Renormalized solutions of semilinear elliptic equations with general measure data

In the paper, we first propose a definition of renormalized solution of semilinear elliptic equation involving operator corresponding to a general (possibly nonlocal) symmetric regular Dirichlet form satisfying the so-called absolute continuity condition and general (possibly nonsmooth) measure data. Then we analyze the relationship between our definition and other concepts of solutions considered in the literature (probabilistic solutions, solution defined via the resolvent kernel of the underlying Dirichlet form, Stampacchia's definition by duality). We show that under mild integrability assumption on the data all these concepts coincide.


Introduction
Let L be the operator associated with a symmetric regular Dirichlet form (E, D(E)) on L 2 (E; m), f : E × R → R be a measurable function and µ be a bounded signed Borel measure on E. In the paper we consider semilinear equations of the form − Lu = f (·, u) + µ in E. (1.1) One of the important problems that arises when studying such equations is the problem of proper definition of a solution. This problem has been dealt with by many authors.
In the present paper we first introduce yet another definition of a solution of (1.1). It is a slight modification of the definition of a renormalized solution introduced in [13] in case µ is smooth. Then we analyze the relationship between this new definition and other concepts of solutions known in the literature. In case L is a uniformly elliptic divergence form operator and f does not depend on u, some definition, now called Stampacchia's definition by duality, was proposed by Stampacchia [24] in 1965. Later on, to deal with equations with more general local operator L, the definitions of entropy solution and renormalized solution were introduced. For a comparison of different forms of these definitions and remarks on other concepts of solutions of equations of the form (1.1) with local operator L and f not depending on u see [6]. Elliptic equations with local operators and nonlinear dependence on general measure data are studied in [7,18].
In case f depends on u most of known results are devoted to the case where µ is smooth. Recall (see [10]) that µ admits a unique decomposition into the smooth (diffuse) part µ d and the concentrated part µ c , i.e. µ d is a bounded Borel measure, which is "absolutely continuous" with respect to the capacity Cap determined by (E, D(E)), and µ c is a bounded Borel measure which is "singular" with respect to Cap. In case L is local and µ is smooth entropy and renormalized solutions of (1.1) are studied in numerous papers (see, e.g., [1,8] and the references given there).
A definition of renormalized solutions applicable to (1.1) with general L associated with a general transient (possibly non-symmetric) Dirichlet form was recently given in [13].
If (E, D(E)) is symmetric and f (·, u) ∈ L 1 (E; m), renormalized solutions in the sense of [13] coincide with probabilistic solutions of (1.1) defined earlier in [12] (see also [14] for equations with operator L associated with a non-symmetric quasi-regular form and [17] for equations with nonlinear dependence on measure data). Recall that a measurable u : E → R is a probabilistic solution of (1.1) in the sense of [12,14] if the following nonlinear Feynman-Kac formula is satisfied for quasi-every x ∈ E. In (1.3), M = (X, P x ) is a Markov process with life time ζ associated with E, E x denotes the expectation with respect to P x and A µ is the continuous additive functional of M associated with µ in the Revuz sense (see Section 2). The equivalence between renormalized and probabilistic solutions allows one to use effectively probabilistic methods in the study of renormalized solutions of (1.1). Also note that if f ∈ L 1 (E; m) then renormalized solutions of (1.1) coincide with Stampacchia's solutions by duality defined in [12,14]. The semilinear case with general, possibly nonsmooth bounded measure µ is much more involved. The study of (1.1) with nonsmooth measure was initiated in 1975 by Brezis and Bénilan in case L is the Laplace operator ∆ (see [2,4] and the references given there for results and historical comments). For some existence and uniqueness results in case L is the fractional Laplacian ∆ α/2 with α ∈ (0, 2) see Chen and Véron [5]. Very recently, Klimsiak [11] started the study of (1.1) in case L corresponds to a transient symmetric regular Dirichlet form satisfying the following absolute continuity condition: (ACR) R α (x, ·) is absolutely continuous with respect to m for each α > 0 and x ∈ E, where R α (x, dy) denotes the resolvent kernel associated with (E, D(E)) (see Section 2.2). Equivalently, (ACT) p t (x, ·) is absolutely continuous with respect to m for each t > 0 and x ∈ E, where p t (x, dy) is the transition function associated with (E, D(E)). The above conditions are satisfied for instance if L is a uniformly divergence form operator or L = ∆ α/2 with α ∈ (0, 2). If the form is transient, then under (ACR) the resolvent kernel R 0 (x, dy) has a density r. In [11] a measurable function u on E is called a solution of (1.1) if for quasi every x ∈ E. In case µ c = 0, the above equation reduces to (1.3), so the definition of [11] reduces to the probabilistic definition of a solution given in [12,14].
In [11] also a partly probabilistic interpretation of (1.4) is given. This suggests that solutions defined via the resolvent density, i.e. by (1.4), may be equivalently defined as renormalized solutions in the same manner as in [13]. In the present paper we show that this is indeed possible. The definition of a renormalized solution adopted in the present paper is a minor modification of the definition of [13]. In our opinion, it is natural, especially from the probabilistic point of view. Moreover, in many cases considered so far in the literature (µ is smooth or µ is nonsmooth and L = ∆ or L = ∆ α/2 , like in [4,5]) the solutions considered there coincide with the renormalized defined in the present paper.
The main result of the paper says that if the form is transient and (ACR) is satisfied then the renormalized solution is a solution in the sense of (1.4), and if u is a solution of (1.1) in the sense of (1.4) and u ∈ L 1 (E; m) then u is a renormalized solution. We find important that, as in the case of smooth measures, this correspondence when combined with probabilistic interpretation of (1.4) given in [11] enables one to study renormalized solutions of (1.1) with the help of probabilistic methods. For results on (1.1) obtained in this way we defer the reader to [11]). Finally, note that at the end of the paper we describe some interesting situations in which solutions of (1.1) in the sense of (1.4) automatically have the property that f (·, u) ∈ L 1 (E; m).

Preliminaries
In the paper E is a separable locally compact metric space and m is a Radon measure on E such that supp[m] = E. By B(E) (resp. B + (E)) we denote the set of all real (resp. nonnegative) Borel measurable functions on E, and by B b (E) the subset of B(E) consisting of all bounded functions.

Dirichlet forms
By (E, D(E)) we denote a symmetric regular Dirichlet form on H = L 2 (E; m) (see [ We will say that some property of points in E holds quasi everywhere (q.e. for short) if the set for which it does not hold is exceptional.
We say that a function u on E is quasi-continuous if there exists a nest {F n } such that u |Fn is continuous for every n ≥ 1. By [9, Theorem 2.1.7], each function u ∈ D e (E) has a quasi-continuous m-version.
Let µ be a signed Borel measure on E, and let |µ| = µ + + µ − , where µ + (resp. µ − ) we denote the positive (resp. negative) part of of µ. We say that µ is smooth if |µ| does not charge exceptional sets and there exists a nest {F n } such that |µ|(F n ) < ∞, n ≥ 1. The set of all smooth measures on E will be denoted by S. By M b we denote the set of all signed Borel measures on E such that µ T V := |µ|(E) < ∞, and by M 0,b the subset of M b consisting of all smooth measures. S + is the subset of S consisting of nonnegative measures. Similarly we define For a complete description of the structure of µ c see [15].

Markov processes
Let E ∪ ∆ be the one-point compactification of E. When E is already compact, we adjoin ∆ to E as an isolated point. We adopt the convention that every function f on E is extended to E ∪ {∆} by setting f (∆) = 0.
By [ life time ζ and cemetery state ∆ whose Dirichlet space is (E, D(E)). This means in particular that for every α > 0 and f ∈ B b (E) ∩ H the resolvent of M, that is the function . In the paper we will assume that M satisfies (ACR) condition formulated in Section 1. By [9, Theorem 4.2.4], for symetric forms considered in the present paper (ACR) is equivalent to (ACT). In general, for non-symmetric forms, (ACT) is stronger than (ACR). Also note that in the literature (ACR) is sometimes called Meyer's hypothesis (L) (see [23,Chapter I,Exercise 10.25] Assume that (E, D(E)) is transient. Then there exists a nonnegative B(E) ⊗ B(E)measurable function r : E × E → R such that r(x, y) = r(y, x), x, y ∈ E and for every Borel set B ⊂ E, In what follows given a positive Borel measure on E, we write For a signed Borel measure µ on E, we set Rµ(x) = Rµ + (x) − Rµ − (x), whenever Rµ + (x) < +∞ or Rµ − (x) < +∞, and we adopt the convention that Rµ(x) = +∞ if Rµ + (x) = Rµ − (x) = +∞.
Denote by M the set of all signed Borel measures µ on E such that R|µ|(x) < +∞ for m-a.e. x ∈ E. By Proposition 2.1, M b ⊂ M. In general, the inclusion is strict (see the remark following [14, Proposition 3.2]).
We define additive functional (AF in abbreviation) and continuous AF of M as in [9, Sections 5.1]. By [9,Theorem 5.1.4], there is a one to one correspondence (called Revuz correspondence) between the set of smooth measures µ on E and the set of positive continuous AFs A of M. It is given by the relation where E m denotes the expectation with respect to the measure P m (·) = E P x (·) m(dx).
In what follows the positive continuous AF of M corresponding to a positive µ ∈ S will be denoted by A µ . If µ in S, then µ + , µ − ∈ S, and we set for q.e. x ∈ E. Indeed, if α > 0 and µ is a measure of finite 0-order energy integral (µ ∈ S (0) 0 in notation; see [9, Section 2.2] for the definition), then (2.1) follows from Exercise 4.2.2 and Lemma 5.1.3 in [9]. The general case follows by approximation. We first let α ↓ 0 to get (2.1) for α ≥ 0 and µ ∈ S where f : E × R → R is a measurable function, f u = f (·, u), µ ∈ M and L is the operator associated with (E, D(E)), i.e. the nonpositive definite self-adjoint operator on H such that where (·, ·) denotes the usual inner product in H (see [9, Corollary 1.3.1]).
The following two definitions of solutions of (3.1) were introduced in [11].
(b) for every exceptional set N ⊂ E, every stopping time T such that T ≥ ζ and every sequence {τ k } ⊂ T such that τ k ր T and E x sup t≤τ k |u(X t )| < ∞ for all x ∈ E \ N and k ≥ 1, we have Any sequence {τ k } with the properties listed in condition (b) will be called the reducing sequence for u, and we will say that {τ k } reduces u. (ii) Assume that u is a probabilistic solution of (1.1). Then for q.e. x ∈ E we have  In what follows for a function u on E and a measure µ on E, we set whenever the integral is well defined, and for k ≥ 0, we write Remark 3.5. (i) By [11,Theorem 3.7], if u is a solution of (1.1) then u is quasicontinuous.
(ii) Let u be a solution of (1.1) with µ ∈ M b . If f u ∈ L 1 (E; m) then by [11,Theorem 3.3], T k u ∈ D e (E) for every k ≥ 0. If, in addition, m(E) < ∞ or E satisfies Poincaré type inequality then T k u ∈ D(E) for k ≥ 0 (see [11,Remark 3.4]).
In closing this section we recall yet another concept of solutions introduced in [11]. We say that u : E → R∪{−∞, +∞} is a solution of (1.1) in the sense of Stampacchia if for every v ∈ B(E) such that |µ|, R|v| < ∞ the integrals (u, v), f u · m, Rv) are finite and (u, v) = (f u , Rv) + µ, Rv .
By [11,Proposition 4.12], if µ ∈ M, then u is a solution of (1.1) in the sense of Stampacchia if and only if it is a solution of (1.1) in the sense of Definition 3.1.

Renormalized solutions
As in Section 3, in this section we assume that (E, D(E)) is transient and (ACR) is satisfied. As for the right-hand side of (1.1), we restrict our considerations to bounded measures.
The following definition extends [13, Definition 3.1] to possibly nonsmooth measures.
Definition 4.1. Let µ ∈ M b (E). We say that u : E → R∪{−∞, +∞} is a renormalized solution of (1.1) if (a) u is quasi-continuous, f u ∈ L 1 (E; m) and T k u ∈ D e (E) for every k ≥ 0, (b) there exists a sequence {ν k } ⊂ M 0,b (E) such that Rν k → Rµ c q.e. as k → ∞, and for every k ∈ N and every bounded v ∈ D e (E), In case µ c = 0, Definition 4.1 reduces to [13,Definition 3.1] with the exception that in [13] in condition (b) it is required that ν k T V → 0. Note that in the case where µ c = 0 the condition Rν k → Rµ c q.e. cannot be replaced by the condition ν k − µ c T V → 0 because the limit, in the total variation norm, of diffuse measures is diffuse. Also, if µ c = 0, then ν k T V 0, because by [16,Lemma 2.5], if ν k T V → 0, then there is a subsequence {ν k ′ } such that Rν k ′ → 0 q.e. We see that the difference between the case µ c = 0 and µ c = 0 is quite similar to that for parabolic equations considered in [19,20] (cf. [19,Definition 4.1] and [20,Definition 3]).
Remark 4.2. (i) Let E ⊂ R d be a bounded domain, and let L be the Laplace operator ∆ on E with zero boundary conditions. By [11,Remark 4.15], if u is a renormalized solution of (1.1), then u is a weak solution in the sense of [4].
(ii) Let α ∈ (0, 2], E ⊂ R d be a bounded domain, and let L be the fractional Laplacian ∆ α/2 on E with zero boundary conditions. By [11,Remark 4.13], if u is a renormalized solution of (1.1), then u is a solution of (1.1) in the sense of [5, Definition 1.1].
The following lemma is a modification of [12,Lemma 5.4]. As compared with [12,Lemma 5.4], we do not assume that µ is smooth, but we additionally require that the form satisfies (ACT).
Likewise, ν, Rg n = g n , Rν . Since Rν ≤ Rµ m-a.e., it follows from the above that µ, Rg n ≥ ν, Rg n , n ≥ 1. Therefore which proves the lemma. (i) If u is a probabilistic solution of (1.1) and f u ∈ L 1 (E; m) then u is a renormalized solution of (1.1).
(ii) If u is a renormalized solution of (1.1) then u is a probabilistic solution of (1.1).
By Itô's formula for convex functions (see, e.g., [22,Theorem IV.66]), for some increasing processes A 1 , A 2 . By [11,Remark 3.10], there is a reducing sequence {τ k } for u. Since M is a local martingale under P x for q.e. x ∈ E, for q.e. x ∈ E there exists a sequence of stopping times {σ n } (possibly depending on x) such that E x t∧σn 0 1 {Y s− ≤0} dM s = 0, t ≥ 0, n ≥ 1. Therefore, by (4.2) and (4.3), for all k, n ≥ 1. Letting n → ∞ we get Similarly, by (4.2) and (4.4), Letting k → ∞ in the above two equalities and using (3.4) shows that for q.e. x ∈ E, By this and Proposition 2.1, E x (A 1 ζ + A 2 ζ ) < +∞ for q.e. x ∈ E. Therefore by [9,Theorem A.3.16] there exists positive AFs of B 1 , B 2 of M such that B i , i = 1, 2, is a compensator of A i under P x for q.e. x ∈ E. The processes B 1 , B 2 are increasing, because A 1 and A 2 are increasing. Since by [9,Theorem A.3.2] the process X has no predictable jumps, it follows from [9, Theorem A.3.5] that B 1 , B 2 are continuous. Thus B 1 , B 2 are increasing continuous AFs of M such that A i − B i , i = 1, 2, is a martingale under P x for q.e. x ∈ E. Let b i ∈ S, i = 1, 2, denote the measure corresponding to B i in the Revuz sense. Then, by (2.1), for q.e. x ∈ E. From this and Lemma 4.3 it follows that b 1 , b 2 ∈ M 0,b . By Itô's formula, for k > 0 we have for some increasing processes A 1,k , A 2,k . By (4.3) and (4.5), whereas by (4.4) and (4.6), By the above two inequalities, Hence E x (A 1,k ζ + A 2,k ζ ) < +∞ for q.e. x ∈ E. Let B 1,k , B 2,k be positive AFs of M such that B i,k , i = 1, 2, is a compensator of A i,k under P x for q.e. x ∈ E. As in case of B 1 , B 2 , we show that B 1,k , B 2,k increasing continuous AFs of M such that A i,k − B i,k , i = 1, 2, is a martingale under P x for q.e. x ∈ E. Let b i,k ∈ S, i = 1, 2, denote the measure corresponding to B i,k in the Revuz sense. Then R( x ∈ E, and hence, by Lemma 4. where Since M k is a martingale under P x for q.e. x ∈ E, from (4.7) it follows that for q.e.
x ∈ E, Since T k u(X t ) → 0 P x -a.s. as t → ∞, E x T k u(X t ) → 0 by the Lebesgue dominated convergence theorem. Therefore from the above equality it follows that Then ν k ∈ M 0,b and for q.e. x ∈ E, On the other hand, by Proposition 3.4, u(x) = R(f u · m + µ d )(x) + Rµ c (x) for q.e.
(ii) Assume that u is a renormalized solution of (1.1). Then T k u is a solution in the sense of duality of the linear equation and hence T k u is a probabilistic solution of the above equation (see the arguments in [13, p. 1924]). Hence for q.e. x ∈ E. Since Rν k → Rµ c q.e., letting k → ∞ in the above equation we see that (1.4) is satisfied for q.e. x ∈ E, i.e. u is a solution of (1.1) in the sense of Definition 3.1. By this and Proposition 3.4, u is a probabilistic solution of (1.1).
Note that by Proposition 3.4, in the formulation of Theorem 4.4 we may replace "probabilistic solution" by "solution in the sense of Definition 3.1", while by [11, Proposition 4.12] we may replace "probabilistic solution" by "solutions in the sense of Stampacchia".
By Theorem 4.4, a probabilistic solution u is a renormalized solution once we know that f u ∈ L 1 (E; m). We close this section with describing some interesting situations in which this condition holds true. Proposition 4.5. Let µ ∈ M b and let f : E × R → R be a measurable function such that f (·, 0) ∈ L 1 (E; m) and for every x ∈ E the mapping R ∋ y → f (x, y) is continuous and nonincreasing. If u is a probabilistic solution of (1.1) then f u ∈ L 1 (E; m).
Following [4,11] we call µ ∈ M a good measure (relative to L and f ) if there exists a probabilistic solution of (1.1).
Proposition 4.6. Assume that f satisfies the assumptions of Proposition 4.5 and µ ∈ M is good relative to L and f . Then there exists a unique renormalized solution of (1.1). Moreover, for every k ≥ 0, Proof. The existence of a solution follows immediately from Theorem 4.4(i) and Proposition 4.5. Uniqueness follows from Theorem 4.4(ii) and [11,Corollary 4.3]. Estimate (4.9) follows from [11,Theorem 3.3], whereas (4.10) from [11,Proposition 4.8].
The following remark shows that the monotonicity assumption imposed on f in Propositions 4.5 and 4.6 can be relaxed in case µ is nonnegative.
The problem of existence of solutions of (1.1) for f satisfying the assumptions of Proposition 4.5 (or more general "sign condition" (4.11)) and the related problem of characterizing the set of good measures are very subtle, and are beyond the scope of the present paper. For many positive results in this direction in the case where A is the Laplace operator we defer the reader to [4,21]. Interesting existence and uniqueness results for equations involving the fractional Laplace operator are to be found in [11,5].