. A Green function is a singular function which satisfies capp(b,)=b1−p,when 0 <b<u(x 0), (1.2) where b={x∈ :u(x) ≥b}. In fact, it follows that Green functions usatisfy capp(b, a)=(b −a)1−p,when 0 ≤a<b≤u(x0), (1.3) where a={x∈ :u(x) >a}and we interpret ∞1−pas 0, see Theorem 9.3. In unweighted Rn, the study of singular and (p-harmonic) Green functions with p= 2 goes back to Serrin [41], [42]. On domains in weighted Rn(with a p-admissible weight) the existence of singular functions follows from Heinonen–Kilpeläinen–Martio [28, Theorem 7.39]. (Instead of (S5) they showed that condition (b.2) in Theorem 7.2 holds, but in view of Theorem 7.2 this establishes the existence of singular functions in our sense.) The classical p-harmonic Green functions defined by (1.1)in unweighted Euclidean domains (and similarly for domains in weighted Rnwith a p-admissible weight) coincide with the Green functions given by Definition 1.2, see Remark 9.4. Uniqueness of Green functions in unweighted Euclidean domains was for p= 2 established by Kichenassamy–Veron [35](see Section 9), but is not really known beyond that. In particular, it remains open in weighted Rn. However, Holopainen [31, Theorem 3.22] proved uniqueness in regular relatively compact domains in n-di- mensional Riemannian manifolds (equipped with their natural measures) when p=n. Moreover, in Balogh–Holopainen–Tyson [2], uniqueness was shown for global Q-harmonic Green functions in Carnot groups of homogeneous dimension Q. In this paper we show the existence of singular functions and also of Green functions satisfying the precise normalization (1.2), or equivalently (1.3), under the following standard assumptions on the metric measure space X; see Section 2for the relevant definitions. We make the following general assumptions in the theorems in the introduction: Assume that 1 <p<∞and that Xis a complete metric space equipped with a doubling measure μsupporting a p-Poincaré inequality. Let ⊂Xbe a bounded domain and let x0∈. We also write Br=B(x0, r) for r>0. These assumptions are fulfilled in weighted Rnequipped with a p-admissible measure, on Riemannian manifolds and Carnot–Carathéodory spaces equipped with their natural measures, and in many other situations, see Sections 2and 13 for further details. Actually, the above assumptions on the space Xcan be relaxed to similar local assumptions. The same applies also to our other results, see Section 11 for details. The following theorem summarizes some of our main results.
A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 6605 Theorem 1.3. (a) There exists a Green function (or equivalently, in view of (b), a singular function) in with singularity at x0if and only if Cp(X \) >0(which is always true if Xis unbounded). (b) If uis a singular function in with singularity at x0, then there is a unique α>0such that αu is a Green function. (c) If uand vare two Green functions in with singularity at x0, then u≃v, (1.4) where the comparison constants depend only on p, the doubling constant and the constants in the Poincaré inequality. If moreover Cp({x0}) >0, then u =vand it is a multiple of the capacitary potential for {x0}in . (d) If uis a Green function (or equivalently, in view of (b), a singular function) in with singularity at x0, then uis bounded if and only if Cp({x0}) >0. When Cp({x0}) =0, Theorem 1.3 (c) gives almost uniqueness of Green functions, and in particular shows that all Green functions have the same growth behaviour near the singularity. As mentioned above, uniqueness of Green functions is not known even in weighted Rn(when Cp({x0}) =0). Proposition 5.3 in our forthcoming paper [14]shows that Cp({x0}) =0if and only if δ ˆ 0ρ μ(Bρ)1/(p−1) dρ =∞ for some (or equivalently all) δ>0, see also Remark 4.7. In unweighted Rn, this happens if and only if p≤n. The next result shows that (1.4)is strong enough to make p-harmonic functions into singular ones, provided that Cp({x0}) =0. Theorem 1.4. Assume that Cp({x0}) =0. Let ube a singular function in with singularity at x0, and let v: →(0, ∞] be a function which is p-harmonic in \{x0}. Then vis a singular function in with singularity at x0if and only if v≃u. Holopainen–Shanmugalingam [32] provided a construction of singular functions (according to their definition); see however Remark 12.2. We show in Proposition 12.3 that, under the assumptions used in [32], the definition therein is essentially equivalent to Definition 1.1, up to a normalization. Hence we also recover the existence of singular functions according to the definition in [32]. Nevertheless, Definition 1.1 seems to be both more general and more flexible, and hence better suited e.g. for studying the existence and uniqueness of singular and Green functions. In particular, the definition in [32] contains explicit superlevel set inequalities, whereas we show in Lemma 9.1 that a precise superlevel set identity is a consequence of the properties assumed in Definition 1.1. The absence of any a priori superlevel set requirements makes it easy to apply our results to general p-harmonic functions with poles, see Theorem 10.1. From the superlevel set property we in turn obtain the following pointwise estimate for Green functions near their singularities.
6606 A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 Theorem 1.5. If uis a Green function in with singularity at x0, then for all r>0such that B50λr ⊂and all x∈∂Br, u(x) ≃capp(Br,) 1/(1−p),(1.5) where the comparison constants depend only on p, the doubling constant and the constants in the Poincaré inequality. Here λis the dilation constant in the p-Poincaré inequality. In weighted Rn(with a p-admissible weight), (1.5)was obtained by Fabes–Jerison–Kenig [23, Lemma 3.1] (for p=2) and Heinonen–Kilpeläinen–Martio [28, Theorem 7.41] (for balls and 1 <p<∞). For p-Laplacian-type equations of the form div A(x, u, ∇u) =B(x,u,∇u) (1.6) in unweighted Rn, with 1 <p<∞, it is due to Serrin [41, Theorem 12], [42, Theorem 1]. In Carnot–Carathéodory spaces, (1.5)was proved by Capogna–Danielli–Garofalo [20, Theorem 7.1]. It was also obtained in some specific cases on metric spaces by Danielli– Garofalo–Marola [22], see Remark 9.5. In [22, Section 6] they obtained some further results for Cheeger singular and Cheeger–Green functions, cf. Section 13. See also Holopainen [31, Section 3] for results on Green functions in regular relatively compact domains in n-dimensional Riemannian manifolds (equipped with their natural measures) when 1 <p≤n. We also establish various useful characterizations for singular functions. Theorems 1.4 and 1.6 contain some of these, but in Sections 7–9we obtain several additional characterizations, which are either more technical to state or which only hold in one of the cases Cp({x0}) =0or Cp({x0}) >0. Theorem 1.6. Assume that Cp(X \) >0and let u: →(0, ∞]. Then the following are equivalent: (a) uis a singular function in with singularity at x0; (b) usatisfies (S1),(S2) and (S5); (c) u(x0) =limx→x0u(x) and usatisfies (S2) and (S5). The outline of the paper is as follows. We begin in Section 2by recalling the basic definitions related to the analysis on metric spaces. In Section 3we establish sharp superlevel set formulas for capacitary potentials. Such a formula was obtained in weighted Rn(with a padmissible weight) in Heinonen–Kilpeläinen–Martio [28, p. 118]. Their argument depends on the Euler–Lagrange equation, which is not available in the metric space setting considered here. Nevertheless, we are able to obtain this formula with virtually no assumptions on the metric space nor on the sets involved, and at the same time the proof is considerably shorter than the one in [28, pp. 116–118]. See Section 3for more details. Section 4contains a discussion about (super)harmonic functions in the metric setting, while in Section 5we obtain, with the help of harmonic extensions and Perron solutions, some finer properties for these functions and, in particular, for capacitary potentials. The actual study of singular and Green functions begins in Section 6, where we record some easy observations concerning singular functions. Sections 7and 8contain proofs for the existence and further properties of singular functions under the respective assumptions that Cp({x0}) =0or
A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 6607 Cp({x0}) >0. Then, in Section 9, we establish a sharp superlevel set property for superharmonic functions and show how this property yields the existence of Green functions. In Section 10 we study the growth behaviour of p-harmonic functions with poles. Local assumptions are discussed in Section 11, and in Section 12 we compare our definitions and results with those in Holopainen– Shanmugalingam [32]. By the theory of Cheeger [21], it is possible to use also a PDE approach to the study of singular and Green functions in metric spaces satisfying the standard assumptions. In Section 13 we show that in this setting the Cheeger–Green functions, based on Definition 1.2, actually satisfy an equation corresponding to (1.1) and hence the situation is analogous to that in (weighted) Rn. Note, however, that Cheeger p-(super)harmonic functions, and thus also the corresponding singular and Green functions, differ in general from those defined by means of upper gradients. Acknowledgment A.B. and J.B. were supported by the Swedish Research Council, grants 2016-03424 and 621- 2014-3974, respectively. J.L. was supported by the Academy of Finland, grant 252108. 2. Preliminaries We assume throughout the paper that 1 <p<∞and that X=(X, d, μ) is a metric space equipped with a metric dand a positive complete Borel measure μsuch that 0 <μ(B) <∞ for all balls B⊂X. The σ-algebra on which μis defined is obtained by the completion of the Borel σ-algebra. It follows that Xis separable. To avoid pathological situations we assume that Xcontains at least two points. Next we are going to introduce the necessary background on Sobolev spaces and capacities in metric spaces. Proofs of most of the results mentioned in this section can be found in the monographs Björn–Björn [8] and Heinonen–Koskela–Shanmugalingam–Tyson [30]. A curve is a continuous mapping from an interval, and a rectifiable curve is a curve with finite length. We will only consider curves which are nonconstant, compact and rectifiable, and thus each curve can be parameterized by its arc length ds. A property is said to hold for p-almost every curve if it fails only for a curve family with zero p-modulus, i.e. there exists 0 ≤ρ∈Lp(X) such that ´γρds=∞for every curve γ∈. We begin with the notion of p-weak upper gradients as defined by Koskela–MacManus [40], see also Heinonen–Koskela [29]. Definition 2.1. A measurable function g:X→[0, ∞] is a p-weak upper gradient of a function f:X→[−∞, ∞] if for p-almost every curve γ:[0, lγ] →X, |f(γ(0)) −f(γ(l γ))|≤ˆ γ gds, where we follow the convention that the left-hand side is ∞whenever at least one of the terms therein is ±∞. If fhas a p-weak upper gradient in Lp loc(X), then it has an a.e. unique minimal p-weak upper gradient gf∈Lp loc(X) in the sense that for every p-weak upper gradient g∈Lp loc(X) of fwe
6608 A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 have gf≤ga.e. Following Shanmugalingam [43], we define a version of Sobolev spaces on the metric space X. Definition 2.2. For a measurable function f:X→[−∞, ∞], let fN1,p(X) =ˆ X |f|pdμ+inf gˆ X gpdμ1/p , where the infimum is taken over all p-weak upper gradients of f. The Newtonian space on Xis N1,p(X) ={f:fN1,p(X) <∞}. The space N1,p(X)/∼, where f∼hif and only if f−hN1,p(X) =0, is a Banach space and a lattice. In this paper we assume that functions in N1,p(X) are defined everywhere, not just up to an equivalence class in the corresponding function space. This is needed for the definition of pweak upper gradients to make sense. For a measurable set A ⊂X, the Newtonian space N1,p(A) is defined by considering (A, d|A, μ|A)as a metric space in its own right. If f, h ∈N1,p loc (X), then gf=gha.e. in {x∈X:f(x) =h(x)}. In particular, gmin{f,c}=gfχ{f<c}for any c∈R. Definition 2.3. The Sobolev capacity of an arbitrary set E⊂Xis Cp(E) =inf uup N1,p(X), where the infimum is taken over all u ∈N1,p(X) such that u ≥1on E. We say that a property holds quasieverywhere (q.e.) if the set of points for which it fails has Sobolev capacity zero. The capacity is the correct gauge for distinguishing between two Newtonian functions. If u ∈N1,p(X), then u ∼vif and only if u =vq.e. Moreover, if u, v∈N1,p loc (X) and u =va.e., then u =vq.e. Both the Sobolev and the variational capacity (defined below in Definition 3.1) are countably subadditive. Definition 2.4. For measurable sets E⊂A ⊂X, let N1,p 0(E;A) ={f|E:f∈N1,p(A) and f=0onA\E}. If A =X, we omit Xin the notation and write N1,p 0(E). Whenever convenient, we regard functions in N1,p 0(E; A) as extended by zero to A \E. The measure μis doubling if there is a constant C>0 such that μ(B(x, 2r))≤Cμ(B(x, r)) (2.1) for all balls B(x, r) ={y∈X:d(x, y) <r}.
A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 6609 The space X(or the measure μ) supports a p-Poincaré inequality if there exist constants C>0 and λ ≥1 such that for all balls B=B(x, r) ⊂X, all integrable functions uon X, and all p-weak upper gradients gof u, − ˆ B |u−uB|dμ≤Cr− ˆ λB gpdμ1/p ,(2.2) where uB:= − ´Bu dμ := ´Bu dμ/μ(B) is the integral average and λB stands for the dilated ball B(x, λr). If Xis complete and μis a doubling measure supporting a p-Poincaré inequality, then functions in N1,p(X) and those in N1,p(), for open ⊂X, are quasicontinuous. This will be important in Theorem 5.2, but affects also how we formulate various statements, such as the definition of the Sobolev capacity above. If X=Rnis equipped with dμ =wdx, then w≥0is a p-admissible weight in the sense of Heinonen–Kilpeläinen–Martio [28]if and only if μis a doubling measure which supports a p-Poincaré inequality, see Corollary 20.9 in [28](which is only in the second edition) and Proposition A.17 in [8]. In this case, N1,p(Rn)and N1,p() are the refined Sobolev spaces defined in [28, p. 96], and moreover our Sobolev and variational capacities coincide with those in [28]; see Björn–Björn [8, Theorem 6.7 (ix) and Appendix A.2] and [9, Theorem 5.1]. The situation is similar on Riemannian manifolds and Carnot–Carathéodory spaces equipped with their natural measures; see Hajłasz–Koskela [27, Sections 10 and 11] and Section 13 below for further details. Throughout the paper, we write YZif there is an implicit constant C>0 such that Y≤CZ. We also write YZif ZY, and Y≃Zif YZY. Unless otherwise stated, we always allow the implicit comparison constants to depend on the standard parameters, such as p, the doubling constant and the constants in the Poincaré inequality. 3. Superlevel identities for capacitary potentials Definition 3.1. If E⊂Aare bounded subsets of X, then the variational capacity of Ewith respect to Ais capp(E, A) =inf uˆ X gp udμ, (3.1) where the infimum is taken over all u ∈N1,p(X) such that u ≥1on Eand u =0on X\A. If no such function uexists then capp(E, A) =∞. One can equivalently take the above infimum over all u ∈N1,p(X) such that u ≥1q.e. on E and u =0q.e. on X\A; we call such uadmissible for the capacity capp(E, A). Since Ais not required to be measurable we cannot take the integral in (3.1) over A, and it is also important that the minimal p-weak upper gradient of uis taken with respect to X. However, if Ais open then the integral and the minimal p-weak upper gradient can equivalently be taken over A.
6616 A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 Conversely, if lim inf r→0 μ(Br) rq>0forsomeq<p, then Cp({x0}) >0. It is also shown in [13] that the power of decay of μ(Br)alone cannot determine whether Cp({x0}) =0. However, Proposition 5.3 in our forthcoming paper [14]shows that Cp({x0}) =0if and only if δ ˆ 0ρ μ(Bρ)1/(p−1) dρ =∞ for some (or equivalently all) δ>0. 5. Perron solutions and boundary behaviour In addition to the general assumptions from the beginning of Section 4, we assume in this section that is bounded and that Cp(X \) >0. Perron solutions will be an important tool for us. Definition 5.1. Given f:∂ →[−∞, ∞], let Uf() be the collection of all superharmonic functions uin that are bounded from below and satisfy lim inf x→yu(x) ≥f(y) for all y∈∂. The upper Perron solution of fis defined by Pf(x)=inf u∈Uf() u(x), x ∈. The lower Perron solution is defined similarly using subharmonic functions or by Pf= −Pf. If Pf=Pf, then we denote the common value by Pf. Moreover, if Pfis realvalued, then fis said to be resolutive (with respect to ). We will often write Pf instead of Pf, and similarly for Pf, P f as well as for Hf . An immediate consequence of Definition 5.1 is that Pf 1≤Pf 2whenever f1≤f2on ∂. It follows from Theorem 7.2 in Kinnunen–Martio [36](or Theorem 9.39 in [8]) that P f ≤ Pf. In each component of , Pf is either p-harmonic or identically ±∞, by Theorem 4.1 in Björn–Björn–Shanmugalingam [16]. (This and all the facts below can also be found in Chapter 10 in [8].) We will need several results from [16, Sections 5 and 6], which we summarize as follows. (Part (a) follows from [16, Theorem 5.1] after multiplying fby a suitable Lipschitz cutoff function.)
A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 6617 Theorem 5.2. (a) If f∈N1,p(G) for some open set G ⊃, then Hf =Pf . (b) If f∈C(∂), then fis resolutive. (c) If fis bounded and as in (a) or (b), and uis a bounded p-harmonic function in such that lim x→yu(x) =f(y) for q.e. y∈∂, then u =Pf . Remark 5.3. In order for (a) to be possible it is important that the Newtonian function fis quasicontinuous, which follows from Theorem 1.1 in Björn–Björn–Shanmugalingam [17](or Theorem 5.29 in [8]). A boundary point x0∈∂ is regular if limx→x0Pf (x) =f(x 0)for every f∈C(∂). We will need the following so-called Kellogg property, see Theorem 3.9 in Björn–Björn– Shanmugalingam [15]. (The definition of regular points is different in [15], but by [16, Theorem 6.1] it is equivalent to our definition.) Theorem 5.4 (The Kellogg property). The set of irregular boundary points has capacity zero. We will also use that regularity is a local property of the boundary, i.e. that x0∈∂ is regular with respect to if and only if it is regular with respect to ∩Bfor every (or some) ball Bx0, see Theorem 6.1 in Björn–Björn [6](or [8, Theorem 11.1]). Moreover, if G ⊂and x0∈∂ ∩∂G is regular with respect to , then it is also regular with respect to G, see [6, Corollary 4.4] (or [8, Corollary 11.3]). Another important tool in this paper is capacitary potentials, which we studied in Section 3 in very general situations. Under our standing assumptions we can say considerably more. In particular, capacitary potentials are unique up to sets of capacity zero, by Theorem 5.13 in Björn– Björn [10]. In fact, it is easy to see that any capacitary potential is a solution to the KχE,0()- obstacle problem, as defined in [8, Section 7], and vice versa. Thus, provided that there is a capacitary potential of (E, ), Theorem 8.27 in [8]shows that there is a unique lsc-regularized capacitary potential u, i.e. such that u∗=uin and u ≡0 on X\. Then ualso coincides with the “capacitary potential” as defined in [8, Definition 11.15], and is therefore superharmonic in , by [8, Proposition 9.4]. We shall sometimes call u|a capacitary potential as well. Recall that a capacitary potential of (E, ) exists if and only if capp(E, ) <∞. We shall need the following two characterizations of capacitary potentials. Lemma 5.5. Let E⊂be relatively closed and let u: →[0, ∞]. Then uis the lsc-regularized capacitary potential of (E, ) if and only if all of the following conditions hold: (a) uis superharmonic in ; (b) uis p-harmonic in G := \E; (c) u =1q.e. on E; (d) u ∈N1,p 0().
6618 A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 Moreover, u =HGuin Gand lim x→yu(x) =0at every regular boundary point y∈∂\E. (5.1) In particular, (5.1)holds for q.e. y∈∂ \E. Proof. If uis an lsc-regularized capacitary potential of (E, ), then it satisfies (c) and (d) by assumption, (a) by the above, and (b) by Theorem 8.28 in [8]. Moreover, it is straightforward to see that within G, uis the lsc-regularized solution of the K0,u(G)-obstacle problem, i.e. u =HGu in G. Hence, (5.1) and the last statement follow from [8, Theorem 11.11 (j)] together with the Kellogg property (Theorem 5.4). Conversely, if u ∈N1,p 0() is p-harmonic in Gthen, by definition, u =HGuin G. If, in addition, u =1q.e. on Ethen u ∈KχE,0() and must therefore be a capacitary potential of (E, ). If it is also superharmonic in , then it is lsc-regularized. Lemma 5.6. Let K⊂be compact and let u: →[0, ∞]. Then uis the lsc-regularized capacitary potential of (K, ) if and only if all of the following conditions hold: (a) uis bounded and p-harmonic in G := \K; (b) u ≡1in int K; (c) lim Gx→yu(x) =χK(y) for q.e. y∈∂G; (d) u(y) =lim inf Gx→yu(x) for all y∈ ∩∂K. Moreover, u =PGχKin G. Proof. Let ube the lsc-regularized capacitary potential of (K, ) and set ˜u=⎧ ⎪ ⎨ ⎪ ⎩ uin \K, 1inK, 0inX\. Then ˜u=uq.e. in and ˜u∈N1,p 0(). Thus u=HGu=HG˜u=PG˜u=PGχKin G, by (4.1) and Theorem 5.2 (a). (In particular, χK∈C(∂G) is resolutive with respect to G.) Hence (a) holds, and so does (c) by the Kellogg property (Theorem 5.4). Since uis the lsc-regularization of ˜u, it satisfies (b) and (d). Conversely, if uis bounded and p-harmonic in Gand satisfies (c) then u =PGχKin Gby Theorem 5.2 (c). Hence, if ualso satisfies (b) and (d), then it is the lsc-regularized capacitary potential of (K, ), by the first part of the proof.
A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 6619 Lemma 5.7. Assume that K⊂is compact and that u: →(0, ∞] is p-harmonic in \K. For an open set Vsuch that K⊂V, consider the following conditions: (b.1)u ∈N1,p 0( \V; X\V); (b.2)uis bounded in \Vand lim x→yu(x) =0for q.e. y∈∂; (5.2) (b.3)uis bounded in \Vand min{u, k} ∈N1,p 0() for every k>0. Then (b.3) ⇒(b.1) ⇔(b.2). Moreover, (5.2)can be equivalently replaced by lim x→yu(x) =0for every regular y∈∂. (5.3) As ucan be defined arbitrarily in Kin (b.1) and (b.2), but not in (b.3), we see that the implication (b.3)⇒(b.1)is not an equivalence. Proof. Extend uby letting u =0 on X\. Let G = \V. (b.3)⇒(b.1) Since uis bounded in \V, we have u =uk:= min{u, k}therein for large k. As uk∈N1,p 0() ⊂N1,p 0( \V; X\V),(b.1 ) follows. (b.1)⇒(b.2)As uis p-harmonic in \Kand u ∈N1,p 0( \V; X\V), it follows from the definition that HGu =uin G. Since uis bounded on ∂V and vanishes on ∂, there is α>0 such that u ≤αv on ∂G, where vis the lsc-regularized capacitary potential for Vin . By the comparison principle (4.1), u ≤αv also in Gand, in particular, uis bounded therein. Now, (5.3) follows from (5.1), applied to v, while (5.2) follows from (5.3) and the Kellogg property (Theorem 5.4). (b.2)⇒(b.1)Let η≥0be a Lipschitz function on Xsuch that η=1on ∂V and η=0 in a neighbourhood of K∪(X \). As uis p-harmonic in \Kand ∂V \K, the function u|∂G =ηu|∂G is continuous. Since (5.2)or (5.3) holds, Theorem 5.2 (c) shows that u =PG(ηu). It follows from the Leibniz rule (see [8, Theorem 2.15]) that ηu ∈N1,p(X). Hence Theorem 5.2 (a) implies that u =HG(ηu) in G, which yields u ∈N1,p 0( \V; X\V). Note that in the generality of Section 3, capacitary potentials are unique up to sets of capacity zero under rather mild conditions, by Theorem 5.13 in [10]. Nevertheless, it is far from clear if we can then always pick a canonical representative in a suitable way. In particular, even if Ais open it is not at all clear if u∗=uq.e., that is whether there always exists an lsc-regularized capacitary potential. Under our standing assumptions in this section it is true that u∗=uq.e., but this is a consequence of the rather deep interior regularity theory for superminimizers. 6. Singular functions In addition to the general assumptions from the beginning of Section 4, we assume in this section that is a bounded domain. Recall properties (S1)–(S5) in Definition 1.1 of singular functions, and that a domain is a nonempty open connected set. In this paper we are interested in singular functions on bounded
6620 A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 domains only. For simplicity, we will often just say that uis a singular function, when we implicitly mean within and with singularity at x0. Note that a singular function must be nonconstant in , as it is positive and (S4) holds. Our first observation, Proposition 6.1, shows that Cp(X \) >0is a necessary condition for the existence of singular functions. (We will later show that it is also sufficient.) Under this condition, the theory of singular functions on bounded domains splits naturally into two cases depending on whether Cp({x0}) =0or Cp({x0}) >0, which we will consider in Sections 7and 8, respectively. But first we deduce some results covering both cases simultaneously. Proposition 6.1. If Cp(X \) =0, then there is no singular function in (or more generally no positive superharmonic function in satisfying (S4)). Proof. It follows directly that Xmust bounded. Let u >0be a superharmonic function in . By Theorem 6.3 in Björn [5](or Theorem 12.3 in [8]), uhas a superharmonic extension to all of X, and by Corollary 9.14 in [8]this extension must be constant. Hence udoes not satisfy (S4) and is, in particular, not a singular function. Proposition 6.2. If Cp(X \) >0then there is no positive p-harmonic function in which satisfies (S5). In particular, a singular function in is never p-harmonic in all of . Proof. Assume that uis a positive p-harmonic function in satisfying (S5). In particular, u ∈ N1,p loc (). Extend uas 0 on X\. Since u ∈N1,p loc () and (S5) holds, we see that u ∈N1,p(X) and hence u ∈N1,p 0(). But then u =Hu =H0 ≡0in , which is a contradiction as uis positive, i.e. no such function exists. Finally, if there is a singular function in , then Proposition 6.1 implies that Cp(X \) >0, and thus the singular function cannot be p-harmonic in by the first part of the lemma. Remark 6.3. There is actually some redundancy in the definition of singular functions. As we shall see, by Theorem 8.5 below, if Cp(X \) >0 then it is enough to assume that usatisfies (S1), (S2) and (S5). However, in the somewhat pathological case Cp(X \) =0, this is not enough as it would not prevent a constant function from being a singular function. To cover also this case it is enough to additionally assume (S4) or to assume that uis nonconstant, or that uis not p-harmonic in . Even though (S3) is thus redundant, we have included it in the definition as it seems such a natural requirement for u. Also, for unbounded domains it seems that one may need to require at least these five properties to obtain a coherent theory of singular functions, but we postpone such a study to a future paper. That (S1) cannot be dropped even if (S3) is replaced by the stronger requirement (S3)u(x0) =sup\{x0}u, follows by considering the function u(x) =1+x, −1<x<0, 2−2x, 0≤x<1,
A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 6621 which is p-harmonic in (−1, 1) \{0} ⊂X:= R. (Note that if (S1) holds, then (S3) ⇔(S3), but without assuming (S1), assuming (S3) might be more natural.) However, if Cp({x0}) =0 then it follows from Theorem 7.2 below that (S1) can be replaced by e.g. u(x0) =∞, and thus Proposition 4.4 shows that, in this case, (S1) can be dropped provided that (S3) is kept. To see that (S2) cannot be dropped we instead let ube the lsc-regularized capacitary potential of (B1, B2)in Rn. That (S5) cannot be dropped follows from Example 7.3 below. We conclude this section by summarizing some useful properties of singular functions. Proposition 6.4. If uis a singular function in with singularity at x0∈, then (a) u(x0) =limx→x0u(x); (b) u ∈N1,p 0( \Br; X\Br)for every r>0; (c) min{u, k} ∈N1,p 0() for every k>0; (d) uis bounded in \Brfor every r>0; (e) lim x→yu(x) =0for q.e. y∈∂, namely for all y∈∂ that are regular with respect to . Note that (b) is just an equivalent way of writing (S5), when is bounded, but not when is unbounded. We therefore prefer to have the formulation (S5) in the definition. Proof. (a) This follows from Proposition 4.4. (b) As is bounded, (b) is equivalent to (S5). (c) Let uk=min{u, k}which is a bounded superharmonic function, and thus a superminimizer, and in particular uk∈N1,p loc (). From (b) it then follows that uk∈N1,p 0(). (d) and (e) These follow from the already proven (b) and Lemma 5.7 applied to K={x0}and V=Br, together with (5.3). 7. Characterizations when Cp({x0}) =0 In addition to the general assumptions from the beginning of Section 4, we assume in Sections 7–9that is a bounded domain such that Cp(X \) >0. In particular, Cp(∂) >0by [8, Lemma 4.5]. As already mentioned, the theory of singular functions (on bounded domains) splits naturally into the two cases Cp({x0}) =0 and Cp({x0}) >0. We postpone the study of the latter case to Section 8and concentrate on the case Cp({x0}) =0in this section. Note first that, when Cp({x0}) =0, it follows from the extension Lemma 4.3 that the requirement of superharmonicity in the definition of singular functions can be replaced by the condition that u(x0) =lim infx→x0u(x). In fact, by the following result, this also forces u(x0) =∞. Lemma 7.1. Assume that Cp({x0}) =0. Also assume that uis a singular function in with singularity at x0, or more generally that u: →(0, ∞] satisfies (S1),(S2) and (S5) in Definition 1.1. Then u(x0) =limx→x0u(x) =∞. That we only assume (S1), (S2) and (S5) will play a role in the proof of Theorem 7.2.
6622 A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 Proof. We already know from Proposition 4.4 that u(x0) =limx→x0u(x). If u(x0)were finite, then uwould be bounded in , and thus u|\{x0}would have a p-harmonic extension to by Lemma 4.3. But this contradicts Proposition 6.2. Singular functions can be characterized in many ways. Our aim is to have as simple and flexible criteria as possible. Note that uis assumed to be positive, and that condition (a.3) can always be guaranteed by redefining uat x0. Theorem 7.2. Assume that Cp({x0}) =0. Let u: →(0, ∞] and consider the following properties: (a.1) uis superharmonic in ; (a.2) u(x0) =limx→x0u(x); (a.3) u(x0) =lim infx→x0u(x); (a.4) u(x0) =∞; and (b.1) u ∈N1,p 0( \Br; X\Br)for every r>0; (b.2) uis bounded in \Brfor every r>0, and lim x→yu(x) =0for q.e. y∈∂; (7.1) (b.3) uis bounded in \Brfor every r>0, and min{u, k} ∈N1,p 0() for every k>0. Let j∈{1, 2, 3, 4}and k∈{1, 2, 3}. Then uis a singular function in with singularity at x0 if and only if uis p-harmonic in \{x0}and usatisfies (a.j) and (b.k). Example 7.3. Let x0=0, x1=(1, 0, ..., 0)and =B(0, 2) \{x1}in (unweighted) Rn, n ≥3, with p=2. Also let v(x) =|x|2−n+|x−x1|2−nand u =v−Pv, where Pv is the Perron solution in . Then, by linearity, uis 2-harmonic in \{x0}and superharmonic in . In fact, u satisfies (S1)–(S4) in Definition 1.1, but not (S5). It also satisfies (a.1)–(a.4), but not (b.1)–(b.3). This shows, in particular, that the boundedness assumptions in (b.2) and (b.3) cannot be dropped. As Cp({x0}) =0, conditions (b.1)–(b.3) allow u(x0)to be arbitrary, which shows that conditions (a.1)–(a.4) cannot be omitted. Proof of Theorem 7.2.If uis a singular function, then uis p-harmonic in \{x0}and satisfies (a.1) by assumption. It further satisfies (a.2), (a.3) and (b.1)–(b.3) by Proposition 6.4, and (a.4) by Lemma 7.1. Conversely, assume that uis p-harmonic in \{x0}and satisfies (a.j) and (b.k)for some jand k. Lemma 5.7 shows that (b.3) ⇒(b.1) ⇔(b.2). The implication (a.2) ⇒(a.3) is trivial, while (a.3) ⇒(a.1) holds by Lemma 4.3 since Cp({x0}) =0. We postpone the case j=4, but otherwise, regardless of the values of j, k∈{1, 2, 3}, we have shown that (a.1), (b.1) and (b.2) are satisfied. Thus (S1) and (S2) are satisfied. As (7.1) holds and Cp(∂) >0, we obtain (S4). Extending uby 0on X\and letting r→0in(b.1)
A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 6623 yields (S5). By Lemma 7.1, u(x0) =∞ =supuand (S3) holds, which concludes the proof that uis a singular function. Finally, consider the case when j=4 and k∈{1, 2, 3}. We have already shown that (b.2) is satisfied. Let ˜u(x) =u(x), x = x0, lim infy→x0u(y), x =x0. Then ˜uis p-harmonic in \{x0}and satisfies (a.3) and (b.2). So by the already established cases, ˜uis a singular function. Lemma 7.1 shows that ˜u(x0) =∞, i.e. u =˜uis a singular function. We are now prepared to prove the existence of singular functions at points having zero capacity. Theorem 7.4. If Cp({x0}) =0, then there is a singular function in with singularity at x0. Proof. Let r0>0be so small that Br0. For 0 <r≤r0, let urbe the lsc-regularized capacitary potential for Brin . Then uris superharmonic in and p-harmonic in \Br, by Lemma 5.5. Let Mr=max∂Br0ur>0, which exists by the continuity of urin \Br(while Mr0=1 as Cp(∂Br0) >0). Also, Mr>0by the strong minimum principle for superharmonic functions since Cp(Br) >0. Let vr=ur/Mr. Then max∂Br0vr=1. Thus we can use Harnack’s convergence theorem (Proposition 5.1 in Shanmugalingam [44]or Theorem 9.37 in [8]) to find a subsequence {vrj}∞ j=1converging locally uniformly in \{x0}to a nonnegative p-harmonic function u. As Cp({x0}) =0, Lemma 4.3 implies that uhas a superharmonic extension to given by u(x0) := lim infx→x0u(x). Clearly u ≤1on ∂Br0, and from the local uniform convergence and the compactness of ∂Br0we conclude that max∂Br0u =1. Thus uis positive in by the strong minimum principle for superharmonic functions. By definition and the comparison principle (4.1), vr=HGvr≤HGur0=ur0in G:= \Br0 for all 0 <r≤r0, and hence 0 ≤u ≤ur0in G. Thus, by Lemma 5.5, 0≤lim inf x→yu(x) ≤lim sup x→y u(x) ≤lim x→yur0(x) =0 for q.e. y∈∂, i.e. (7.1) holds. Since uis p-harmonic, and thus continuous, in \{x0}, it is bounded in the compact set Br0\Brfor every r>0. As also 0 ≤u ≤1in G = \Br0, we see that uis bounded in \Brfor every r>0. We have thus shown that uis a positive p-harmonic function in \{x0}, which satisfies (a.1) and (b.2), and hence uis a singular function by Theorem 7.2. Remark 7.5. In the above proof we constructed a singular function using capacitary potentials of balls. This is just for convenience, but there is nothing special about balls in this case. Indeed, if we let G1⊃G2⊃... be open sets such that G1and ∞ k=1Gk={x0}, then we can instead use the capacitary potentials for Gk. It is an open question, even in weighted Rn(with
6624 A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 a p-admissible weight), whether all such constructions lead to the same singular function (upon proper normalization as in (1.2)). We conclude this section with a simple nonintegrability result for singular functions. Part (c) is mainly interesting as contrasting Proposition 8.4 below, see also Theorem 8.6. In our forthcoming paper [14], we will give much more precise results on the Ltintegrability and nonintegrability of uand gufor singular and Green functions u, where t>0. Proposition 7.6. Assume that Cp({x0}) =0and that uis a singular function in with singularity at x0. Extend uby letting u =0on X\. Then the following are true: (a) u /∈N1,p(Br)is true for every r>0; (b) ´Brgp udμ =∞for every r>0; (c) u /∈N1,p 0(). Proof. Parts (a) and (b) follow directly from Proposition 6.2 or Lemma 7.1, together with Proposition 4.5. Part (c) then follows directly from (a). 8. Characterizations when Cp({x0}) >0 Recall the standing assumptions from the beginning of Section 7. We now turn to the case when the singularity point x0has positive capacity. As we shall see, singular functions are unique in this case, up to multiplication by positive constants. By Theorem 8.2 below, there is also an explicit representative for singular functions, namely the capacitary potential for {x0}in . Lemma 8.1. Assume that Cp({x0}) >0, and let ube a p-harmonic function in \{x0}. Then lim infx→x0u(x) <∞. In particular, if limx→x0u(x) =: u(x0)exists, then u(x0) ∈R. Proof. If lim infx→x0u(x) =∞, then there is a connected open neighbourhood G ⊂of x0 such that u >0in G \{x0}. The definition of Perron solutions implies that u/k ≥PG\{x0}χ{x0} for all k>0. Letting k→∞shows that PG\{x0}χ{x0}≡0, which contradicts Cp({x0}) >0 and the Kellogg property (Theorem 5.4). Hence lim infx→x0u(x) <∞. Applying this also to −ushows that when u(x0) := limx→x0u(x) exists it must be real. The following is an existence and uniqueness result (up to normalization) for singular functions when Cp({x0}) >0. Theorem 8.2. Assume that Cp({x0}) >0, and let vbe the lsc-regularized capacitary potential for {x0}in . Then a function uis a singular function in with singularity at x0if and only if there is a constant 0 <b<∞such that u =bv in . Moreover, b=u(x0) =limx→x0u(x) in that case. In particular, vis a singular function in with singularity at x0. Proof. Let u =bv. By definition, uis nonnegative and bounded. Lemma 5.5 shows that uis pharmonic in \{x0}and superharmonic in . As Cp(∂) >0 and Cp({x0}) >0, we conclude
A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 6625 from Lemma 5.6 (c) that infu =0 and u(x0) =b=supu. In particular, u ≡ 0, and so u >0in by the strong minimum principle for superharmonic functions. Thus, uis a singular function. Conversely assume that uis a singular function. Proposition 4.4 and Lemma 8.1 imply that b:= u(x0) =limx→x0u(x) <∞. Thus uis a bounded superharmonic function in some neighbourhood Brof x0, and in particular u ∈N1,p(Br/2). Together with (S5) this shows that u ∈N1,p 0() and Lemma 5.5 implies that u =bv in . Also when Cp({x0}) >0, singular functions can be characterized in many ways. Theorem 8.3. Assume that Cp({x0}) >0. Let u: →(0, ∞] and consider the properties (a.j) and (b.k) from Theorem 7.2. Let j∈{1, 2}and k∈{1, 2, 3}. Then uis a singular function in with singularity at x0if and only if uis p-harmonic in \{x0}and usatisfies (a.j) and (b.k). Note that compared with Theorem 7.2 (for the case when Cp({x0}) =0) conditions (a.3) and (a.4) are omitted here. By Theorem 8.2, condition (a.4) is never satisfied for singular functions when Cp({x0}) >0, so it cannot be included here. To see that (a.3) cannot be included, consider the function u(x) =1+x, −1<x≤0, 2−2x, 0<x<1, which is p-harmonic in (−1, 1) \{0} ⊂X:= Rand satisfies (a.3), (b.2) and (b.1), but not (a.2), and hence not (a.1) either, by Proposition 4.4. In particular, uis not a singular function. Also (b.3) fails as functions in N1,p(R)are continuous. The above ualso shows that (a.j) cannot be dropped if k∈{1, 2}. We do not know if (a.j) is redundant when (b.3) is assumed. That (b.1)–(b.3) cannot be dropped follows by considering the constant function u ≡1. Proof of Theorem 8.3.If uis a singular function, then uis p-harmonic in \{x0}and satisfies (a.1) by assumption. It further satisfies (a.2) and (b.1)–(b.3) by Proposition 6.4. Conversely, assume that uis p-harmonic in \{x0}and usatisfies (a.j) and (b.k)for some j∈{1, 2}and k∈{1, 2, 3}. Lemma 5.7 shows that (b.3) ⇒(b.1) ⇔(b.2). If (a.2) holds, then u(x0) =limx→x0u(x) <∞by Lemma 8.1. Hence, in view of (b.2), u is bounded in . Lemma 5.6, together with (a.2) and (7.1) from (b.2), implies that u =u(x0)v, where vis the lsc-regularized capacitary potential for {x0}in . In particular, uis superharmonic in , and thus (a.2) ⇒(a.1). Hence, regardless of the values of jand k, we have shown that (a.1), (b.1) and (b.2) hold, and so (S1), (S2) and (S5) are satisfied. As (7.1) holds and Cp(∂) >0, we obtain (S4). It remains to show that (S3) holds. If u(x0)were ∞then this would be immediate, so we may assume that u(x0) <∞. Proposition 4.4 implies that limx→x0u(x) =u(x0)and hence u =P\{x0}(u(x0)χ{x0}), by (7.1) and Theorem 5.2 (c). Thus u ≤u(x0)in , and hence (S3) holds. The following result shows that if we strengthen (b.1) in a suitable way, we do not even need to assume (a.1) or (a.2).
6632 A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 v(x) =a(x +1), −1<x<0, 1−x, 0≤x<1,with a>0, are p-harmonic in \{x0}and comparable to each other, but only the one with a=1is superharmonic and singular in with singularity at x0. 10. p-harmonic functions with poles Assume in this section that Xis complete and that μis doubling and supports a p-Poincaré inequality. We also fix x0∈Xand write Br=B(x0, r) for r>0. We shall now apply our results to general p-harmonic functions with poles. Note that there is no relation between Gand Uin the theorem below. Theorem 10.1. Let Gand Ube arbitrary open sets containing x0, such that Gis bounded and Cp(X \G) >0. Let uand vbe p-harmonic functions in U\{x0}such that u(x0):= lim x→x0 u(x) =∞ and v(x0):= lim x→x0 v(x) =∞.(10.1) Then the following are true: (a) Cp({x0}) =0; (b) there is a bounded domain x0and a≥0such that u −ais a singular function in with singularity at x0; (c) there is r0>0such that if 0 <r<r 0and x∈∂Br, then u(x) ≃capp(Br,G) 1/(1−p),(10.2) where the comparison constants depend on Gand u, but not on r; (d) there is r0>0such that u≃vin Br0, where the comparison constants depend on uand v. Note that also the radius r0, for which (c) and (d) hold, depends on u(and v). This is easily seen by considering the function |x|(p−n)/(p−1)−cin Rn, p<n, for various constants c≥0. However, Theorem 10.1 (c)–(d) shows that all p-harmonic functions with a given pole (i.e. such that (10.1) holds) have growth of the same order near the pole. For elliptic quasilinear equations (1.6) on unweighted Rn, this is a classical result due to Serrin [42, Theorem 1]. On the contrary, results in Björn–Björn [7]show that the so-called quasiminimizers (rather than minimizers) of the p-energy integral ´gp udμ can have singularities of arbitrary order, depending on the quasiminimizing constant. Quasiminimizers were introduced by Giaquinta and Giusti [25], [26]as a natural unification of elliptic equations with various ellipticity constants.
A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 6633 Proof. Let r>0 be such that BrUand Cp(X \Br) >0, and let M(r) =max ∂Br ufor 0 <r≤r. Let a=max{M(r), 0}, ={x∈Br:u(x) >a}and ¯u=u −a. By the strong maximum principle for p-harmonic functions, must be connected. It is easy to see that ¯usatisfies (a.2) and (b.1) in Theorem 7.2 (with ¯uin place of u). As ¯uis p-harmonic in \{x0}, it follows from Theorem 8.5 that ¯uis a singular function in , i.e. (b) holds. Thus (a) follows from Theorem 8.6. Let next r0>0be so small that B50λr0⊂. By the strong minimum principle for superharmonic functions, infBr0u >a≥0 and thus u≥u−a≥Cu in Br0, with C>0 depending on aand infBr0u. Theorems 1.3 (b) and 1.5, applied to ¯u, then yield u(x) ≃¯u(x) ≃capp(Br,) 1/(1−p) whenever x∈∂Brand 0 <r<r 0,(10.3) where the comparison constants depend on u, aand r0. This proves (c) for G =. Also (d) follows directly from this, with the same choice of r0. Now consider a general open set Gin (c). We may choose rabove so small that Br⊂G. It follows that ⊂G. For 0 <r≤r0, let urbe the capacitary potential for Brin G, and set ar=max∂ ur. Then 0 <a r≤ar0<1. Also let vr=ur−ar 1−ar . Then vr=1in Brand vr≤0 on X\. Hence capp(Br,)≤ˆ X gp vrdμ≤1 1−arpˆ X gp urdμ ≤1 1−ar0pˆ X gp urdμ=1 1−ar0p capp(Br,G). As capp(Br, G) ≤capp(Br, ), we see that (10.2) follows from (10.3). 11. Local assumptions In this section we investigate to which extent our results hold in more general metric measure spaces than those assuming our three standing assumptions: completeness, doubling measure and p-Poincaré inequality. We start by introducing the local assumptions. Definition 11.1. The measure μis doubling within a ball B0if there is C>0 (depending on B0) such that μ(2B) ≤Cμ(B) holds for all balls B⊂B0.
6634 A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 Similarly, the p-Poincaré inequality holds within a ball B0if there are constants C>0 and λ ≥1 (depending on B0) such that (2.2) holds for all balls B⊂B0, all integrable functions uon λB, and all p-weak upper gradients gof uwithin λB. We also say that any of the above two properties is local if for every x0∈Xthere is r0 (depending on x0) such that the property holds within B(x0, r0). If a property holds within every ball B(x0, r0)then it is called semilocal. Note that if μis semilocally doubling and Cis independent of x0and r0, then μis doubling according to (2.1). The situation is similar for Poincaré inequalities. The following result from Björn–Björn [11] makes it possible to generalize the results in this paper to spaces with local assumptions. Recall that a space is proper if every bounded closed subset is compact. Theorem 11.2. (Proposition 1.2 and Theorem 1.3 in [11]) If Xis proper and connected, and μ is locally doubling and supports a local p-Poincaré inequality, then μis semilocally doubling and supports a semilocal p-Poincaré inequality. Examples in [11]show that properness cannot be replaced by completeness, and connectedness cannot be dropped from Theorem 11.2. Moreover, if μsupports a semilocal Poincaré inequality, then Xis connected. So, for the rest of this section, we assume that Xis proper and connected, and that μis locally doubling and supports a local p-Poincaré inequality. As in Keith–Zhong [34, Theorem 1.0.1], a better semilocal q-Poincaré inequality with some q<pholds also in this case, by Theorem 5.3 in [11]. In [11, Section 10], it was explained how the potential theory of p-harmonic functions, specifically the results in Chapters 7–14 in [8], hold under these assumptions, with the exception of the Liouville theorem. The same is true for the results in this paper, it is only the dependence of constants on the different associated parameters that needs to be carefully investigated. If X is bounded, then the semilocal assumptions are global and hence our standing assumptions are equivalent to the local assumptions above in this case. If Xis unbounded, we let (as before) be a bounded domain and find a ball B0⊃. Since X is unbounded, the condition Cp(X \) >0is automatically satisfied. Let CPI, λand Cμbe the constants in the p-Poincaré inequality and the doubling condition within 2B0. The weak Harnack inequalities then hold for every ball Bsuch that 50λB ⊂and with a constant depending only on p, CPI, λand Cμ, coming from 2B0as above. Thus all our estimates depend on these parameters instead of the constants in the global assumptions, which perhaps do not hold on X. 12. Holopainen–Shanmugalingam’s definition In this section we compare our results with the following definition of singular functions from Holopainen–Shanmugalingam [32]. (See below for the precise assumptions on X.) Definition 12.1. (Definition 3.1 in [32]) Let ⊂Xbe a relatively compact domain. A function u:X→[0, ∞] is a singular function in the sense of Holopainen–Shanmugalingam, or an HS- singular function, in with singularity at x0∈if
A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 6635 (HS1) uis p-harmonic in \{x0}and positive in ; (HS2) u|X\=0 q.e.; (HS3) u ∈N1,p(X \B(x0, r)) for all r>0; (HS4) lim x→x0 u(x) =capp({x0}, )1/(1−p) (in particular, lim x→x0 u(x) =∞if capp({x0}, ) =0); (HS5) For 0 ≤a<b<supu, p−1 p2(p−1) (b −a)1−p≤capp(b, a)≤p2(b −a)1−p,(12.1) where b={x∈ :u(x) ≥b}and a={x∈ :u(x) >a}. The existence of such a function, when x0∈ ⊂Xand is a relatively compact domain, was given in [32, Theorem 3.4] under the assumptions that Xis connected, locally compact, noncompact and satisfies the so-called LLC property, and that μis locally doubling and supports a local q-Poincaré inequality for some 1 ≤q<p<∞, cf. Remark 2.4 in [32]. These local assumptions are as defined in [32] and are stronger than those in Section 11. In fact, they coincide with those called semiuniformly local in Björn–Björn [11]. Remark 12.2. From the proof of [32, Theorem 3.4] it is not clear why the function called gon p. 322 therein satisfies (HS3) in the case when Cp({x0}) =0. This can be justified, at least under the assumptions in this paper, in a similar way as we do in Lemma 5.7, using Perron solutions and the uniqueness result in Theorem 5.2 (c). These tools were however not available at that time. In the definition of HS-singular functions above, the value u(x0)can be rather arbitrary. In particular, uis not required to be superharmonic in . However, in order for (HS5) to be satisfied, one must have 0 <u(x 0) ≤capp({x0}, )1/(1−p) (which automatically holds if Cp({x0}) =0). In view of (HS4) it is natural to let u(x0) := limx→x0u(x), and we do so from now on. We obtain the following relation to our Definitions 1.1 and 1.2. Proposition 12.3. Assume that Xis a proper connected metric space, and that μis locally doubling and supports a local p-Poincaré inequality. Let ⊂Xbe a bounded domain such that Cp(X \) >0, and let x0∈and u:X→[0, ∞]. (a) If uis an HS-singular function in with singularity at x0, and u(x0) =limx→x0u(x), then u|is a singular function in in the sense of Definition 1.1. (b) If uis a Green function in with singularity at x0in the sense of Definition 1.2, then its zero extension ˜u(given by letting ˜u=0on X\) is an HS-singular function in with singularity at x0. In particular there is an HS-singular function in with singularity at x0. Proof. By the discussion in Section 11 the results in this paper hold under these assumptions on X. Part (a) follows from Theorem 1.6, while part (b) follows from the definition of Green functions and Theorem 9.3 (which yields (HS4)). Finally, the existence follows from Theorem 1.3 (a).
6636 A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 Requiring the superlevel set estimates in (12.1) with those explicit constants, is a weaker type of (pseudo)normalization than in our definition of Green functions. However, it was natural in [32]as it used the best estimates available at the time. We also remark that while proving the existence of HS-singular functions, Holopainen and Shanmugalingam [32, formula (8), p. 322] showed that estimate (1.5) holds, for xclose enough to x0, for the HS-singular functions obtained by their construction. Here, we obtain it for all Green functions. Recall that in this generality, it is not known whether Green functions are unique when Cp({x0}) =0. 13. Cheeger–Green functions Recall the standing assumptions from the beginning of Section 4. In this section we also assume that is bounded and that Cp(X \) >0. Theorem 4.38 in Cheeger [21]shows that, under our standing assumptions, the metric space Xcan be equipped with a coordinate structure in such a way that each Lipschitz function uin Xhas a vector-valued “gradient” Du, defined a.e. in X. Since Lipschitz functions are dense in N1,p(X), this gradient can be extended uniquely to N1,p(X), by Franchi–Hajłasz–Koskela [24, Theorem 10] or Keith [33]. Then |Du| ≃gua.e. in Xfor all u ∈N1,p(X), where the comparison constants are independent of u. Here and throughout this section | ·|is an inner product norm on some RN, related to the Cheeger structure. Both | ·|and the dimension Ndepend on x∈X, but there is a bound on N, which only depends on the doubling constant and the constants in the Poincaré inequality. By adding dummy coordinates, it can thus be assumed that Du(x) ∈RN, with the same dimension Nfor all x. In a general metric space Xthere is some freedom in choosing the Cheeger structure. In (weighted) Rnwe will however always make the natural choice Du =∇u, where ∇udenotes the Sobolev gradient from Heinonen–Kilpeläinen–Martio [28]. In this case |Du| =gu, by Proposition A.13 in [8]. If the weight won Rnsatisfies w1/(1−p) ∈L1 loc(Rn)(in particular, if it is a Muckenhoupt Apweight) then the Sobolev gradient ∇uis also the distributional gradient. It was shown by Hajłasz and Koskela that gu=|∇u|also on Riemannian manifolds [27, Proposition 10.1] and Carnot–Carathéodory spaces [27, Proposition 11.6 and Theorem 11.7], equipped with their natural measures. Cheeger (super)minimizers and Cheeger p-harmonic functions are defined by replacing gu and gu+ϕin Definition 4.1 with |Du|and |D(u +ϕ)|. Similarly, the Cheeger variational capacity of E⊂, denoted Ch-capp(E, ), and the related capacitary potentials are defined as in Section 3but with gureplaced by |Du|. Then all the results we have obtained in the previous sections hold also for the corresponding Cheeger singular and Cheeger–Green functions, which are defined as in Definitions 1.1 and 1.2, with obvious modifications. See Appendix B.2 in [8]for more comments, details and references on Cheeger p-harmonic functions in general, and Danielli–Garofalo–Marola [22, Section 6] for some specific results for Cheeger singular and Cheeger–Green functions. Due to the additional vector structure of the Cheeger gradient it is possible to make the following definition, which has no counterpart in the case of general scalar-valued upper gradients.
A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 6637 Definition 13.1. A function u ∈N1,p loc () is a(super)solution in if ˆ |Du|p−2Du ·Dϕ dμ ≥0 for all (nonnegative) ϕ∈Lipc(), (13.1) where ·is the inner product giving rise to the norm | ·|, and Lipc() denotes the family of Lipschitz functions with compact support in . For solutions, one can equivalently replace ≥by =in (13.1), which follows directly after testing also with −ϕ. It can be shown that a function is a (super)solution if and only if it is a Cheeger (super)minimizer, the proof is the same as for Theorem 5.13 in Heinonen–Kilpeläinen–Martio [28]. In weighted Rnwith a p-admissible weight and the choice Du =∇u, we have gu=|Du| =|∇u| a.e. which implies that (super)minimizers, Cheeger (super)minimizers and (super)solutions coincide, and are the same as in [28]. Similar identities hold also on Riemannian manifolds and Carnot–Carathéodory spaces equipped with their natural measures. The following result is contained in Proposition 5.1 in Björn–Björn–Latvala [12], see also Proposition 3.5 and Remark 3.6 in Björn–MacManus–Shanmugalingam [19]. Proposition 13.2. For every supersolution uin there is a Radon measure ν∈N1,p 0()such that for all ϕ∈N1,p 0(), ˆ |Du|p−2Du ·Dϕ dμ =ˆ ϕdν, (13.2) where ·is the inner product giving rise to the norm | ·|. Next we show that the Cheeger–Green functions are exactly the weak solutions of the p- Laplace equation with the Dirac measure on the right-hand side and with zero boundary values, as in the case of Rnconsidered in Remark 9.4. Theorem 13.3. Let ube a Cheeger–Green function in with singularity at x0. Then ˆ |Du|p−2Du ·Dϕ dμ =ϕ(x0)for all ϕ∈Lipc(), (13.3) that is, pu =−δx0in the weak sense. Conversely, assume that vis an (extended real-valued) continuous function in such that |Dv| ∈Lp−1(),(S5) in Definition 1.1 is satisfied, and vis a solution of (13.3). Then vis a Cheeger–Green function. Note that the assumption |Dv| ∈Lp−1() in the second part of the statement is natural, since it guarantees that the integral in (13.3)is well-defined, and it moreover holds for all superharmonic functions, by Theorem 5.6 in Kinnunen–Martio [37](or [8, Corollary 9.55]).
6638 A. Björn et al. / J. Differential Equations 269 (2020) 6602–6640 Proof. Assume first that Cp({x0}) =0. Write uk=min{u, k}for k>0. Then uk∈N1,p 0() by Proposition 6.4 (c), and ukis a supersolution. Let νk∈N1,p 0()be the corresponding Radon measures given by Proposition 13.2. Since ukis Cheeger p-harmonic in \k, νkis supported on k. Hence, by testing (13.2)for νkwith ϕ=uk, we obtain that ˆ |Duk|pdμ =ˆ ukdνk=kνk(k). (13.4) On the other hand, the function uk/k is the Cheeger capacitary potential of (k, ), by Lemma 5.5. Thus it follows from the normalization (1.2)of Cheeger–Green functions that ˆ |Duk|pdμ =kpCh-capp(k,)=kpk1−p=k, (13.5) and so νk(k) =1for all k>0. Let ϕ∈Lipc() and let ε>0. Choose k0>0so large that |ϕ(x) −ϕ(x0)| <εfor all x∈k0 (and hence also for all x∈kwhenever k≥k0); note that this is possible by Theorem 1.5 and Proposition 6.4 (d). Then (13.2) and the fact that νk() =νk(k) =1 yield ˆ |Duk|p−2Duk·Dϕ dμ −ϕ(x0) =ˆ ϕdν k−ϕ(x0) ≤ˆ k |ϕ−ϕ(x0)|dνk≤ε for all k≥k0. Since |Du| ∈Lp−1() by Theorem 5.6 in Kinnunen–Martio [37](or [8, Corollary 9.55]) and ϕ∈Lipc(), we see that |Duk|p−2Duk·Dϕ≤|Duk|p−1|Dϕ|≤|Du|p−1Dϕ∞∈L1() for all k>0. As Duk→Du a.e. in , we hence obtain by dominated convergence that ˆ |Du|p−2Du ·Dϕ dμ −ϕ(x0) ≤ε. Since this holds for all ε>0, the claimed identity (13.3) follows when Cp({x0}) =0. Next, consider the case when Cp({x0}) >0. Then we know by Theorem 8.2 that u ∈N1,p 0() and uis Cheeger p-harmonic in \{x0}. Let νbe the measure provided for uby Proposition 13.2. Since uis Cheeger p-harmonic in \{x0}, νmust be supported on {x0}and hence ´ϕdν=ϕ(x0)ν({x0})for all ϕ∈N1,p 0(). Testing (13.2) with ϕ=uthen shows as in (13.5) and (13.4) that u(x0)1−p=Ch-capp(u(x0),)=1 u(x0)pˆ |Du|pdμ =u(x0)1−pν({x0}), i.e. ν({x0}) =1, which proves the claim when Cp({x0}) >0.
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