ON THE WEAK CONTINUITY OF ELLIPTIC OPERATORS AND APPLICATIONS TO POTENTIAL THEORY

Transcription

1 ON THE WEAK CONTINUITY OF ELLIPTIC OPERATORS AND APPLICATIONS TO POTENTIAL THEORY Neil S. Trudinger Xu-Jia Wang Centre for Mathematics and Its Applications Australian National University Canberra, ACT 0200 Australia Abstract. In this paper, we establish weak continuity results for quasilinear elliptic and subelliptic operators of divergence form, acting on corresponding classes of subharmonic functions. These results are analogous to our earlier results for fully nonlinear k-hessian operators. From the weak continuity, we derive various potential theoretic results including capacity estimates, potential estimates and the Wiener criterion for regular boundary points. Our methods make substantial use of Harnack inequalities for solutions. Running title: Weak continuity of elliptic operators Key words: Elliptic operator, weak continuity, subharmonic, subelliptic, capacity, potential estimates, Wiener criterion. AMS mathematical subject classification: 35J67, 35H20, 35B05. Supported by Australian Research Council 0

2 ON THE WEAK CONTINUITY OF ELLIPTIC OPERATORS AND APPLICATIONS TO POTENTIAL THEORY Neil S. Trudinger Xu-Jia Wang 1. Introduction There are numerous papers on the local behaviour of solutions, subsolutions, and supersolutions of second order, quasilinear elliptic, divergence structure, partial differential equations, as well as extensions to degenerate elliptic and subelliptic equations. In this paper we show that many properties, including the quasi-continuity of subharmonic functions, asymptotic behaviour of fundamental solutions, removability of singularities, Wiener criteria for regular boundary points, can be derived via Harnack inequalities for solutions. The two main ingredients in our approach, which follows on from our treatment of the fully nonlinear Hessian operators in [45, 46, 47], together with that of Labutin [30], are a weak convergence result and a pointwise potential estimate for subharmonic functions. The weak convergence result is analogous to that for Hessian operators [45, 46] while the potential estimate extends that found for quasilinear elliptic operators by Kilpelainen and Maly [27]. Weak convergence results in the theory of nonlinear partial differential equations have been addressed in the survey [13]. In this paper we treat nonlinear elliptic, divergence structure, operators for which Harnack type inequalities are available. To state our result, we need an appropriate notion of subharmonic function. In the fully nonlinear case [46], we employed the viscosity notion based on differential inequalities for smooth test functions. For quasilinear divergence structure equations the traditional definition, using the homogeneous equation, is more convenient. Accordingly, we shall say that a continuous function u, in a domain in Euclidean n-space, IR n, is harmonic with respect to an operator L, or L-harmonic in short, if it is a solution of the homogeneous equation L[u] = 0 (in an appropriate weak sense). An upper semicontinuous function u in is subharmonic with respect to L, or briefly, L-subharmonic, if the set {u = } has Lebesgue measure zero and if for any open set O (i.e., O ) and any L-harmonic function v such that v u on O, it follows v u in O. In the following we will denote by USC() the set of Supported by Australian Research Council 1

3 upper semi-continuous functions such that the set {u = } has Lebesgue measure zero, by H() the set of all L-harmonic functions, and by SH() the set of all L-subharmonic functions. For an L-subharmonic function, we will assign a nonnegative measure µ = µ[u] to u such that µ[u] = L[u] when the latter is an integrable function. Then our weak convergence result states that if {u j } is a sequence of locally uniformly upper bounded L-subharmonic functions which converges to an L-subharmonic function u a.e., then µ[u j ] µ[u] weakly as measures. Our approach to the weak convergence depends on the divergence structure of the operators and basic pointwise properties. As a result, it applies to a large class of divergence structure, elliptic operators. The fundamental examples are quasilinear operators of the form, (1.1) L[u] = diva(x, Du), where A : IR n IR n is a vector valued function, measurable in the first variables and continuous in the second, satisfying the structure conditions: (1.2) (1.3) A(x, ξ) a 0 ξ p 1, A(x, ξ) ξ ξ p, and the monotonicity condition (1.4) (ξ ζ) (A(x, ξ) A(x, ζ)) > 0, for all x, ξ, ζ IR n, (ξ ζ), for positive constants a 0 and p with p > 1. Typical examples are the p-laplacian operators (1.5) p = div( Du p 2 Du) for p > 1, which have been extensively studied in the literature. The monotonicity condition (1.4) is a weak ellipticity condition, which coincides with ellipticity for the linear case (1.6) L[u] = D i (a ij (x)d j u), where (1.4) is equivalent to A = [a ij ] > 0 in. An important generalisation of the structure conditions (1.2) (1.3) involves the incorporation of weight functions, thereby embracing classes of degenerate elliptic linear operators. An exhaustive treatment of degenerate elliptic quasilinear equations under such hypotheses is presented in the monograph [23]. For other related results, see [16, 17]. Further extensions permit non-isotropic structure in the gradient variables, as in [34]. 2

4 Our methods also apply to subelliptic quasilinear operators, as studied recently by Garofalo and co-workers [4, 5, 6, 10, 11, 18]. Here the gradient operator D is replaced by a system of vector fields X = (X 1,, X m ) with (1.7) X i u = b ij D j u, i = 1, m, and b ij C (IR n ), i = 1,, m, j = 1,, n. The corresponding operator L is defined by m (1.8) L[u] = Xi A i (x, Xu), i=1 where A = (A 1,, A m ) : IR m IR m and X i is the formal adjoint of X i given by (1.9) Xi v = D i (b ij v). If u is locally integrable in, together with its distributional derivatives X i u and the functions A i (, X i ), then L[u] in (1.8) is well defined in the distribution sense, which we employ for our notion of L-harmonic function. We suppose that the system X satisfies the rank condition, (1.10) rank Lie[X 1,, X m ] = n at each point of, where Lie[X 1,, X m ] denotes the Lie algebra generated by X 1,, X m and successive commutators, while the function A is subjected to conditions (1.2), (1.3), (1.4), with ξ, ζ IR m, instead of IR n. The quasilinear elliptic operator (1.1) then corresponds to the special case m = n, X i = D i u, i = 1, 2,, n. A simple example in IR 3, where one commutator is needed to achieve the rank condition (1.10), is given by (1.11) X 1 u = D 1 u, X 2 u = D 2 u + x 1 D 3 u. A special case of (1.8) is the subelliptic p-laplacian operator, (1.12) L[u] = X i ( Xu p 2 X i u), which corresponds to the functional (1.13) J p (u) = Xu p = [ m (X i u) 2] p/2. Note that the regular subelliptic Laplacian is a linear degenerate elliptic operator of the form (1.14) L[u] = X i (X i u) = D i (b ik b jk D j u). 3 i=1

5 Our proof also applies to fully nonlinear elliptic operators, including the k-hessian operators, (1.15) F k [u] = and the k-curvature operators, (1.16) H k [u] = i 1 < <i k λ i1 λ ik i 1 < <i k κ i1 κ ik, where 1 k n, (λ 1,, λ n ) are the eigenvalues of the Hessian matrix D 2 u, and (κ 1,, κ n ) are the principal curvatures of the graph of u. Instead of Harnack inequalities we have interior gradient estimates for Hessian and curvature equations. The k-hessian operators F k [u] can also be expressed as the sum of the k th -principal minors of the Hessian matrix D 2 u. When k = 1, it is the Laplacian, when k = n, it is the Monge-Ampere operator. The weak convergence result for Hessian operators was proved in [46] by a series of integral estimates. By these integral estimates we introduced further mixed Hessian measures and proved their weak convergence [47]. These weak convergence results were employed to develop a nonlinear potential theory, (quite complete compared with the Newtonian potential theory), for the Hessian operators [47, 30]. We remark that both operators (1.15) and (1.16) can be written in divergence form, (1.17) L[u] = diva(du, D 2 u). In this paper we focus specifically on quasilinear subelliptic operators of the form (1.8). The special case (1.1), where Xu = Du, shall be referred to as the elliptic case. We leave it to the reader to check that our arguments also embrace quasilinear degenerate elliptic equations, including the type considered in [23]. However, we need to treat curvature equations, even the mean curvature case, separately since various modifications are required due to their inhomogeneity [49]. The weak convergence result will be proved in Section 3, (Theorem 3.1). Prior to this, we present some basic material on quasilinear subelliptic equations in Section 2. We first develop our approach to weak convergence in Section 3, through the technique of Perron liftings, which as remarked earlier can be applied rather generally. However we conclude Section 3, with a different proof based on gradient estimates, Lemma 3.9, which for subelliptic operators are similar to the elliptic case [23]. Applications to potential theory are treated in the remaining sections, following to some extent previous routes [2, 23, 27, 30, 47]. In Section 4, we introduce for a subset E, the capacity (1.18) C(E, ) = sup{µ[u](e) u SH(), 0 u 1}, 4

6 which is equivalent to the standard p-capacity (Theorem 4.4), (1.19) C p (E, ) = inf{ Xu p u C 0 (), u 1 on E} for E. Among other results, we prove also, in Theorem 4.2, that the supremum in (1.18) for closed E is achieved by the potential ˆPE semicontinuous regularisation of the function SH(), which is the upper (1.20) P E = sup{w SH() w 0 in, w 1 in E}. We also extend well known results [23] such as the quasicontinuity of subharmonic functions (Theorem 4.1) and the capacitability of Borel sets (Theorem 4.3) to the subelliptic case. We remark here that the capacity (1.18) is more useful in our investigation of the operator L, which differs from that in [23]. The treatment in [23], based on the capacity (1.19), was extended to the subelliptic case in [36]. A further significant result in quasilinear elliptic potential theory is the following pointwise estimate (called Wolff potential estimate) for L-subharmonic functions, due to Kilpelainen and Maly [27]. Letting µ = µ[u] denote the measure extension of L[u], we set (1.21) W µ p (x, r) = r 0 µ(b t (x)) t n p dt t, where for x, t > 0, B t (x) denotes the Euclidean ball of centre x and radius t. Then for quasilinear elliptic operators (1.1) satisfying the structure conditions (1.2) and (1.3), and u SH(B 2R (x)), u 0, we have (1.22) C 1 W p µ (x, R) u(x) C{W p µ (x, 2R) + sup u }, B R (x) where C is a positive constant depending only on n, p and a 0. In Section 5, we extend the potential estimate (1.22) to subelliptic operators, Theorem 5.1, utilising a new proof, based on the Harnack inequality, which is simpler, in the elliptic case, than that in [27]. As an application of the pointwise estimate, we treat in Section 6 the boundary regularity of Perron solutions. That is, we consider for a continuous function u 0, whether the Perron solution (1.23) u = sup{w SH() w u0 on } is continuous on. We introduce the concept of thin set (Theorem 6.1) and prove the Wiener criterion for the regularity of boundary points (Theorem 6.2), which, for quasilinear 5

7 elliptic equations, was originally established in [27]. We also prove that the set of irregular boundary points has null capacity, (Theorem 6.3). Finally in Section 7 we consider further applications of the pointwise estimates. We give necessary and sufficient conditions on µ[u] such that u is continuous or Hölder continuous, and a result on the asymptotic behaviour of the fundamental solutions to the operator (1.8), Theorem 7.2. We also investigate the removability of singularities of solutions to L[u] = 0, extending previous results in Newtonian and quasilinear elliptic potential theory [8, 42, 50]. From the weak convergence result follows the existence of solutions to the Dirichlet problem with f a nonnegative Radon measure. We will treat the Dirichlet problem separately in [48] where we extend the weak convergence to signed measures and discuss the uniqueness of solutions. To conclude this introduction, we remark that our results in this paper suggest that a potential theory can be developed for an elliptic operator of divergence form if the corresponding equation satisfies a Harnack inequality (or has interior regularity). The divergence form is essential as the weak convergence does not hold in general for elliptic operators (see discussion at the end of section 3). For divergence form elliptic operators without interior regularity, such as the complex Monge-Ampére equation, the results are weaker than those in this paper [2, 29]. For partial results for non-divergence operators, see [25]. 2. Quasilinear subelliptic equations In this section, we gather together the basic properties of quasilinear subelliptic equations of the form (1.8), which we need for further development. The vector fields X 1,, X m will be subject to the rank condition (1.10), while the function, (x, ξ) IR m A(x, ξ) IR m, is measurable in x and continuous in ξ, satisfying the structure conditions: (2.1) (2.2) (2.3) A(x, ξ) a 0 ξ p 1, A(x, ξ) ξ ξ p, (ξ ζ) (A(x, ξ) A(x, ζ)) > 0 for all x, ξ, ζ IR m, (ξ ζ), for positive constants a 0 and p (> 1). Subelliptic equations have been studied considerably [15, 37, 40, 41], ever since the fundamental paper of Hörmander [22], where he proved that a linear subelliptic equation, 6

8 satisfying the rank condition (1.10) is also hypoelliptic, that is distribution solutions are smooth. Various applications of subelliptic equations are indicated in [3]. For p > 1, equation (1.12) arises naturally in studying CR manifolds and quasi-regular mappings on stratified Lie groups. The rank condition (1.10) for vector fields was introduced in [7, 9]. It is proven that the Euclidean metric and the C-C (Carnot-Caratheodory) metric induced from the vector fields X are mutually Hölder continuous; (see [15, 20, 37]). For any two points x, y, the Carnot-Caratheodory distance d(x, y) = d X (x, y) is defined by (2.4) d(x, y) = inf{t > 0 a sub-unitary γ : [0, T ] IR n with γ(0) = x, γ(t ) = y}, where a piecewise C 1 curve γ(t) : [0, T ] IR n is said to be sub-unitary, with respect to the system of vector fields X, if for every ξ IR n and t (0, T ), (2.5) (γ (t) ξ) 2 m (X i (γ(t)) ξ) 2. i=1 Let B R (x) denote the ball {y IR n d(x, y) < R}, and let be a bounded domain in IR n. Then there exist positive constants C, Q and R 0, depending on X and, such that (2.6) B tr (x) Ct Q B R (x) for any x, t (0, 1), and R < R 0, where B r denotes the Lebesgue volume; (see [37]). The number Q ( n) will be chosen as the least integer such that (2.6) holds, and is called the homogeneous dimension of X in. We remark that for example (1.11), we have Q = 4, (n + 1) and R 0 = +. Following [4], we introduce, for p 1, a Sobolev space S 1,p (), defined by (2.7) S 1,p () = {u L p () Xi u L p, i = 1,, m}, where X i u is understood in the sense of distributions. It is a Banach space under the norm ( 1/p (2.8) u S 1,p () = ( u p + Xu )) p, being the completion of C S 1,p () under (2.8), [18]. Furthermore, the following Sobolev type embedding holds (2.9) S 1,p 0 () Lq () for p < Q, q pq/(q p), which is compact when q < pq/(q p), where S 1,p 0 () is the completion of C0 () under (2.8). For p > Q, we have q =. We refer the reader to [4, 18] for proofs. There is also a Poincaré inequality [24], (2.10) u u B p CR p B R (x) 7 B R (x) Xu p

9 for all u S 1,p (), R R 0, x, where C and R 0 are positive constants depending on X,, p and X, respectively, and u B = 1 u. B R (x) B R (x) Note that the Sobolev inequality in [4], corresponding to (2.9), (2.11) ( B ) 1 p/q u pq/(q p) C B Xu p, for u S 1,p 0 (B), B = B r(x), x, is also subject to the restriction R < R 0, where R 0 is the constant in (2.6). In the well known Euclidean case, X i u = D i u, we have of course, Q = n and R 0 =. From the Sobolev embedding (2.9) and the Poincaré inequality (2.10), follow the existence and uniqueness of weak solutions to the Dirichlet problem (2.12) L[u] = f in, u = u 0 on, where is any bounded domain in IR n, u 0 S 1,p (), f [ S 1,p () ], the dual space of S 1,p (). A function u S 1,p () (or more generally u S 1,p loc ()) is called a weak solution (subsolution, supersolution) of the equation, L[u] = f in, if for any ψ C0 (), ψ 0, (2.13) A(x, Xu) Xψ + ( f, ψ ) = 0 ( 0, 0). Uniqueness for (2.12) follows from the monotonicity condition (2.3), since (Xu Xv) (A(x, Xu) A(x, Xv)) = 0 if u and v are two solutions of (2.12) and the resultant identity Xu = Xv implies Du = Dv from (1.10). Consequently u v S 1,p 0 () must be a constant, which by virtue of the Poincaré inequality (2.10) is zero. The existence follows from standard results in monotone operator theory [31], since by (2.2), L is a coercive operator in the space S 1,p 0 (). Note that coercivity in the small balls follows immediately from (2.11). In the general case, we need a positive bound from below for the functional J p in (1.13) on the set {u S 1,p 0 () u L p () = 1}. Using the compactness of the embedding S 1,p 0 () Lp () and the direct method in the calculus of variations, we see that this is equivalent to the uniqueness of the trivial solution to the Dirichlet problem (2.12), when L is the subelliptic p-laplacian (1.12), f = 0, u 0 = 0. This argument also shows that the ball B in the Sobolev inequality (2.11) can be replaced by the whole domain. 8

10 From the theory of variational inequalities, (see for example [28], 3.1 or [23], Appendix I), one also has the existence and uniqueness of solutions to the corresponding obstacle problem. That is, for any measurable function g : (, ] and u 0 S 1,p () such that the set (2.14) K = K g,u0 = {w S 1,p () w g a.e. and w u0 S 1,p 0 ()} is nonempty, there is a unique solution u K of the variational inequality (2.15) A(x, Xu)X(u v) 0 v K, where g is the upper obstacle and u 0 the boundary function. Obviously the solution u is also a subsolution of the equation, L[u] = 0 in, by the definition (2.13), and a solution when g =. Local estimates for solutions of quasilinear subelliptic equations follow analogously to the elliptic case, ([19], [42]). Inserting a test function ψ into (2.13) of the form (2.16) ψ = η p u where η 0, C 0 (), we obtain under the structural hypotheses (2.1), (2.2), a Cacciopoli type inequality for solutions of the homogeneous equation, Lu = 0, namely, (2.17) η p Xu p C Dη p u p, where C depends on a 0. From the Poincaré and Sobolev inequalities, (2.10), (2.11), one has the weak Harnack inequality [4]. Let u S 1,p () be a non-negative weak supersolution of the homogeneous equation, L[u] = 0 in. Then, for B R (x), q < q 0 = Q(p 1)/(Q p) if p Q, R R 0, 0 < σ < τ < 1, we have (2.18) ( ) 1/q 1 u q C ess inf B τr u, BσR B τr where R 0 depends on X,, as before, and C depends on p, q, σ, τ, X,, and a 0. For p > Q, we have a full Harnack inequality with sup BτR u on the left hand side of (2.18). For the elliptic case, (2.18) is established in [44], following the earlier work of Moser [35] and Serrin [42]; (see also [19] and [23]). For our approach, through subsolutions and subharmonic functions, we will use the corresponding weak Harnack inequality for non-positive weak subsolutions, namely (2.19) ( 1 B τr B τr ( u) q ) 1/q C ess sup BσR u. 9

11 From the weak Harnack inequality (2.18), we have local Hölder estimates for weak solutions of the equation, L[u] = 0, with respect to both C-C and Euclidean metrics. Namely for any subdomain, there exist constants C, α > 0, depending on X,, and a 0 such that (2.20) u(x) u(y) sup x,y d α (x, y) C sup u, u(x) u(y) sup x,y x y α C sup u. We also have a full Harnack inequality for non-negative weak solutions of the homogeneous equation, L[u] = 0. namely for B R (x), R R 0, 0 < τ < 1, (2.21) sup B τr u C inf B τr u. with R 0 as before and C > 0 depending on X,, τ, and a 0. Moreover the Harnack inequality (2.21) also holds in shells B τr B σr, 0 < σ < τ < 1, with C also depending on σ, [4,5]. 3. Weak convergence In this section we establish various convergence results for quasilinear subelliptic operators of the form (1.8), culminating in the weak convergence result, Theorem 3.1. Since an L-subharmonic function is locally upper bounded, for simplicity we suppose that the subharmonic functions we treat in this section are nonpositive. First we give two lemmas on the relation between subsolutions and L-subharmonic functions. These two results are proved in [23] for quasilinear elliptic equations. equations. The same proofs extend to subelliptic Lemma 3.1. For any u SH(), there is a sequence of subsolutions {u j } S 1,p loc () C() of the equation (3.1) L[u] = 0 in such that u j u a.e.. We outline the proof here. Let {w j } be a sequence of C () functions such that w j u pointwise. Let u j be the solution of the obstacle problem (2.15) with upper obstacle g = w j such that u j = w j on. Then w j u j u, u j u, and u j is a subsolution of (1). The continuity of u j follows from the weak Harnack inequality, (2.19). 10

12 Lemma 3.2. Let u S 1,p loc () be a subsolution of (3.1). Let (3.2) u(x) = ess lim y x u(y). Then u is L-subharmonic and u = u a.e.. Obviously u is upper semicontinuous. The L-subharmonicity of u follows by definition. The fact that u = u a.e. follows from the weak Harnack inequality (2.19). A similar argument will also be used in the proof of Lemma 3.4 below, so we omit the details here. From Lemma 3.2 we may suppose all subsolutions of (3.1) are upper semicontinuous. Therefore a function u is a subsolution of (3.1) if and only if u SH() S 1,p loc (). If u is a subsolution of (3.1), then L[u] is a nonnegative distribution, and so it is a nonnegative measure, which we denote by µ[u]. It follows (3.3) ϕdµ[u] = XϕA(x, Xu) ϕ C 0 (). Let u S 1,p loc (), u 0, be a subsolution of (3.1). For δ > 0 small, let uδ = u in δ = {x d(x, ) > δ}, and u δ be the unique solution of L[u] = 0 in ω := δ such that u δ = u on δ, u = 0 on, i.e., (3.4) u δ = sup{w SH() w u in δ, w 0 on }. In other words, u δ S 1,p 0 () is the unique solution of the obstacle problem (2.15) with obstacle g = uχ δ and boundary condition u 0 = 0. In the next lemma, we use the function u δ to establish a uniform bound for the measure µ in compact subsets of. Lemma 3.3. Let u S 1,p loc (), u 0, be a subsolution of (3.1). Then for any E, (3.5) µ[u](e) C where C depends on δ = d(e, ) and sup δ/3 2δ/3 u δ. Proof. There is no loss of generality in assuming is a smooth domain. Since u δ S 1,p (), u = u δ near E, we have (3.6) µ[u](e) = µ[u δ ](E) ϕdµ[u δ ] = A(x, Xu δ ) Xϕ ( C suppdϕ 11 Xu δ p ) 1 1/p C

13 by (2.17) since L[u δ ] = 0 in suppdϕ, where ϕ C 0 ( δ/3 ) is a nonnegative function such that ϕ = 1 in 2δ/3. It follows that if u SH() L (), then u S 1,p loc () and for any subdomain, (3.7) u S 1,p ( ) C, where C > 0 depends only on n, p,,, and u L (). Indeed, by Lemma 3.1 it suffices to prove (3.7) for u S 1,p loc (). Let uδ be as above, where δ = 1 2 dist(, ). We have Xu δ p C Xu δ A(x, Xu δ ) = C ( u δ )dµ[u δ ] C. Hence (3.7) follows since u = u δ in δ. Alternatively, we may deduce (3.7) directly by replacing u by u inf u in (2.16). For any locally upper bounded measurable function u, we can define (3.8) ũ(x) = lim sup u, r 0 B r (x) such that ũ is upper semicontinuous. The function ũ is called the upper semicontinuous regularisation of u. The following lemma shows that subharmonicity is essentially preserved by almost everywhere convergence. Lemma 3.4. Suppose {u j } SH() such that u j converges to a function u pointwise with {u = } = 0. Let ũ be the upper semicontinuous regularisation of u. Then ũ = u a.e. and ũ is L-subharmonic. Proof. First suppose {u j } SH() such that u j u pointwise. Then u is upper semicontinuous and ũ = u. To see that u is L-subharmonic suppose ϕ is L-harmonic and ϕ u on O, where O is an open subset of. Then for any ε > 0, by the monotonicity of {u j } and by the upper semicontinuity of u j we have ϕ+ε > u j on O for j large enough. It follows that ϕ+ε u j in O for j large and hence ϕ u in O. That is, u is L-subharmonic. Next suppose {u j } is monotone increasing. Replacing u j by max{u j, t} for any given constant t > 0, we may suppose {u j } is uniformly bounded and u j S 1,p loc (). To prove ũ = u a.e., we suppose to the contrary that the set {u < ũ} has positive Lebesgue measure. We claim there is a point x 0, and positive constants ε, δ > 0 such that (3.9) {x u(x) < ũ(x0 ) ε} B rj (x 0 ) δ B rj (x 0 ) 12

14 for a sequence r j 0. Indeed, choose ε > 0 such that the set {u(x) < ũ(x) ε} has positive Lebesgue measure, where. For any r > 0 small, let {B r (x j ) j = 1,, j r } be a set of disjoint C-C metric balls such that all x j are rational numbers and no other balls with radius r can be located in {x IR n d(x, ) < r} without intersecting these balls. Then {B 3r (x j )} is a finite cover of. If {u(x) < ũ(x) ε} B r (x j ) δ B r (x j ) for all these balls, by (2.6) we have (3.10) {u(x) < ũ(x) ε} {u(x) < ũ(x) ε} B 3r (x j ) Cδ, which is a contradiction when δ is small enough. Hence for any r > 0 small, there is a ball B r (x) with rational x as the centre such that {u < ũ ε} B r (x) δ B r (x). Since the set (0, r 0 ) is larger than the set of all rational points in, we obtain (3.9). For any ε 1 << ε, there exists r > 0 small enough such that (3.11) sup{ũ(x) x B 2r (x 0 )} ε 1 + ũ(x 0 ). Let j be large enough so that sup u j ũ(x 0 ) ε 1. B r (x 0 ) Applying the weak Harnack inequality (2.19) to we obtain v = u j sup ũ, B 2r (x 0 ) ( ) 1/q 1 (3.12) sup v C v q, q (0, q 0 ). B r B 2r B 2r On the left hand side we have sup Br v 2ε 1. But on the right hand side we have, by (3.9) and since u u j, 1 v q Cδε q. B 2r B 2r (x 0 ) We reach a contradiction when ε 1 is sufficiently small. Hence ũ = u a.e.. 13

15 To see that ũ is L-subharmonic let O be an open set and let ϕ be L-harmonic such that ϕ u on O, then ϕ u i on O and by definition we have ϕ u i in O. It follows ϕ ũ in O. For a general sequence {u j } SH(), let w k,j = max{u k,, u j }. Then for fixed k, w k,j w k pointwise for a nonpositive function w k and its upper semicontinuous regularisation w k SH(). Moreover, w k u a.e.. Hence u = ũ a.e. and ũ SH(). By the weak Harnack inequality (2.19) and the upper semicontinuity it is easy to see that if u 0 is L-subharmonic, then ( ) 1/q 1 (3.13) u(x) = lim u q, 0 < q < q 0 r 0 B r (x) B r (x) for any x. From (3.13) we see that if u, v SH() such that u = v a.e., then u v. Therefore from the above proof it follows that if {u j } SH() and u SH() such that u j u a.e., then u j u pointwise. In our next lemma we introduce the important technique of Perron lifting. Lemma 3.5. For any u SH() and any open set ω, let the Perron lifting of u over ω, u ω, be the upper semicontinuous regularisation of (3.14) u = sup{w SH() w u in ω}. Then u ω SH() and u ω is L-harmonic in ω. Proof. By Choquet s lemma, (see Lemma 8.3 in [23]), there is a monotone increasing sequence (u j ) SH() with u j u in ω such that u j u and the upper semicontinuous regularisation ũ = u ω. Hence by Lemma 3.4 we have u ω SH(). To prove u ω is L-harmonic in ω we have by Lemma 3.1 a monotone decreasing sequence u j SH()) S 1,p loc () such that u j u. By the solvability of the Dirichlet problem (2.12), there exists u j S1,p loc () such that u j = u j in ω, u j u j, and u j is L- harmonic in ω. Obviously u j is L-subharmonic and u j u, where u is given by (3.14). Hence u j u u and so u u ω. On the other hand, since u is L-subharmonic and u = u on ω, we have u u ω by definition. It follows u = u ω and hence u ω is L-harmonic in ω. Remark 3.1. In addition to the Perron lifting defined in (3.14), we also use the following modification when ω (say, ω = δ ). Let u SH(), u 0, and let u ω 0 be the upper semicontinuous regularisation of (3.15) u 0(x) = sup{w SH() w u in ω, w 0 on }. 14

16 Then u ω 0 is L-subharmonic in, L-harmonic in ω, and u ω 0 = 0 on, in the sense that u ω 0 S 1,p 0 near. If furthermore u S 1,p loc (), then uω 0 is the solution of the obstacle problem (2.15) with upper obstacle g = uχ E, E = ω, and boundary condition u 0 = 0, and so u ω S 1,p 0 (). Similarly the Perron lifting uω, defined in Lemma 3.5, lies in S 1,p loc () if u S 1,p loc (). Note that we have already used the modified lifting uδ = u ω 0 when ω = δ, δ > 0, in Lemma 3.3. Let ω be an open set with smooth boundary. Let { {x ω d(x, ω) > t} if t 0, (3.16) ω(t) = {x d(x, ω) < t} if t > 0. Then ω(0) = ω, and ω(t) depends continuously on t near 0. For u SH(), the Perron lifting u ω(t), defined in Lemma 3.5, is monotone increasing. That is (3.17) lim t δ uω(t) (x) u ω(δ) (x) lim t δ + uω(t) (x) x. Let q be as in (2.18). Then u ω(t) L q (), as a function of t, is monotone and bounded. Hence u ω(t) L q is continuous for almost all t. Since u ω(t) is continuous in ω(t), it follows (3.18) lim t δ u ω(t) (x) = u ω(δ) (x) for a.e. δ ( δ 0, δ 0 ) and all x, where δ 0 is a small positive constant. We remark that (3.18) holds for all δ ( δ 0, δ 0 ) if u S 1,p (), since u + u S 1,p 0 (ω(δ)) and u± are L-harmonic in ω(δ), where u + = lim t δ + u ω(t) and u = lim t δ u ω(t). Using the extended domain ω(t), we now derive a relation between convergence of sequences of subharmonic functions and their Perron liftings. Lemma 3.6. Suppose (u j ) SH() such that u j u SH() a.e.. Suppose (3.18) holds at δ = 0. Then u ω j uω a.e. Proof. By Lemma 3.4 we may suppose by choosing a subsequence that u ω j converges a.e. to a function v SH() such that v = u in ω. By definition we have u ω v, namely, (3.19) u ω lim j u ω j. For t > 0, let D t = ω(t) ω and let u t j = ud t j. By choosing a subsequence we have u t j ut a.e. for some function u t SH(), and u t = lim j ut j lim j u j = u. 15

17 We claim that for any ε 1 > 0, (3.20) u t j u ε 1 in ω = ω(t/2) ω(t/4) as long as j is large enough. Indeed, by the interior Hölder continuity (2.20) of solutions of (3.1), we have u t j ut uniformly in ω. Hence the claim follows since u t u. From (3.20) it follows since t > 0. It follows when j is large enough. Hence Together with (3.19) we obtain (3.21) u ω(t) lim j u ω(t) j (u t j) ω(t/2) u ω(t/2) ε 1 u ω ε 1 u ω(t) j (u t j) ω(t/2) u ω ε 1 lim j u ω(t) j u ω. lim j u ω(t) j u ω. The lemma follows by the assumption that u ω(t) is continuous at t = 0. Let u S 1,p loc () be a subsolution of (1). Then as indicated earlier, L[u] defines a nonnegative measure, which we denote by µ[u]. An important property of µ[u] is that for any subdomain ω with smooth boundary, µ[u](ω) depends only on the value of u near ω. That is, (3.22) µ[u](ω) = ω dµ[u] = lim ϕ ε dµ[u] ε 0 ω = lim A(x, Xu)Xϕ ε. ε 0 supp(xϕ ε ) where ϕ ε C 0 (ω) such that ϕ ε = 1 in {x ω d(x, ω) > ε}. We also need the following simple convergence results for L-harmonic functions. Lemma 3.7. Let (u j ) be a sequence of L-harmonic functions converging to u uniformly. Then (3.23) Xu j Xu a.e. and (3.24) XϕA(x, Xu j ) XϕA(x, Xu) 16 ϕ C ().

18 Proof. For any, let η C0 () such that η 0, η = 1 in. We have (A(x, Xu j ) A(x, Xu))(Xu j Xu) (A(x, Xu j ) A(x, Xu))X((u j u)η) + (A(x, Xu j ) A(x, Xu))(u j u)xη = (A(x, Xu j ) A(x, Xu))(u j u)xη 0 by (3.7). Hence (3.23) holds. To prove (3.24) we have for any ε > 0, letting E with E < ε such that Xu j Xu uniformly in E, Xϕ(A(x, Xu j ) A(x, Xu)) ε 1 + ( A(x, Xu j ) + A(x, Xu) ) E ε 1 + C E (r 1)/r ( Xu j (p 1)r + Xu (p 1)r ) ε 2, where r Finally we are ready to extend the measure µ from subsolutions to subharmonic functions. p p 1 in view of (3.7). Hence (3.24) holds. Lemma 3.8. Let {u j } SH() S 1,p loc () be a sequence of L-subharmonic functions which converges to u SH() a.e., then the sequence of measures {µ[u j ]} converges to a Radon measure µ weakly. Proof. By Lemma 3.3, µ[u j ] are locally uniformly bounded in. Hence there is a subsequence of µ[u j ] which converges weakly to a measure µ. We need to prove that µ is independent of the choice of the subsequences of {u j }, (as in [45], Theorem 1.1). For this purpose we suppose {u j }, {v j } SH() S 1,p loc () such that both sequences converge to u a.e. and such that µ[u j ] µ, µ[v j ] ν weakly as measures. For any ball B r (x 0 ) such that B 2r (x 0 ), we need to prove µ(b r ) = ν(b r ), or equivalently to prove for any σ > 0, E (3.25) µ(b r ) ν(b r+σ ), ν(b r ) µ(b r+σ ). Let ω = B r+σ B r. We may suppose u ω(δ), as a function of δ, is continuous at δ = 0, otherwise by (3.18) we may choose r r and σ σ such that u ω(δ) (with ω = Ber+eσ Ber) is continuous at δ = 0. By Lemma 3.6 we have u ω j uω and v ω j uω a.e.. By the interior Hölder continuity (2.20) we have furthermore u ω j, vω j uω uniformly in B r+3σ/4 B r+σ/4. We claim that (3.26) lim j µ[uω j ](B r ) = lim j µ[vω j ](B r ) 17

19 for any r (r + σ/4, r + 3σ/4). Suppose for a moment that (3.26) holds. Observe that (3.27) lim j µ[vω j ](B r ) lim j µ[v j](b r+σ ) ν(b r+σ ), and µ[u j ](B r ) = µ[u ω j ](B r ) µ[u ω j ](B r ). We obtain, by (3.26) and (3.27), Hence (3.25) holds. µ(b r ) lim j µ[u j](b r ) lim j µ[uω j ](B r ) = lim j µ[vω j ](B r ) ν(b r+σ ). To prove (3.26) let ϕ C 0 (B r +t) be a nonnegative cut-off function such that ϕ = 1 in B r, t < σ/8. Then we have (3.28) since Xϕ is supported on B r +t B r other hand, we have (3.29) µ[u ω j ](B r ) ϕdµ[u ω j ] = A(x, Xu ω j )Xϕ A(x, Xu ω )Xϕ and u ω j, uω are L-harmonic in B r+σ B r. On the µ[vj ω ](B r +t) ϕdµ[vj ω ] = A(x, Xvj ω )Xϕ A(x, Xu ω )Xϕ. Hence lim j µ[u ω j ](B r ) lim j µ[v ω j ](B r +t) for any t > 0 small. Sending t 0 and exchanging u j and v j we obtain (3.26). By Lemma 3.8 we can define for any u SH(), a Radon measure µ = µ[u] being the weak limit of any sequence {µ[u j ]} such that u j SH() S 1,p loc () and u j u a.e.. We can now state our weak convergence result as follows. 18

20 Theorem 3.1. For any u SH(), there exists a Radon measure µ[u] such that (i) µ[u] = L[u] if u S 1,p loc () and, (ii) if (u j ) SH() is a sequence of L-subharmonic functions which converges to u SH() a.e, then µ[u j ] µ[u] weakly as measures. Proof. The first part follows by definition. To see the second part, we have, for any ϕ C0 (), by Lemmas 3.1 and 3.8 there is a sequence ũ j SH() S 1,p loc () such that ũ j u a.e. and such that (3.30) lim ϕdµ[ũ j ] = lim ϕdµ[u j ]. j j The left hand side is equal to ϕdµ[u] by Lemma 3.8. That is, µ[u j] µ[u] weakly as measures. Theorem 3.1 can be extended to quasilinear elliptic operators on manifolds. An alternative approach Our approach above to convergence results for L-subharmonic functions rested on reduction to properties of the corresponding L-harmonic functions by means of Perron liftings. As indicated in the introduction this procedure extends to embrace various types of nonlinear equations including Hessian and curvature equations. In our papers [46] and [47], we employed a more direct but technically more complicated approach based on gradient estimates for the subharmonic functions themselves. For quasilinear subelliptic equations, satisfying (2.1), (2.2) and (2.3), it turns out that the gradient estimates are relatively simple, as in the elliptic case [23], and readily yield a more direct proof of Theorem 3.1, which we present now. This approach was also employed by us in our treatment of quasilinear elliptic equations involving signed measures in [48]. To initiate the procedure, we first need an extension of the vector fields X i to L- subharmonic functions, corresponding to that for the gradient in the elliptic theory [48]. Using the fact that if u is L-subharmonic in, its truncation u N = max{u, N} S 1,p loc (), we see that (3.31) Xu = lim N Xu N is well defined a.e.. Note that Xu coincide with the regular distributional derivatives if either u L () or u S 1,p loc () for some p 1 but in general Xu L1 loc (). We can now prove the gradient estimate Lemma 3.9. For any u SH(), u 0, we have (1 + u ) γ S 1,p loc () and Xu Lr loc (), where γ < p 1 p and r < Q(p 1) Q 1. 19

21 Proof. Since the result is local, we may replace by a small sub-ball. By approximation (Lemma 3.1) and replacing u by u δ (defined in (3.4)), we need only prove relevant estimates for u S 1,p 0 (). For any θ > 0, by Lemma 3.3 we have ( ) 1 1 (1 + u ) θ dµ[u] < C. Integrating by parts, we obtain (3.32) X(1 + u ) 1 ε p C ( 1 ) 1 (1 + u θ dµ[u] < C, ) where ε = 1 p (1 + θ). Hence (1 + u )γ S 1,p loc () and by the Sobolev embedding (2.11), we have (3.33) By Hölder s inequality, (3.34) where ( Xu r u q < C, q = r = X(1 + u ) 1 ε p ) r/p ( Qp(1 ε), (p < Q). Q p Qp(p 1 θ) Q(p 1 θ) + (Q p)(1 + θ). Lemma 3.9 holds since θ > 0 can be arbitrarily small. (1 + u ) q ) 1 r/p < C, Using the estimate (3.34) we can then extend Lemma 3.7 to almost everywhere convergence of L-subharmonic functions. Lemma If {u j } SH() such that u j u SH() a.e., then the convergence results (3.23) (3.24) hold. Proof. To prove (3.23) let h j,k (x) = (A(x, Xu j ) A(x, Xu k ))(Xu j Xu k ) and let E j,k = {x δ hj,k (x) > ε}. We have E j,k E j,k { u j u k > ε 2 } + 1 ε E j,k { u j u k <ε 2 } h j,k. Let w j,k = { uj u k if u j u k < ε 2, ε 2 u j u k u j u k otherwise. 20

22 For any cut-off function η C0 () such that η = 1 in δ, we have h j,k (A(x, Xu j ) A(x, Xu k ))X(w j,k η) E j,k { u j u k <ε 2 } Cε 2. + (A(x, Xu j ) A(x, Xu k ))w j,k Xη w j,k η(dµ[u j ] + dµ[u k ]) + C w j,k ( Xu j p 1 + Xu k p 1 ) It follows E j,k 0 as j, k. Hence (3.23) holds. By Lemma 3.9, (3.24) follows similarly as before. S 1,p From Lemma 3.10 we obtain Theorem 3.1 immediately. Indeed, let {u j } SH() loc () such that u j u SH() a.e.. Then by (3.24), (3.35) ϕdµ[u j ] = XϕA(x, Xu j ) XϕA(x, Xu) ϕ C0 (), so that we can define the measure µ[u] by (3.36) ϕdµ[u] = XϕA(x, Xu). The advantage of our first proof, which reduces, in view of (3.22), the weak convergence for sequences of L-subharmonic functions to that of L-harmonic functions, is that it can be applied to many other operators for which a formula like (3.24) for subharmonic functions is difficult to prove, or not available. One example is the quasilinear elliptic operators with nonstandard growth condition [34]. Other examples include Hessian operators and curvature operators already mentioned. The divergence structure is crucial in both proofs. In fact, the weak convergence result is not true in general for elliptic operators of nondivergence form. For example, it is not true for the Pucci operator, (3.37) L[u] = Λ σ + i + λ σ i where Λ > λ > 0 are positive constants, (σ 1,, σ n ) are eigenvalues of the Hessian matrix D 2 u, σ + i = max(σ i, 0) and σ i = min(σ i, 0). Another example is the one dimensional operator given by { 2u x (0, 1], (3.38) L[u] = u x (1, 2). 21

23 4. Capacities Most potential theoretic results on quasilinear elliptic operators [23] can be extended to the quasilinear subelliptic operator (1.8) [36]. In this section we will prove a few basic results on capacities for later applications. Our treatment is based on the weak convergence result, Theorem 3.1, and differs somewhat from that in [23, 36]. First we need a convergence result (corresponding to Theorem 2.6 in [47]). Lemma 4.1. Let {u j }, {v j } SH(), be uniformly bounded sequences in S 1,p (). Suppose u j u monotonely and v j v a.e. Then u j u in S 1,p loc (), and (4.1) u j µ[v j ] uµ[v] weakly as measures. Proof. Since the results are local, by Remark 3.1 we may suppose u j = v j = 0 on and µ[u j ] = µ[v j ] = 0 in δ. Then u j, v j converges to u, v locally uniformly in δ. First we prove (4.1) holds. For any cut-off function η 0, η C 0 (), we claim that (4.2) lim j ηu j dµ[v j ] ηudµ[v] Indeed, if {u j } is monotone increasing, we have (4.3) lim j ηu j dµ[v j ] lim j ηudµ[v j ] ηudµ[v] by the upper semicontinuity of u. If {u j } is monotone decreasing, we have, for any given i, ηu i dµ[v] lim j ηu i dµ[v j ] lim j by the upper semicontinuity of u i. Let i we obtain ηudµ[v] lim i ηu i dµ[v] ηu j dµ[v j ] by the upper semicontinuity of u. Recall that if u j u SH() a.e., u j u pointwise (see note before Lemma 3.5). Hence (4.2) holds. Next we show for any u SH() S 1,p (), (4.4) uµ[v j ] uµ[v] 22

24 weakly as measures. Indeed, for any function η C0 (), we have ηu(dµ[v j ] dµ[v)] = X(ηu)(A(x, Xv j ) A(x, Xv)) X(ηu)(A(x, Xv j ) A(x, Xv)) + ε E ε,m C( X(ηu) p ) 1/p + ε E ε,m where E ε,m = { Xv j Xv < ε} { Xv < M}. The last integral converges to zero since by Lemma 3.10, E ε,m 0 when ε 0 and M. Hence (4.4) holds. By virtue of (4.4) it is easy to prove the reverse of (4.2) holds. If {u j } is monotone decreasing, we have, by (4.4), ηu j dµ[v j ] ηudµ[v] ηu(dµ[v j ] dµ[v]) 0. If {u j } is monotone increasing, we have, again by (4.4), lim ηu j dµ[v j ] lim ηu i dµ[v j ] = j j for any fixed i. Sending i we have ηu i dµ[v] = X(ηu i )A(x, Xv) ηu i dµ[v] X(ηu)A(x, Xv) since v S 1,p () and Xu i Xu weakly in L p (). Hence (4.1) holds. To prove u j u in S 1,p loc (), we suppose to the contrary that there is a subsequence, which we still denote by {u j }, and a subdomain such that u j u S 1,p ( ) δ > 0 for all j. Since Xu j Xu a.e. (Lemma 3.10), there is a sequence E i with E i 0 such that X(u j u) L p (E i ) δ for all j, and Xu j Xu uniformly in E i for any given i. It follows lim X(u j u) (A(x, Xu j ) A(x, Xu)) j ( = lim Xuj A(x, Xu j ) Xu A(x, Xu) ) j E i [ lim X(u j u) p C Xu p] δ/2 j E i E i if i is large enough. On the other hand, from (4.1) we have (4.5) X(u j u)(a(x, Xu j ) A(x, Xu)) 0. We reach a contradiction. This completes the proof. Note that Lemma 4.1 is not true if the monotonicity condition is removed. 23

25 Let be a bounded domain in IR n and E a Borel set. capacity C(E) of E, relative to, by We define the (inner) (4.6) C(E) = C(E, ) = sup{µ[u](e) u SH(), 0 u 1}, The capacity satisfies the following properties: (i) for any E 1, E 2, C(E 1 E 2 ) C(E 1 ) + C(E 2 ); (ii) if E 1 E 2 and 1 2, then C(E 1, 1 ) C(E 2, 2 ); and (iii) if E 1 E 2 are Borel subsets of, then C( m E m ) = lim m C(E m ). Indeed, (i) and (ii) are obvious. To see (iii) let u SH(), 0 u 1, be chosen such that C(E) µ[u](e) + ε, where E = m E m. Since lim m µ[u](e m ) lim m C(E m ), we have C(E) lim m C(E m ). The other inequality follows from (ii). From Lemma 4.1, we can then deduce a further convergence result. Lemma 4.2. Let {u j } SH() be a monotone sequence which converges to a function u SH() almost everywhere. Then u j u pointwise except a set of capacity zero. Furthermore, for any ε > 0, there is a Borel set E ε with C(E ε ) < ε such that u j u locally uniformly in E ε. Proof. It suffices to prove the second assertion and we need only to prove that for any η C 0 (), η 0, (4.7) a j = sup{ η u j u dµ[v] v SH(), 0 v 1} 0 as j. First we suppose that {u j } is uniformly bounded in S 1,p (). Let v j SH(), 0 v j 1 in, such that η(u j u)dµ[v j ] 1 2 a j. By Lemma 4.1 and (3.7) we see that the left hand side converges to zero. Hence (4.7) holds. In the general case by Remark 3.1 we may suppose u j = 0 on and µ[u j ] = 0 near. By the monotonicity of the sequence {u j } we need only to prove either (4.8) lim N C(E N ) 0 if u j is monotone decreasing, or lim C(E i,n ) 0 N 24

26 for any fixed i, if u j is monotone increasing, where E N = {u < N} and E i,n = {u i < N}. We claim that for any Borel set E, and any u SH() such that u 1 in E and u 0 on, (4.9) C(E) µ[u](). To prove (4.9) let us first consider the case when E is a closed subset. We need to prove that for any w SH() with 0 w 1, there holds µ[w](e) µ[u](). Replacing u by (1+δ)u, and w by (1 δ)(w+ 1 2 ) 1 2, for some δ > 0 small, we may suppose E {u < 1} and 0 > w > 1 in. Let w = max(u, w). Then w = w near E and w = u near. It follows µ[w](e) = µ[ w](e) µ[ w]() = µ[u](). Hence (4.9) holds for closed E. If E is not closed, we can choose a sequence of closed sets E j E. Then (4.9) follows from (iii) above. By (4.9) we have C(E N ) µ[n 1 u]() 0, since N 1 u 0 uniformly near as N. Similarly we have C(E i,n ) 0 as N. Hence (4.8) holds. We can now prove the quasicontinuity of L-subharmonic functions. Theorem 4.1. Let be a bounded domain and u SH(). Then for any ε > 0, there exists an open subset O with C(O) ε such that the restriction of u to O is continuous. Proof. Let {u k } C() be a sequence of L-subharmonic function such that u k u pointwise as k. For any given j > 1, by Lemma 4.2 we have C({u k > u + 2 j } δj ) < 2 j ε as long as k is large enough, say, k k j, where δ j > 0, δ j 0 as j. Let O = j=1 {u k j > u + 2 j }. Then C(O) < ε, u kj u locally uniformly in O, and u is continuous when restricted on O. For any set E, let (4.10) P E = P E, = sup{w SH() w 0 in, w 1 on E}, and let ˆP E = ˆP E, be the upper semicontinuous regularisation of P E, that is (4.11) ˆPE (x) = lim sup P E. r 0 B r (x) ˆP E is called the potential of E in. 25

27 Theorem 4.2. We have ˆP E = P E, except on a set of capacity zero. Furthermore if E is closed, the potential ˆPE is a maximiser for capacity, that is, (4.12) C(E) = µ[ ˆP E ](E). Proof. By Choquet s lemma ([23], p.158), there is a monotone increasing sequence u j with u j 0 in and u j 1 on E such that u j P E a.e.. The first assertion then follows from Lemma 4.2 and Lemma 3.4. To prove (4.12) we have µ[ ˆP ] = 0 in E by Lemma 3.5, where ˆP = ˆP E. Hence by (4.9) we have C(E) µ[ ˆP ]() = µ[ ˆP ](E). On the other hand, by definition, we have µ[ ˆP ](E) C(E), and hence (4.12) holds. Since µ[ ˆP ] = 0 in E, from Theorem 4.2 we also have (4.13) C(E) = µ[ ˆP ](E { ˆP = 1}) = µ[ ˆP ]({ ˆP = 1}). Next we introduce an outer capacity. For any subset E we define (4.14) C (E) = C (E, ) = inf{c(o) E O, O open}. Then the outer capacity satisfies (i), (ii) above, and (iv) if E 1 E 2 are compact Borel subsets of, then C ( j E j ) = lim m C (E m ). For Borel sets, the outer capacity agrees with the capacity. Theorem 4.3. For any Borel set E we have C(E) = C (E). Proof. By definition we have C(E) C (E) for any subset E. We need only to prove that for a Borel set E, (4.15) C(E) C (E). First we consider the case when E is a closed set. By Choquet s lemma [23], there is a monotone increasing sequence (u j ) SH() such that u j ˆP = ˆP E a.e.. It follows, by (4.9), (4.16) C (E) lim j µ[u j]() = µ[ ˆP ]() = µ[ ˆP ](E) C(E). It follows that for any Borel set E, C (E) = 0 if C(E) = 0. Indeed, let {K j } be a sequence of closed subsets such that K j E. Since C (K j ) = C(K j ) = 0, there exists, for ε > 0, an open set O j K j such that C (O j ) 2 j ε. Hence C (E) C ( j O j ) j C (O j ) ε, 26

28 so C (E) = 0. For an arbitrary Borel set E, let Ê = E { ˆP E = 1}. By Theorem 4.2, C(E Ê) = 0. Hence it suffices to prove (4.17) C (Ê) C(E). Let {E j } be a sequence of closed sets such that E j E. Then ˆP j is monotone decreasing and so ˆP j ˆP pointwise (see remarks before Lemma 3.5), where ˆP j = ˆP Ej, and ˆP = ˆP E,. It follows, by (4.9) and (4.12), (4.18) C({ ˆP = 1}) µ[ ˆP ]() = lim j µ[ ˆP j ]() Hence This completes the proof. = lim j µ[ ˆP j ](E j ) = lim j C(E j) = C(E) C (Ê) lim C({ ˆP < 1 + δ}) = C({ ˆP = 1}) C(E). δ 0 In view of (4.9) we also have, for any Borel set E, (4.19) C(E) = inf{µ[u]() u SH(), u = 0 on, u 1 on E}. The standard capacity in elliptic potential theory is defined by (see [1]) (4.20) C p (E) = C p (E, ) = inf{ Xu p u S 1,p 0 (), u 1 on E}. It is readily seen that C p satisfies the properties (i)-(iii) above. We prove C p and C are equivalent. Theorem 4.4. For any Borel set E we have (4.21) αc(e) C p (E) βc(e), for some positive constants α and β independent of E. If L is the subelliptic p-laplacian operator (1.12), then C(E) = C p (E). Proof. It suffices to prove the case when E is closed. First we prove C = C p when L is the subelliptic p-laplacian operator. Suppose the minimum in (4.20) is given by u. Then u solves the variational inequality Xu p 2 XuX(v u) 0 27

29 for any v S 1,p 0 () such that v χ E, where χ E is the characteristic function. Hence u is subharmonic with respect to the subelliptic p-laplacian. Hence C p (E, ) can be defined equivalently by (4.22) C p (E, ) = inf{ Xu p u SH() S 1,p 0 (), u 1 on E}. The infimum (4.22) is achieved by u = ˆP E, the potential of E in with respect to the subelliptic p-laplacian operator. Let Ê = {u = 1}. Then C p(e) = C p (Ê) from the definition (4.22) and C(E) = C(Ê) by (4.13). We have, since µ[u] = 0 on {u > 1}, (4.23) C p (E) = Xu p = ( u)dµ[u] = µ[u](ê) = C(E). To prove (4.21) let v = ˆP E be the potential of E in with respect to the operator L. Since µ[v] is supported on E and u = 1 on E, we have C(E) = = ( C ( u)dµ[v] XuA(x, Xv) Xu p ) 1/p ( ) (p 1)/p XvA(x, Xv) where XvA(x, Xv) = ( v)dµ[v] = C(E), Xu p = C p (E). Hence C(E) αc p (E). Similarly we can prove C p (E) βc(e). For any Borel set E B r/2, let u = ˆP E,B2r. Applying the Harnack inequality (2.21) to u + 1 in the shell B 2r B r/2, we have u 1 + a on B r for some a > 0 independent of E and r. Hence u + a 1 (u + 1) 0 on B r. By (4.9) and the first part of Theorem 4.2 it follows C p (E, B r ) µ p [(1 + a 1 )u](b r ) = (1 + a 1 ) p 1 C p (E, B 2r ), where µ p is the measure corresponding to the subelliptic p-laplacian operator. By Theorem 4.4 we obtain (4.24) C(E, B r ) const.c(e, B 2r ). 28

30 The capacity C p (B r, B R ), 0 < r < R < R 0, has been computed in [5]. Let r/r (δ, 1 δ) for some δ > 0. One has (4.25) C 1 β(x, r) C p (B r, B R ) Cβ(x, r), where C > 0 is independent of x, r, r p B r (x) if 1 < p < Q(x), (4.26) β(x, r) = 1 if p = Q(x), r Q(x) p if p > Q(x), and Q(x) [n, Q] is the homogeneous dimension at x of X 1,, X m in B R. It is proven [37] that there exist constants C, R 0 > 0 independent of x, and a = a(x) > 0 such that (4.27) a r Q(x) B r (x) Cr Q(x) for r R 0. Hence (4.28) C p (B r, B R ) r Q(x) p. For later applications we need to extend the definitions of capacities in (4.19) (4.20). For a Borel set E and t > 0, we define Then C t (E, ) = inf{µ[u]() u SH(), u = 0 on, u t on E} Cp(E, t ) = inf{ Xu p u S 1,p 0 (), u t on E}. (4.29) αc t (E) 1 t Ct p(e) βc t (E), with the same constants α and β as in Theorem 4.4. Let a < b be two constants. We define ˆP = ˆP a,b E, as the upper semicontinuous regularisation of the function (4.30) P a,b E, (x) = sup{w SH() w b in, w a on E}. Then ˆP E, = ˆP 1,0 E,, µ[ ˆP ] is supported on E if E is closed, and (4.31) µ[ ˆP ](E) = C b a Cb a p (E) = C C b a (E), where C, C > 0 depend only on α, β. If the operator L satisfies the homogeneous condition A(x, tξ) = t t p 2 A(x, ξ), one has C t (E) = t p 1 a,b C(E) and ˆP E, = b + (b a) ˆP E,. 29