Semilinear elliptic systems with measure data

We study the Dirichlet problem for systems of the form -\Delta u^k=f^k(x,u)+\mu^k, x\in\Omega, k=1,...,n, where \Omega\subset R^d$ is an open (possibly nonregular) bounded set, \mu^1,...,\mu^n are bounded diffuse measures on \Omega, f=(f^1,...,f^n) satisfies some mild integrability condition and the so-called angle condition. Using the methods of probabilistic Dirichlet forms theory we show that the system has a unique solution in the generalized Sobolev space i.e. space of functions having fine gradient. We provide also a stochastic representation of the solution.


Introduction
Let Ω ⊂ R d , d ≥ 2, be an open bounded set. In the present paper we study the existence and uniqueness of solutions of systems of the form − 1 2 ∆u k = f k (x, u) + µ k in Ω, k = 1, . . . , n, u k = 0 on ∂Ω, k = 1, . . . , n, (1.1) where f = (f 1 , . . . , f n ) : Ω × R n → R n is a Carathéodory function and µ 1 , . . . , µ n belong to the space M 0,b of bounded diffuse measures on Ω (see Section 2). Let qL 1 loc (Ω) denote the space of locally quasi-integrable functions (see Section 2). In the scalar case (n = 1) it is known that if x → sup |u|≤r |f (x, u)| ∈ qL 1 loc (Ω) for every r ≥ 0, (1.2) y → f (x, y) is continuous on R for a.e. x ∈ Ω (1. 3) and f (x, u) · u ≤ 0 for a.e. x ∈ Ω and every u ∈ R, (1.4) then there exists a solution of (1.1) (see [13]; see also [1] for equations with general Leray-Lions type operators). One of the crucial ingredient in the proof of the existence result for (1.1) is the following Stampacchia estimate f (·, u) L 1 (Ω;m) ≤ µ T V (1.5) (see [15]), which holds true under (1.4) for every solution of (1.1). An attempt to generalize the existence result to n = 2 has been made in [13]. It is proved there that if Ω is smooth, f does not depend on x, is continuous on R 2 and monotone componentwise, i.e. f 1 (·, v), f 2 (u, ·) are nonincreasing and f 1 (0, v) = f 2 (u, 0) = 0 for every (u, v) ∈ R 2 , then there exists a unique solution of (1.1). In [13], as in the scalar case, the key step in proving the existence of solutions is Stampacchia's estimate, which is derived by using the componentwise character of monotonicity of f and by introducing the important notion of quasi-integrability of functions. Note also that in [13,Remark 7.1] the authors raise the question of existence of solutions to (1.1) for f satisfying weaker than monotonicity sign condition with respect to each coordinate, i.e. for f such that f 1 (·, u) · v ≤ 0, f 2 (u, ·) · u ≤ 0 for every (u, v) ∈ R 2 . We answer positively the question raised in [13]. Actually, using quite different than in [13] methods of proof we show existence and uniqueness results for more general systems.
Following [12] we will call (A4) the angle condition. In [12] more general than (1.1) elliptic systems with the perturbed Leray-Lions type operator are considered. As a matter of fact the assumptions in [12] when adjusted to our problem say that the perturbation satisfies some strong growth conditions and stronger then (A4) condition (A5) There exists α > 0 such that for every y ∈ R n and a.e. x ∈ R d , f (x, y), y ≤ −α|y| 2 , which we will call the uniform angle condition.
In the paper we show that if the right-hand side of (1.1) satisfies (A5) then Stampacchia's estimate (1.5) holds true for any solution of (1.1), which immediately implies that any solution of (1.1) belongs to the Sobolev space W 1,q 0 (Ω) with 0 < q < d d−1 . Under (A4) no analogue of Stampacchia's estimate appears to be available. Consequently, it seems that in general f (·, u) / ∈ L 1 loc (Ω). Therefore the first problem we have to address is to describe the regularity space for u and then formulate suitable definition of a solution of (1.1). We propose two equivalent definitions: the probabilistic and analytic one.
Let X = (X, P x ) denote the Wiener process killed upon leaving Ω and let ζ denote its life-time. In the probabilistic definition by a solution we mean a quasi-continuous in the restricted sense function u : Ω → R such that the following stochastic equation is satisfied for quasi-every (with respect to the Newtonian capacity) x ∈ Ω. Here τ is an arbitrary stopping time such that 0 ≤ τ < ζ, M is some local martingale additive functional of X (as a matter of fact M corresponds to the gradient of u) and A µ is a positive continuous additive functional of X associated with the measure µ via Revuz duality (see Section 2). In the analytic definition (see Section 4), a quasi-continuous function u : Ω → R is a solution of (1.1) if for every k = 1, . . . , n and l ≥ 0, where is the Dirichlet form associated with the operator ( 1 2 ∆, H 1 0 (Ω)), {G l , l ≥ 1} is a suitable family of finely open sets depending on u such l≥1 G l = Ω q.e. and (1.7) makes sense.
Since we are looking for solutions of (1.1) in Sobolev type spaces, our minimal regularity requirement for them is quasi-continuity or, equivalently, continuity in the fine topology (see [7]). Quasi-continuity provides some information on local, in terms of fine topology, regularity of functions and allows one to control their behavior on finely open (closed) sets of the form {u < t} ({u ≤ t}), t ≥ 0. It is therefore natural to try to derive a priori estimates for solutions of (1.1) in Sobolev spaces on finely open sets to make sense of the analytic definition and then prove existence of solutions of (1.1). In the present paper we prefer, however, a stochastic approach to the problem. The main reason for adopting the stochastic approach is that (1.6) is simpler to investigate than (1.7), because (1.7) is in fact a family of variational equations on finely open domains G k which depend on the solution. Equations of the form (1.7) were considered for example in [6,9]. It seems that direct analysis of equations (1.7) would generate many technical difficulties in using the fine topology, while the stochastic approach avoids them, because in the latter approach the fine topology is hidden in a very convenient way in probabilistic notions of the Dirichlet forms theory we are using in our proofs.
We prove that under (A1)-(A4) there exists a probabilistic solution u of (1.1), f (·, u) ∈ qL 1 loc (Ω) and u belongs to the generalized Sobolev spaceḢ 1 loc (Ω) of functions having fine gradient (see [9]). The spaceḢ 1 loc (Ω) is wider than the space T 1,2 of Borel functions whose truncations on every level belong to H 1 0 (Ω), which was introduced in [1] to cope with elliptic equations with L 1 data. The solution u satisfies (1.7), because we show that in general, if u ∈Ḣ 1 loc (Ω) then u is a probabilistic solution of (1.1) iff it is a solution of (1.1) in the analytic sense. We also prove that if we replace (A4) by x ∈ Ω and every y, y ′ ∈ R n , then the solution of (1.1) is unique. Let us note that besides proving our existence result under the general angle condition (A4), in contrast to [13] we allow f to depend on x, the dimension of the system is arbitrary and we impose no assumption on the regularity of Ω.

Preliminary results
In the whole paper we adopt the convention that for given class of real functions Let Ω be a nonempty bounded open subset in R d , d ≥ 2. By m we denote the Lebesgue measure on R d . Let (E, D[E]) be the Dirichlet form defined by (1.8) and let for m-a.e. x ∈ Ω, where {p t , t ≥ 0} is the semigroup generated by (E, D[E]) and ζ = inf{t ≥ 0, X t = ∆}, where ∆ is a one-point compactification of the space Ω and E x denote the expectation with respect to P x (see [7]). We also admit the convention that u(∆) = 0. It is well known that X is the Brownian motion killed upon leaving Ω so we sometimes use the letter B t instead of X t . From (2.1) it follows that {p t , t ≥ 0} is a semigroup of contractions. By {R α , α > 0} we denote the resolvent generated by {p t , t ≥ 0}. Since E is transient (see [7, Example 1.5.1]), R 0 is also well defined. Write R = R 0 . From (2.1) it follows that for m-a.e. x ∈ Ω.
By cap we denote the capacity associated with E, i.e. cap : 2 Ω → R + ∪ {+∞} is a subadditive set functions defined as A nonnegative Borel measure µ on Ω is called smooth if it charges no set of zero capacity and there exists an ascending sequence {F n } of closed subsets of Ω such that µ(F n ) < ∞ for n ≥ 1 and for every compact set K ⊂ Ω, By S we denote the set of all smooth measures on Ω. By M + 0,b we denote the space of finite smooth measures, M 0,b = M + 0,b − M + 0,b . Elements of M 0,b are called diffuse or soft measures (see [3,4]).
Let B(Ω) (B + (Ω)) denote the set of all real (nonegative) Borel measurable functions on Ω. For A ⊂ Ω we write A ∈ B(Ω) if 1 A ∈ B(Ω). It is known (see [7, Section 5.1]) that for every µ ∈ S there exists a unique positive continuous additive functional A µ (PCAF for short) such that for every f, h ∈ B + (Ω), (Ω) and h ∈ B + (Ω), and On the other hand, for every PCAF A of X there exists a unique smooth measure µ such that (2.4) holds with A µ replaced by A. The measure µ is called the Revuz measure associated with PCAF A. Using (2.4) one can extend the resolvent R to S by putting Similarly one can extend R α , α > 0. The right-hand side of (2.5) is finite q.e. for every finite measure µ ∈ S. This follows from (3.6) and Lemma 4.1 and Proposition 5.13 in [11]. Moreover, by Theorem 2.2.2, Lemma 2.2.11 and Lemma 5.1.3 in [7], for every µ ∈ H −1 (Ω) and v ∈ H 1 0 (Ω), whereṽ is a quasi-continuous m-version of v. By C(Ω) we denote the space of all quasi-continuous functions on Ω. Let us recall that u : Ω → R is quasi-continuous if for every ε > 0 there exists an open set G ε ⊂ Ω such that cap(G ε ) < ε and u |Ω\Gε is continuous. It is known that u is quasi-continuous iff the process t → u(X t ) is continuous on [0, ζ), P x -a.s. for q.e. x ∈ Ω (see [7,Section 4.2]).
C 0 (Ω) is the space of quasi-continuous functions on Ω in the restricted sense. Let us note that a Borel measurable function u on Ω when considered as the function on where C 0 (Ω) is the closure in C(Ω) of the space C c (Ω) of all continuous functions with compact support in Ω. A Borel function u on Ω is called quasi-continuous in the restricted sense if for every ε > 0 there exists an open set G ε ⊂ Ω such that cap(G ε ) < ε and u |(Ω∪{∆})\Gε is continuous.
The following lemma shows that if u ∈ C 0 (Ω) then u(x) tends to zero if x tends to the boundary of Ω along the trajectories of the process X.
We say that a Borel measurable function f : Ω → R is locally quasi-integrable in the analytic sense if for every compact K ⊂ U and every ε > 0 there exists an open set U ε ⊂ Ω such that cap(U ε ) < ε and f |K\Uε ∈ L 1 (K \ U ε ). We say that a Borel measurable function f : Ω → R is quasi-integrable in the analytic sense if in the above definition one can replace K by Ω. [11] it follows (see Remark 4.4) that f is quasi-integrable in the analytic sense iff f ∈ qL 1 (Ω) (this is true for bounded domains). Moreover, if f is locally quasi-integrable in the analytic sense then f ∈ qL 1 loc (Ω). The reverse implication is also true. Indeed, let f ∈ qL 1 loc (Ω) and f ≥ 0. Then by the very definition of the space dθ is a PCAF of X and its associated Revuz measure is f · m (see [7, Section 5.1]). Since the associated Revuz measure is smooth, there exists an ascending sequence {F n } of closed subsets of Ω such that f |Fn ∈ L 1 (F n ; m) for every n ≥ 1 and (2.3) holds for every compact set K ⊂ Ω. From this one can easily deduce that f is locally quasi-integrable in the analytic sense.
The notion of quasi-integrability in the analytic sense was introduced in [13]. In [13] the authors do not distinguish between local quasi-integrability and quasi-integrability, and quasi-integrabiity in the sense of [13] coincides with local quasi-integrability in the analytical sense defined in the present paper.
By T we denote the set of all stopping times with respect to the filtration {F t , t ≥ 0} (see (2.1)). Let us recall (see [7,Section 5 x ∈ Ω. By M we denote the space of all MAFs of X. By M loc we denote the set of all local additive functionals of X (see [7, page 226]) for which there exists an ascending sequence {G n , n ≥ 1} of finely open subsets of Ω such that n≥1 G n = Ω q.e., a sequence {M n } ⊂ M and N ⊂ Ω such that cap(N )=0 and for every n ≥ 1 and x ∈ Ω \ N , From now on we admit the following notation for every measurable function u : Ω → R n . Following [10,11] let us consider the class (FD) consisting of all functions u ∈ B(Ω) with the property that the process t → u(X t ) is of Doob's class (D) under the measure P x for q.e. x ∈ Ω, i.e. for q.e. x ∈ Ω the family {u(X τ ), τ ∈ T } is uniformly integrable under P x .
Definition. We say that u : Ω → R n is a probabilistic solution of (1.1) if (a) u is of class (FD), In the sequel we admit the convention that for q.e. x ∈ Ω. Indeed, by (2.5), E x A ζ < ∞ for q.e. x ∈ Ω. Since every stopping time with respect to Brownian filtration is predictable, there exists a sequence {τ n } ⊂ T such that 0 ≤ τ n < T ∧ ζ and τ n ր T ∧ ζ. Taking τ n in place of τ in (2.8) and letting n → +∞ we get (2.9) (the integral involving f (u) is well defined since f (u) ∈ qL 1 loc (Ω)). On the other hand, if (2.9) is satisfied, then replacing T by an arbitrary stopping time τ such that 0 ≤ τ < ζ we get (2.8).
Remark 2.5. If f satisfies (A4) and u is a solution of (1.1) then u vanishes on the boundary of Ω in the sense of Sobolev spaces. Indeed, by the Itô-Tanaka formula (see [2]) and (A4), for any τ ∈ T such that 0 ≤ τ < ζ, Let {τ k } be a sequence of stopping times such that 0 ≤ τ k < ζ, k ≥ 1, and τ k → ζ. Such a sequence exists since every stopping time with respect to Brownian filtration is predictable (see [14,Theorem 4,Chapter 3]). It is clear that u(X τ k ) → 0 as k → +∞, P x -a.s. for q.e. x ∈ Ω. This when combined with the fact that u is of class (FD) implies that for q.e. x ∈ Ω. By [11], T k (v) ∈ H 1 0 (Ω) for every k > 1, which forces u to vanish on the boundary of Ω.
In Section 4 we give a different, analytic definition of a solution of (1.1) and we prove that actually it is equivalent to the probabilistic definition. Before doing this we would like to present the motivation behind the two definitions. We begin with a concise presentation of famous Dynkin's formula.
Let B ⊂ Ω be a Borel set and let (see [5,6,9]). Here and in the sequel for a given function u ∈ H 1 0 (Ω) we always consider its quasi-continuous version. It is clear that H 1 0 (B) is a closed subspace of the Hilbert space H 1 0 (Ω). Therefore Let G be a finely open subset of Ω and let (E G , D[E G ]) denote the restriction of the form defined by (1.8) to G, i.e.
Applying the operator H G to both sides of the above equation and using Dynkin's formula we get As a consequence, This equation expresses the property that if u is a solution of (1.1) on Ω then for every finely open set G, u |G is a solution of (1.1) on G with the boundary condition u |G = u on ∂G.
In general, if µ ∈ M 0,b , it is natural to look for solutions of (1.1) in the class of quasi-continuous functions vanishing at the boundary of Ω and, roughly speaking, such that they coincide with functions from the space H 1 0 (Ω) on each set G from some family of finely open set which covers Ω. Therefore it is natural to require the solution u of (1.1) to satisfy (2.15) or (2.14) for each set G from some family of suitably chosen (depending on u in general) finely open sets. It is not easy to deal with such families of equations. Fortunately, we can obtain (2.15) from (2.9), and, in view of Remark 2.4, from (2.8) if we know that f (u) ∈ qL 1 (Ω). Indeed, by standard arguments one can replace T in (2.9) by τ G with arbitrary finely open set G ⊂ Ω and then putting t = 0 and taking expectation one can get which in view of (2.12) and (2.13) gives (2.15). The stochastic equations (2.8), (2.9) are much more convenient to work with than systems of the form (2.15). One of the major advantage of (2.8) (resp. (2.9)) lies in the fact that it is well defined whenever f ∈ qL 1 loc (Ω) (resp. f (u) ∈ qL 1 (Ω)) and µ ∈ M 0,b . Moreover, (2.8), (2.9) allow one to apply stochastic analysis methods to study partial differential equations and are well suited for dealing with the fine topology.

Existence and uniqueness of probabilistic solutions
We begin with the uniqueness result. Proof. Assume that u 1 , u 2 are solutions of (1.1) and M 1 , M 2 are local MAFs associated with u 1 , u 2 , respectively. Then denoting u = u 1 − u 2 and M = M 1 − M 2 we have for q.e. x ∈ Ω. By the Itô-Tanaka formula and (A4 ′ ), for q.e. x ∈ Ω. Without loss of generality we may assume that · 0û (X θ ) dM θ is a true martingale (otherwise one can apply the standard localization procedure). Therefore putting t = 0 and taking the expectation with respect to P x we conclude that for q.e. x ∈ Ω. Let {τ k } be a sequence of stopping times such that 0 ≤ τ k < ζ, k ≥ 1, and τ k → ζ. Since u is of class (FD) and u ∈ C 0 (Ω), replacing τ by τ k in (3.1) and then letting k → +∞ we conclude that |u| = 0 q.e. ✷ Remark 3.2. In general, the class C 0 (Ω) is too large to ensure uniqueness of a solution of (1.1) under (A4 ′ ). To see this, let us set n = 1, Ω = B(0, 1) ≡ {x ∈ R d ; |x| < 1} and Then u ∈ C 0 (Ω) and from the Fukushima decomposition (see [7, Theorem 5.5.1]) it follows that for some M ∈ M loc . Thus, u is a solution of (1.1) with f ≡ 0, µ ≡ 0. Obviously, the other solution of the above equation is v ≡ 0. In fact, it is known (see [3]) that u is a renormalized solution of (1.1) with f ≡ 0 and µ = σ d−1 δ 0 (which is not a smooth measure for d ≥ 2), where σ d−1 is the measure of ∂B(0, 1).
Let us recall that for a given additive functional A of X its energy is given by t ≡ u(X t ) − u(X 0 ) admits the so-called Fukushima decomposition, i.e. for q.e. x ∈ Ω, Proof. That ∇u satisfies (3.2) follows immediately from the fact that ∇u ∈ L 2 (Ω; m) (see (5.2.21) in [7]). Let {u n } ⊂ C ∞ 0 (Ω) be such that u n → u strongly in H 1 0 (Ω). By On the other hand, by (5.2.8) in [7], Then A 1 , A 2 are PCAFs of X and their associated Revuz measures are 1 B · m and m, respectively. Since 1 B · m = m, it follows from uniqueness of the Revuz correspondence that A 1 t = A 2 t , t ≥ 0, P x -a.s. for q.e. x ∈ Ω, which leads to the desired result. ✷ Let FS q , q > 0, denote the set of all functions u ∈ C(Ω) such that for q.e. x ∈ Ω, |u(X t )| q < +∞, and letḢ 1 loc (Ω) denote the space of all Borel measurable functions on Ω for which there exists a quasi-total family {U α , α ∈ I} (i.e. cap(Ω \ ( α∈I U α )) = 0) of finely open subsets of Ω such that for every α ∈ I there exists a function u α ∈ H 1 0 (Ω) such that u = u α q.e. on U α . For any function u ∈Ḣ 1 loc (Ω) one can define its gradient as Theorem 3.5. Assume that (A1)-(A4) are satisfied. Then there exists a solution u of (1.1) such that u ∈ FS q for q ∈ (0, 1), u ∈Ḣ 1 loc (Ω) and for q.e. x ∈ Ω.
Put f n = T n (f ), n ∈ N. Then f n is bounded and satisfies (A4). Let {F n } be a generalized nest such that µ n = 1 Fn · µ ∈ H −1 (Ω), n ∈ N (for the existence of such family see [7,Theorem 2.2.4]). It is well known that there exists a solution u n ∈ H 1 0 (Ω) of (1.1) with f n in place of f and µ n in place of µ. By [11], u n ∈ FS 2 and there exists M n ∈ M 2 such that for q.e. x ∈ Ω,

As in [7, page 201]) one can check that the CAFs
are of zero energy. It follows from uniqueness of the Fukushima decomposition and Lemma 3.3 that for q.e. x ∈ Ω, Since u n is of class (FD) and u ∈ C 0 (Ω), in much the same way as in the proof of (2.11) we show that By [11], v ∈ FS q , q ∈ (0, 1), for every k > 0, T k (v) ∈ H 1 0 (Ω) and v is of class (FD). Let G k = {|v| < k}. Since v is quasi-continuous, G k is finely open. Moreover, the family {G k } is quasi-total. Let us put By Itô's formula, for q.e. x ∈ Ω we have By the definition of τ k , (3.3) and (A4), for q.e. x ∈ Ω, Since Ω is bounded, there exists R > 0 such that Ω ⊂ B(0, R). Hence for every x ∈ Ω (for the last inequality see, e.g., [8, page 253]). From (2.4) and (3.5) it follows in particular that for every f ∈ B + (Ω) and µ ∈ S, where · T V denotes the total variation norm. By (3.4) and (3.6), Since G k is finely open, it follows from [7, Theorem 4.2.2] that E G k is a regular Dirichlet form on L 2 (G k ; m) and the semigroup {p G k t , t ≥ 0} is determined by the process X G k (see (2.13)). Therefore for every f ∈ L 2 (G k ; m), Moreover, by [7,Theorem 4.4.4], E G k is transient and D[E G k ] = H 1 0 (G k ). Therefore from (3.7) it follows that sup n≥1 R G k (|∇u n | 2 ) L 1 (Ω;m) < +∞. (3.8) On the other hand, by [7, Lemma 5.1.10], (Ω) and |u n (x)| ≤ k on G k for q.e. x ∈ Ω, we conclude from the above estimate that sup n≥1 Ω |∇(u n · R G k 1)| 2 (y) dy < +∞. (3.9) Since L 2 (Ω; m) has the Banach-Saks property, it follows from (3.9) that one can choose a subsequence (still denoted by {n}) such that σ n ({∇(u n · R G k 1)}) is convergent in L 2 (Ω; m) for every k ≥ 1. By [7, Theorem 2.1.4], one can find a further subsequence (still denoted by {n}) such that {σ n ({u n R G k 1})} is convergent q.e. for every k ≥ 1.
on Ω. Set u(s, x) = lim sup n→+∞ σ n ({u n })(s, x) for (s, x) ∈ Ω. Since R G k 1 ∈ H 1 0 (Ω), u is quasi-continuous. Using this and the fact that R G k 1 > 0 q.e. on G k we see that one can find a quasi-total family {G k } such that for every k ≥ 1, as n → +∞. Therefore we may define a measurable function w : Ω → R n × R d such that w |G k = lim n→+∞ σ n ({∇u n }) in L 2 (G k ; m) for every k ≥ 1. Let us fix α > 0 and ν ∈ S (0) 00 (see [7,Section 2]). Then Similarly, which converges to 0 as n → ∞. Using the above two inequalities and the Borel-Cantelli lemma one can show that P x ( T 0 |w(X t )| 2 dt < +∞, T ≥ 0) = 1 for q.e. x ∈ Ω and there exists a subsequence (still denoted by {n}) such that for every T > 0, in ucp on [0, T ] with respect to P x for q.e. x ∈ Ω (see the proof of [7, Theorem 5.2.1]). Furthermore, by (3.9), the Rellich-Kondrachov theorem and the fact that R G k 1 > 0 q.e. on G k and {G k } is a quasi-total family we conclude that there exists a subsequence (still denoted by {n}) such that u n → u m-a.e. Hence, by Lemma 3.4, for n ≥ k and 0 ≤ τ < ζ we have which for q.e. x ∈ Ω converges to 0 as n → +∞. From this we conclude that for q.e.

Analytic solutions
To formulate the definition of a solution of (1.1) in the analytic sense we will need the following lemma.
is the resolvent associated with the perturbed form (see [7]). It follows in particular that φ is quasi-continuous. Put Since f is positive and φ is quasi-continuous, {G k } is a finely open quasi-total family. We have To see this, it suffices to take T 3 = T 1 × T 2 and {W (α,β) = U α ∩ V β , α ∈ T 1 , β 1 ∈ T 2 }.
From the the definition of the spaceḢ 1 loc (Ω), Lemma 4.1 and Remark 4.2 it follows that for given u ∈Ḣ 1 loc (Ω), f ∈ qL 1 loc (Ω) and µ ∈ S there exists a finely open quasi-total family {U α , α ∈ T } such that for every α ∈ T , u = u α q.e. on U α for some u α ∈ H 1 0 (Ω), 1 Uα f ∈ L 2 (Ω; m), 1 Uα · µ ∈ H −1 . In what follows we say that {U α , α ∈ T } is a finely open quasi-total family for the triple (u, f, µ) ∈Ḣ 1 loc (Ω) × qL 1 loc (Ω) × S if (4.1) is satisfied for every α ∈ T . Proposition 3.1 and Remark 3.2 suggest that the classḢ 1 loc (Ω) ∩ C 0 (Ω) is too large to get uniqueness of analytic solutions of (1.1) under (A4 ′ ), and secondly, that the uniqueness holds if we restrict the class of solutions to functions which additionally are of class (FD). Unfortunately, we do not know how to define the class (FD) analytically. Therefore to state the definition of a solution of (1.1) in a purely analytic way we introduce a class U , which is a bit narrower than (FD).
where v is a solution in the sense of duality (see [15]) of the problem for some ν ∈ M 0,b }.
Proof. Using the Itô-Tanaka formula we get the first inequality in (2.10). From this inequality, the fact that u is of class (FD), u ∈ C 0 (Ω) and (A5) one can conclude that for every τ ∈ T such that 0 ≤ τ < ζ, for q.e. x ∈ Ω. The desired inequality now follows from Fatou's lemma and [11,Lemma 5.4]. ✷