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Linköping University Post Print Nonlinear balayage on metric spaces Anders Björn, Jana Björn, Tero Mäkäläinen and Mikko Parviainen N.B.: When citing this work, cite the original article. Original Publication:

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Linköping University Post Print Nonlinear balayage on metric spaces Anders Björn, Jana Björn, Tero Mäkäläinen and Mikko Parviainen N.B.: When citing this work, cite the original article. Original Publication: Anders Björn, Jana Björn, Tero Mäkäläinen and Mikko Parviainen, Nonlinear balayage on metric spaces, 2009, Nonlinear Analysis, (71), 5-6, Copyright: Elsevier Science B.V., Amsterdam. Postprint available at: Linköping University Electronic Press Nonlinear balayage on metric spaces Anders Björn Department of Mathematics, Linköpings universitet, SE Linköping, Sweden; Jana Björn Department of Mathematics, Linköpings universitet, SE Linköping, Sweden; Tero Mäkäläinen Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI University of Jyväskylä, Finland; Mikko Parviainen Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, FI Helsinki University of Technology, Finland; Abstract. We develop a theory of balayage on complete doubling metric measure spaces supporting a Poincaré inequality. In particular, we are interested in continuity and p- harmonicity of the balayage. We also study connections to the obstacle problem. As applications, we characterize regular boundary points and polar sets in terms of balayage. Key words and phrases: Balayage, boundary regularity, continuity, doubling measure, metric space, nonlinear, obstacle problem, Perron solution, p-harmonic, polar set, Poincaré inequality, potential theory, superharmonic. Mathematics Subject Classification (2000): Primary: 31C45; Secondary: 31C05, 35J Introduction Balayage is one of the most useful tools in linear potential theory and has been used to obtain many important results therein. Heinonen, Kilpeläinen and Martio were the first to use nonlinear balayage for studying A-harmonic functions on R n in [22], [23] and [24]. The purpose of this paper is to develop the nonlinear balayage theory on metric spaces. Analysis and nonlinear potential theory on metric measure spaces have undergone a rapid development during the last decade, see e.g. Haj lasz [19], Heinonen Koskela [25], Koskela MacManus [34], Haj lasz Koskela [20], Cheeger [16], Shanmugalingam [37], Kinnunen Martio [28], [29] and more recently Keith Zhong [26]. Using upper gradients, which were introduced by Heinonen and Koskela in [25], it is possible to define (Newtonian) Sobolev-type spaces on general metric spaces. Variational inequalities can then be used to define p-harmonic and superharmonic 1 2 Anders Björn, Jana Björn, Tero Mäkäläinen and Mikko Parviainen functions (see Section 3). In our generality there are no corresponding partial differential equations, which causes some difficulties. Nevertheless, under rather mild assumptions on the metric space, a large part of the theory of p-harmonic and superharmonic functions on weighted R n has been extended to metric spaces, see, e.g., Shanmugalingam [38], [39], Kinnunen Martio [30], Kinnunen Shanmugalingam [32], [33], Björn Björn Shanmugalingam [9], [10] and Björn Björn [5]. Examples of spaces satisfying our assumptions include weighted R n, manifolds, Heisenberg groups and more general Carnot groups and Carnot Carathéodory spaces, see, e.g., [5], [10] and Haj lasz Koskela [20]. Balayage is a regularized infimum of the family of superharmonic functions lying above a given obstacle. First, we use the fundamental convergence theorem from Björn Björn Parviainen [8] to show that regularizing changes the infimum only on a set of capacity zero and that the resulting function is superharmonic. This makes it possible for us to develop the theory of balayage in a way different from Heinonen Kilpeläinen Martio [24], where a substantial part of the balayage theory was developed before proving that the infimum only needs to be regularized on a set of capacity zero. We generalize the balayage results from [22], [23] and [24] to metric spaces, but in most cases our proofs are different. Sets of capacity zero in potential theory correspond to sets of measure zero in the study of L p -spaces and can sometimes be disregarded. In linear potential theory there are two ways of defining the balayage, depending on if sets of capacity zero are ignored or not, and it is almost immediate that they are equivalent. This equivalence is then used to obtain many important consequences. In the nonlinear case it is not known whether the two definitions, which we call R- and Q-balayage, see Section 4, always coincide. A partial result on their equality was obtained in Heinonen Kilpeläinen [23] in R n. We extend this result to metric spaces and also provide other sufficient conditions for when the two types of balayage coincide. This is particularly useful in our characterizations of polar sets by means of barriers in Section 8. We develop the theories of R- and Q-balayage in parallel, proving results for both types of balayage where possible. In most cases we are able to obtain results for the Q-balayage, but in connection with Perron solutions we can only obtain some parts for the R-balayage. On metric spaces, obstacle problems have earlier been used instead of balayage to prove various results in nonlinear potential theory. In Section 5, we study the relationship between balayage and obstacle problems. We also study the continuity of balayage and show that even for irregular obstacles, the balayage is p-harmonic in the set where it lies strictly above the obstacle, see Section 6. As an application of the theory of balayage, in Section 7 we provide two types of characterizations of regular boundary points in terms of balayage. These complement the large number of characterizations obtained in Björn Björn [5]. Finally we use balayage for calculating capacities. Our results are also used in Mäkäläinen [35] to obtain a characterization of removable singularities for Hölder continuous Cheeger p-harmonic functions on metric spaces. Many of our results are new also in R n. The results and proofs given in this paper hold also for Cheeger p-harmonic functions, as discussed in e.g. Björn MacManus Shanmugalingam [14] and Björn Björn Shanmugalingam [9], and for A-harmonic functions as defined on pp of Heinonen Kilpeläinen Martio [24]. Acknowledgement. We would like to thank Olli Martio for letting us use his notes [36] in this research. The first two authors were supported by the Swedish Science Research Council. This research belongs to the European Science Foundation Networking Programme Harmonic and Complex Analysis and Applications HCAA. Nonlinear balayage on metric spaces 3 2. Preliminaries We assume throughout the paper that 1 p and that X = (X, d, µ) is a complete metric space endowed with a metric d and a positive complete Borel measure µ which is doubling, i.e. there exists a constant C µ 1 such that for all balls B = B(x 0, r) := {x X : d(x, x 0 ) r} in X, 0 µ(2b) C µ µ(b), where λb = B(x 0, λr). It follows that X is proper, i.e. that closed bounded sets are compact. In this paper, a path in X is a rectifiable nonconstant continuous mapping from a compact interval. A path can thus be parametrized by arc length ds. We follow Heinonen Koskela [25] introducing upper gradients as follows (they called them very weak gradients). Definition 2.1. A nonnegative Borel function g on X is an upper gradient of an extended real-valued function f on X if for all paths γ : [0, l γ ] X, f(γ(0)) f(γ(l γ )) g ds (2.1) whenever both f(γ(0)) and f(γ(l γ )) are finite, and g ds = otherwise. If g is a γ nonnegative measurable function on X and if (2.1) holds for p-a.e. path, then g is a p-weak upper gradient of f. By saying that (2.1) holds for p-a.e. path, we mean that it fails only for a path family with zero p-modulus, see Definition 2.1 in Shanmugalingam [37]. It is implicitly assumed that g ds is defined (with a value in [0, ]) for p-a.e. path. γ The p-weak upper gradients were introduced in Koskela MacManus [34]. They also showed that if g L p (X) is a p-weak upper gradient of f, then one can find a sequence {g j } j=1 of upper gradients of f such that g j g in L p (X). If f has an upper gradient in L p (X), then it has a minimal p-weak upper gradient g f L p (X) in the sense that for every p-weak upper gradient g L p (X) of f, g f g a.e., see Corollary 3.7 in Shanmugalingam [38]. Next we define a version of Sobolev spaces on the metric space X due to Shanmugalingam [37]. Cheeger [16] gave an alternative definition which leads to the same space, when p 1. Definition 2.2. Whenever u L p (X), let ( u N 1,p (X) = X γ 1/p u p dµ + inf g dµ) p, g X where the infimum is taken over all upper gradients of u. The Newtonian space on X is the quotient space N 1,p (X) = {u : u N 1,p (X) }/, where u v if and only if u v N 1,p (X) = 0. Definition 2.3. The capacity of a set E X is the number C p (E) = inf u p N 1,p (X), where the infimum is taken over all u N 1,p (X) such that u 1 on E. 4 Anders Björn, Jana Björn, Tero Mäkäläinen and Mikko Parviainen By truncation, the infimum can be taken over u such that 0 u 1. The capacity is countably subadditive. For this and other properties as well as equivalent definitions of the capacity we refer to Kilpeläinen Kinnunen Martio [27], Kinnunen Martio [28], [29], and Björn Björn [7]. We say that a property holds quasieverywhere (q.e.) if the set of points for which the property does not hold has capacity zero. The capacity is the correct gauge for distinguishing between two Newtonian functions. Indeed, if u N 1,p (X), then u v if and only if u = v q.e. in X. Moreover, if u, v N 1,p (X) and u = v a.e., then u v. The following consequence of Mazur s lemma will be useful. For a proof see Björn Björn Parviainen [8]. Lemma 2.4. Assume that {u i } i=1 is bounded in N 1,p (X) and that u i u q.e. Then u N 1,p (X) and X gu p dµ lim inf gu p i i dµ. X We assume further that X supports a weak (1, p)-poincaré inequality, i.e. there exist constants C 0 and λ 1 such that for all balls B X, all integrable functions f on X and for all upper gradients g of f, B ( 1/p f f B dµ C(diam B) g dµ) p, (2.2) λb where f B := B f dµ := B f dµ/µ(b). By the Hölder inequality, it is easy to see that if X supports a weak (1, p)- Poincaré inequality, then it supports a weak (1, q)-poincaré inequality for every q p. A deep theorem of Keith and Zhong [26] shows that X even supports a weak (1, p)-poincaré inequality for some p p, which was earlier a standard assumption for the theory of p-harmonic functions on metric spaces. In the definition of the Poincaré inequality we can equivalently assume that g is a p-weak upper gradient. Under these assumptions, Lipschitz functions are dense in N 1,p (X), and the functions in N 1,p (X) are quasicontinuous, see Shanmugalingam [37] and Björn Björn Shanmugalingam [11]. This means that in the Euclidean setting, N 1,p (R n ) is the refined Sobolev space. We need a Newtonian space with zero boundary values defined as follows for an open set Ω X, N 1,p 0 (Ω) = {f Ω : f N 1,p (X) and f = 0 in X \ Ω}. One can replace the assumption f = 0 in X \Ω with f = 0 q.e. in X \Ω without changing the obtained space. We say that f N 1,p loc (Ω) if for every x Ω there is r x such that f N 1,p (B(x, r x )). This is clearly equivalent to saying that f N 1,p (V ) for every open V Ω. By saying that V Ω we mean that V is a compact subset of Ω. 3. Minimizers and superharmonic functions Let us recall that we assume that X is a complete metric space supporting a weak (1, p)-poincaré inequality and that µ is doubling. Assume also from now on that Ω is a nonempty open set which is either unbounded or is such that C p (X \ Ω) 0. (See Section 9 for the exceptional case when X is bounded and C p (X \ Ω) = 0.) Nonlinear balayage on metric spaces 5 Definition 3.1. A function u N 1,p 1,p loc (Ω) is a minimizer in Ω if for all ϕ N0 (Ω) we have that gu p dµ g p u+ϕ dµ. (3.1) ϕ 0 A function u N 1,p loc (Ω) is a superminimizer in Ω if (3.1) holds for all nonnegative ϕ N 1,p 0 (Ω). By Proposition 3.2 in A. Björn [3] it is enough to test (3.1) with (all and nonnegative, respectively) ϕ Lip c (Ω). We follow Björn Björn [7] in the definition of the obstacle problem. This definition is a special case of the definition used by Farnana [18] for the double obstacle problem. Definition 3.2. Let V X be a nonempty bounded open set with C p (X \ V ) 0. Let f N 1,p (V ) and ψ : V [, ]. Then we define ϕ 0 K ψ,f (V ) = {v N 1,p (V ) : v f N 1,p 0 (V ) and v ψ q.e. in V }. Furthermore, a function u K ψ,f (V ) is a solution of the K ψ,f (V )-obstacle problem if gu p dµ gv p dµ for all v K ψ,f (V ). We also let K ψ,f = K ψ,f (Ω). V V Kinnunen Martio [30] made a similar definition but with q.e. replaced by a.e., which was sufficient for their purposes. Classical Sobolev functions in Euclidean spaces are defined only up to a.e. equivalence classes, so the a.e. obstacle problem is the only reasonable interpretation in that case. On the other hand, Newtonian functions are defined up to q.e. equivalence classes and correspond to the fine representatives of Sobolev functions. Hence, the q.e. definition is more natural for them. If ψ N 1,p loc (Ω), then the two types of obstacle problems coincide, but more generally there are differences, see the discussion in Farnana [18]. In particular, if E Ω has zero measure but positive capacity, then our definition of the obstacle problem leads to the capacitary potential of E in Ω, whereas solutions of the a.e. obstacle problem are trivial. In several of our results, e.g. in Theorem 5.3 and Proposition 5.6, it will be important that we work with the definition above. In nonlinear potential theory, even in the Euclidean case, obstacle problems and Sobolev spaces are a useful tool. In the classical linear theory, these notions, being essentially replaced by potentials, are often not visible at all, cf. Armitage Gardiner [1] or Doob [17]. We shall use the ess lim inf-regularization u (x) = ess lim inf u(y) := lim ess inf y x R 0 B(x,R) u. (3.2) It is easily verified that u is indeed lower semicontinuous. If Ω is bounded and K ψ,f, then there is a solution u of the K ψ,f -obstacle problem, and the solution is unique up to equivalence in N 1,p (Ω). The proof of this fact is slightly more involved than the proof of Theorem 3.2 in [30] for the a.e.- obstacle problem, see either Farnana [18] or Björn Björn [7]. Moreover u = u q.e. and u is the unique ess lim inf-regularized solution of the K ψ,f -obstacle problem. A function u is a superminimizer if and only if it is a solution of the K u,u (Ω )- obstacle problem for every nonempty open subset Ω Ω. On the other hand, if Ω is bounded, then a solution of the K ψ,f -obstacle problem is a superminimizer, and a 6 Anders Björn, Jana Björn, Tero Mäkäläinen and Mikko Parviainen superminimizer u N 1,p (Ω) is a solution of the K u,u -obstacle problem. Moreover, if u is a superminimizer then u = u q.e. and u is superharmonic (see below). If ψ, the obstacle problem reduces to the usual Dirichlet problem. By Proposition 3.8 and Corollary 5.5 in Kinnunen Shanmugalingam [32] a minimizer can be modified on a set of capacity zero so that it becomes locally Hölder continuous. A p-harmonic function is a continuous minimizer. For f N 1,p (V ), we define H V f to be the continuous solution of the K,f (V )-obstacle problem. Definition 3.3. A function u : Ω (, ] is superharmonic in Ω if (i) u is lower semicontinuous; (ii) u is not identically in any component of Ω; (iii) for every nonempty open set V Ω and all functions v Lip(X), we have that H V v u in Ω whenever v u on V. A function u : Ω [, ) is subharmonic if u is superharmonic. This definition is equivalent to the definitions used in Heinonen Kilpeläinen Martio [24] and Kinnunen Martio [30], see A. Björn [2]. If u and v are superharmonic, α 0 and β R, then αu + β and min{u, v} are superharmonic, but in general u + v is not. Superharmonic functions are ess lim infregularized, and a function in N 1,p loc (Ω) is superharmonic if and only if it is an ess lim inf-regularized superminimizer. However, there are superharmonic functions not belonging to N 1,p loc (Ω), and thus they are not superminimizers, see also a discussion in Björn Björn Parviainen [8]. A superharmonic function u satisfies the strong minimum principle: If u attains its minimum in Ω at some point x Ω, then u is constant in the component containing x. For the facts above on superminimizers, superharmonic functions and obstacle problems we refer to [30]. The following comparison lemma is proved for the a.e.-obstacle problem in Björn Björn [5], Lemma 5.4, the proof is the same in our case, see also Farnana [18], where the corresponding result is proved for the more general double (q.e.)-obstacle problem. Lemma 3.4. Assume that Ω is bounded. Let ψ j : Ω R and f j N 1,p (Ω) be such that K ψj,f j, and let u j be the ess lim inf-regularized solution of the K ψj,f j - obstacle problem, j = 1, 2. Assume that ψ 1 ψ 2 q.e. in Ω and that (f 1 f 2 ) + N 1,p 0 (Ω), then u 1 u 2 in Ω. We will need two results for superminimizers and superharmonic functions from Björn Björn Parviainen [8]. Proposition 3.5. If u is superharmonic in Ω and bounded from above by an (Ω)-function, then u is a superminimizer. N 1,p loc For the second result, called the fundamental convergence theorem, we first need to define the lim inf-regularization of a function f : Ω R as ˆf(x) = lim inf r 0 Ω B(x,r) f, x Ω. It follows that ˆf f, and it is easy to show that ˆf is lower semicontinuous. Theorem 3.6. (The fundamental convergence theorem) Let F be a nonempty family of superharmonic functions in Ω. Assume that there is a function f N 1,p loc (Ω) such that u f a.e. in Ω for all u F. Let w = inf F. Then the following are true: (a) ŵ is superharmonic; (b) ŵ = w in Ω; (c) ŵ = w q.e. in Ω. Nonlinear balayage on metric spaces 7 We will also use Choquet s topological lemma. We say that a family of functions U is downward directed if for each u, v U there is w U with w min{u, v}. Lemma 3.7. (Choquet s topological lemma) Let U = {u γ : γ I} be a nonempty family of functions u γ : Ω R. Let u = inf U. If U is downward directed, then there is a decreasing sequence of functions v j U with v = lim j v j such that ˆv = û. Proof. The proof of Lemma 8.3 in Heinonen Kilpeläinen Martio [24] generalizes directly to metric spaces. Just remember that our metric space X is separable. See also Björn Björn [7]. One way of solving the Dirichlet problem for p-harmonic functions is by using the Perron method, which was studied in Björn Björn Shanmugalingam [10] in the metric space setting. Definition 3.8. Assume that Ω is bounded. Let f : Ω R. Let U f be the set of all superharmonic functions u on Ω bounded from below such that lim inf u(y) f(x) Ω y x for all x Ω. The upper Perron solution of f is defined by P f(x) = inf u U f u(x), x Ω. Similarly, we define L f to be the set of all subharmonic functions u on Ω bounded from above such that lim sup u(y) f(x) Ω y x for all x Ω, and the lower Perron solution of f is P f(x) = sup u L f u(x), x Ω. If P f = P f, then we set P Ω f = P f = P f, and f is said to be resolutive. In Theorem 6.1 in Björn Björn Shanmugalingam [10], it is shown that if f C(Ω), then f is resolutive. Moreover, if f N 1,p (X), then f is resolutive and P f = Hf, by Theorem 5.1 in [10]. Definition 3.9. Assume that Ω is bounded. A point x 0 Ω is regular if lim P f(y) = f(x 0 ) Ω y x 0 for all f C( Ω). The set Ω is regular, if all x 0 Ω are regular. If x 0 Ω is not regular, then it is irregular. In Theorems 4.2 and 6.1 in Björn Björn [5], regular boundary points were characterized in several ways by means of barriers and obstacle problems. We recall the characterizations in ord
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