Renormalized solutions of semilinear elliptic equations with general measure data
aa r X i v : . [ m a t h . A P ] F e b Renormalized solutions of semilinear elliptic equations withgeneral measure data
Tomasz Klimsiak and Andrzej Rozkosz
Abstract
In the paper, we first propose a definition of renormalized solution of semilin-ear elliptic equation involving operator corresponding to a general (possibly non-local) symmetric regular Dirichlet form satisfying the so-called absolute continuitycondition and general (possibly nonsmooth) measure data. Then we analyze therelationship between our definition and other concepts of solutions considered inthe literature (probabilistic solutions, solution defined via the resolvent kernel ofthe underlying Dirichlet form, Stampacchia’s definition by duality). We show thatunder mild integrability assumption on the data all these concepts coincide.
Keywords:
Semilinear elliptic equation, Dirichlet form and operator, measure data, renormal-ized solution.
Mathematics Subject Classification (2010) . Primary: 35D99. Secondary: 35J61, 60H30.
Let L be the operator associated with a symmetric regular Dirichlet form ( E , D ( E )) on L ( E ; m ), f : E × R → R be a measurable function and µ be a bounded signed Borelmeasure 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 problemof 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). Itis 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 andother concepts of solutions known in the literature.In case L is a uniformly elliptic divergence form operator and f does not dependon u , some definition, now called Stampacchia’s definition by duality, was proposedby Stampacchia [24] in 1965. Later on, to deal with equations with more generallocal operator L , the definitions of entropy solution and renormalized solution wereintroduced. For a comparison of different forms of these definitions and remarks onother concepts of solutions of equations of the form (1.1) with local operator L and T. Klimsiak: Institute of Mathematics, Polish Academy of Sciences, ´Sniadeckich 8, 00-956Warszawa, Poland, and Faculty of Mathematics and Computer Science, Nicolaus Copernicus Uni-versity, Chopina 12/18, 87-100 Toru´n, Poland. E-mail: [email protected]. Rozkosz: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University,Chopina 12/18, 87-100 Toru´n, Poland. E-mail: [email protected]. not depending on u see [6]. Elliptic equations with local operators and nonlineardependence 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 issmooth. Recall (see [10]) that µ admits a unique decomposition µ = µ d + µ c (1.2)into the smooth (diffuse) part µ d and the concentrated part µ c , i.e. µ d is a boundedBorel measure, which is “absolutely continuous” with respect to the capacity Capdetermined by ( E , D ( E )), and µ c is a bounded Borel measure which is “singular” withrespect to Cap. In case L is local and µ is smooth entropy and renormalized solutionsof (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 witha general transient (possibly non-symmetric) Dirichlet form was recently given in [13].If ( E , D ( E )) is symmetric and f ( · , u ) ∈ L ( E ; m ), renormalized solutions in the sense of[13] coincide with probabilistic solutions of (1.1) defined earlier in [12] (see also [14] forequations 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 followingnonlinear Feynman-Kac formula u ( x ) = E x (cid:16) Z ζ f ( X t , u ( X t )) dt + Z ζ dA µt (cid:17) (1.3)is satisfied for quasi-every x ∈ E . In (1.3), M = ( X, P x ) is a Markov process withlife 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 (seeSection 2). The equivalence between renormalized and probabilistic solutions allowsone to use effectively probabilistic methods in the study of renormalized solutions of(1.1). Also note that if f ∈ L ( E ; m ) then renormalized solutions of (1.1) coincide withStampacchia’s solutions by duality defined in [12, 14].The semilinear case with general, possibly nonsmooth bounded measure µ is muchmore involved. The study of (1.1) with nonsmooth measure was initiated in 1975 byBrezis and B´enilan in case L is the Laplace operator ∆ (see [2, 4] and the referencesgiven there for results and historical comments). For some existence and uniquenessresults in case L is the fractional Laplacian ∆ α/ with α ∈ (0 ,
2) see Chen and V´eron[5]. Very recently, Klimsiak [11] started the study of (1.1) in case L corresponds to atransient symmetric regular Dirichlet form satisfying the following absolute continuitycondition:(ACR) R α ( x, · ) is absolutely continuous with respect to m for each α > x ∈ E ,where R α ( x, dy ) denotes the resolvent kernel associated with ( E , D ( E )) (see Section2.2). Equivalently,(ACT) p t ( x, · ) is absolutely continuous with respect to m for each t > x ∈ E ,where p t ( x, dy ) is the transition function associated with ( E , D ( E )). The above condi-tions are satisfied for instance if L is a uniformly divergence form operator or L = ∆ α/ α ∈ (0 , R ( x, dy )has a density r . In [11] a measurable function u on E is called a solution of (1.1) if u ( x ) = Z E r ( x, y ) f ( y, u ( y )) dy + Z E r ( x, y ) µ ( dy ) (1.4)for quasi every x ∈ E . In case µ c = 0, the above equation reduces to (1.3), so thedefinition 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 thatsolutions defined via the resolvent density, i.e. by (1.4), may be equivalently defined asrenormalized solutions in the same manner as in [13]. In the present paper we show thatthis is indeed possible. The definition of a renormalized solution adopted in the presentpaper 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 sofar in the literature ( µ is smooth or µ is nonsmooth and L = ∆ or L = ∆ α/ , likein [4, 5]) the solutions considered there coincide with the renormalized defined in thepresent paper.The main result of the paper says that if the form is transient and (ACR) is satisfiedthen 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 ( E ; m ) then u is a renormalized solution. We findimportant that, as in the case of smooth measures, this correspondence when combinedwith probabilistic interpretation of (1.4) given in [11] enables one to study renormalizedsolutions of (1.1) with the help of probabilistic methods. For results on (1.1) obtainedin this way we defer the reader to [11]). Finally, note that at the end of the paperwe describe some interesting situations in which solutions of (1.1) in the sense of (1.4)automatically have the property that f ( · , u ) ∈ L ( E ; m ). In the paper E is a separable locally compact metric space and m is a Radon measureon 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.For u : E → R we set u + ( x ) = max { u ( x ) , } , u − ( x ) = max {− u ( x ) , } . By ( E , D ( E )) we denote a symmetric regular Dirichlet form on H = L ( E ; m ) (see [9,Section 1.1] for the definition). In case ( E , D ( E )) is transient, by ( D e ( E ) , E ) we denotethe extended Dirichlet space of ( E , D ( E )) (see [9, Section 1.5]).In the paper, we define capacity Cap as in [9, Section 2.1]. Recall that an increasingsequence { F n } of closed subsets of E is called nest if Cap( E \ F n ) → n → ∞ . Asubset N ⊂ E is called exceptional if Cap( N ) = 0. We will say that some property ofpoints in E holds quasi everywhere (q.e. for short) if the set for which it does not holdis exceptional.We say that a function u on E is quasi-continuous if there exists a nest { F n } suchthat u | F n is continuous for every n ≥
1. By [9, Theorem 2.1.7], each function u ∈ D e ( E )has a quasi-continuous m -version. 3et 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 denotethe set of all signed Borel measures on E such that k µ k T V := | µ | ( E ) < ∞ , and by M ,b the subset of M b consisting of all smooth measures. S + is the subset of S consisting ofnonnegative measures. Similarly we define M + b , M +0 ,b . By [10, Lemma 2.1], for every µ ∈ M b there exists a unique pair ( µ d , µ c ) ∈ M b × M b such that µ d ∈ M ,b , µ c is concentrated on some exceptional Borel subset of E and (1.2) is satisfied. If µ isnonnegative, so are µ d , µ c . For a complete description of the structure of µ c see [15]. Let E ∪ ∆ be the one-point compactification of E . When E is already compact, weadjoin ∆ 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 [9, Theorems 4.2.8, 7.2.3] there exists a unique (up to equivalence) m -symmetricHunt process M = (Ω , F , ( F t ) t ≥ , ( X t ) t ≥ , ζ, ( P x ) x ∈ E ∪ ∆ ) with state space E , life time ζ and cemetery state ∆ whose Dirichlet space is ( E , D ( E )). This means in particularthat for every α > f ∈ B b ( E ) ∩ H the resolvent of M , that is the function R α f ( x ) = E x Z ∞ e − αt f ( X t ) dt, x ∈ E is a quasi-continuous m -version of G α f .Let R α ( x, dy ) denote the kernel on ( E, B ( E )) defined as R α ( x, B ) = R α B ( x ). Inthe 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) isequivalent 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 everyBorel set B ⊂ E , R ( x, B ) = Z B r ( x, y ) m ( dy ) , x ∈ E. In fact, r ( x, y ) = lim α ↓ r α ( x, y ), where r α ( x, y ) is the density of R α ( x, dy ) constructedin [9, Lemma 4.2.4] (see remarks in [3, p. 256]). We call r the resolvent density.In what follows given a positive Borel measure on E , we write R α µ ( x ) = Z E r α ( x, y ) µ ( dy ) , Rµ ( x ) = Z E r ( x, y ) µ ( dy ) , x ∈ E, α > . 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 ) = + ∞ . Proposition 2.1.
Assume that ( E , D ( E )) is transient and (ACR) is satisfied. If µ ∈M b then R | µ | ( x ) < + ∞ for q.e. x ∈ E .Proof. See [11, Proposition 3.2]. 4enote 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 (seethe 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 (calledRevuz correspondence) between the set of smooth measures µ on E and the set ofpositive continuous AFs A of M . It is given by the relationlim t → + t E m Z t f ( X s ) dA s = Z E f ( x ) µ ( dx ) , f ∈ B + ( E ) , where E m denotes the expectation with respect to the measure P m ( · ) = R E P x ( · ) m ( dx ).In what follows the positive continuous AF of M corresponding to a positive µ ∈ S willbe denoted by A µ . If µ in S , then µ + , µ − ∈ S , and we set A µ = A µ + − A µ − . Note thatif µ ∈ S + then for every α ≥ R α µ ( x ) = E x Z ζ e − αt dA µt = E x Z ∞ e − αt dA µt (2.1)for q.e. x ∈ E . Indeed, if α > 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 fromExercise 4.2.2 and Lemma 5.1.3 in [9]. The general case follows by approximation. Wefirst let α ↓ α ≥ µ ∈ S (0)0 , and then we use the 0-order version of[9, Theorem 2.2.4] (see remark following [9, Corollary 2.2.2]) to get (2.1) for any α ≥ µ ∈ S + . We assume that ( E , D ( E )) is transient and (ACR) is satisfied. Consider the problem − Lu = f u + µ, (3.1)where f : E × R → R is a measurable function, f u = f ( · , u ), µ ∈ M and L is theoperator associated with ( E , D ( E )), i.e. the nonpositive definite self-adjoint operatoron H such that D ( L ) ⊂ D ( E ) , E ( u, v ) = ( − Lu, v ) , u ∈ D ( L ) , v ∈ D ( E ) , 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]. Definition 3.1.
We say that a measurable function u : E → R ∪ {−∞ , + ∞} is asolution of (1.1) if f u · m ∈ M and (1.4) is satisfied for q.e. x ∈ E . Definition 3.2.
We say that a measurable u : E → R ∪ {−∞ , + ∞} is a probabilisticsolution of (1.1) if(a) f u · m ∈ M and there exists an AF M of M such that such that for q.e. x ∈ E the process M is an ( F ) t ≥ -local martingale under P x and u ( X t ) = u ( X ) − Z t f u ( X s ) ds − Z t dA µ d s + Z t dM s , t ≥ , P x -a.s. (3.2)5b) for every exceptional set N ⊂ E , every stopping time T such that T ≥ ζ andevery 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 E x u ( X τ k ) → Rµ c ( x ) , x ∈ E \ N. (3.3)Any sequence { τ k } with the properties listed in condition (b) will be called thereducing sequence for u , and we will say that { τ k } reduces u . Remark 3.3. (i) By [11, Renark 3.10], if µ c = 0, then the above definition reduces tothe definition introduced in [12].(ii) Assume that u is a probabilistic solution of (1.1). Then for q.e. x ∈ E we have E x u + ( X τ k ) → Rµ + c ( x ) , E x u − ( X τ k ) → Rµ − c ( x ) . (3.4)Indeed, if u is a solution of (1.1) then by [11, Theorem 6.3], Lu + ∈ M . In differentwords, u + is a solution of the equation Lu + = ν with some ν ∈ M . Hence, by condition(b) of Definition 3.2, E x u + ( X τ k ) → Rν c ( x ) for q.e. x ∈ E . But by [11, Theorem 6.3],( Lu + ) c = ( Lu ) + c . Hence ν c = ( f u · m + µ ) + c = µ + c , which proves the first convergencein (3.4). The second convergence follows from the first one and (3.3). Proposition 3.4.
Let µ ∈ M . A measurable u : E → R ∪ {−∞ , + ∞} is a solution of (1.1) in the sense of Definition if and only if it is a solution of (1.1) in the senseof Definition .Proof. See [11, Proposition 3.12].In what follows for a function u on E and a measure µ on E , we set h µ, u i = Z E u ( x ) µ ( dx )whenever the integral is well defined, and for k ≥
0, we write T k u ( x ) = max { min { u ( x ) , k } , − k } , x ∈ E. Remark 3.5. (i) By [11, Theorem 3.7], if u is a solution of (1.1) then u is quasi-continuous.(ii) Let u be a solution of (1.1) with µ ∈ M b . If f u ∈ L ( E ; m ) then by [11, Theorem3.3], T k u ∈ D e ( E ) for every k ≥
0. If, in addition, m ( E ) < ∞ or E satisfies Poincar´etype inequality then T k u ∈ D ( E ) for k ≥ u : E → R ∪{−∞ , + ∞} is a solution of (1.1) in the sense of Stampacchiaif for every v ∈ B ( E ) such that h| µ | , R | v |i < ∞ the integrals ( u, v ), f u · m, Rv ) are finiteand ( u, v ) = ( f u , Rv ) + h µ, Rv i . By [11, Proposition 4.12], if µ ∈ M , then u is a solution of (1.1) in the sense ofStampacchia if and only if it is a solution of (1.1) in the sense of Definition 3.1.6 Renormalized solutions
As in Section 3, in this section we assume that ( E , D ( E )) is transient and (ACR) issatisfied. As for the right-hand side of (1.1), we restrict our considerations to boundedmeasures.The following definition extends [13, Definition 3.1] to possibly nonsmooth mea-sures. Definition 4.1.
Let µ ∈ M b ( E ). We say that u : E → R ∪{−∞ , + ∞} is a renormalizedsolution of (1.1) if(a) u is quasi-continuous, f u ∈ L ( E ; m ) and T k u ∈ D e ( E ) for every k ≥ { ν k } ⊂ M ,b ( E ) such that Rν k → Rµ c q.e. as k → ∞ ,and for every k ∈ N and every bounded v ∈ D e ( E ), E ( T k u, v ) = h f u · m + µ d , ˜ v i + h ν k , ˜ v i . (4.1)Note that in the case of local operators, the above definition is essentially [6, Defi-nition 2.29]. A similar in spirit definition of renormalized solutions of parabolic equa-tions with local Leray-Lions type operators is considered in [19, Definition 4.1] (in case µ c = 0) and [20, Definition 3] (in the case of general bounded measures).In case µ c = 0, Definition 4.1 reduces to [13, Definition 3.1] with the exceptionthat in [13] in condition (b) it is required that k ν k k T V →
0. Note that in the casewhere µ c = 0 the condition Rν k → Rµ c q.e. cannot be replaced by the condition k ν k − µ c k T V → µ c = 0, then k ν k k T V
0, because by [16, Lemma 2.5], if k ν k k T V → { ν k ′ } such that Rν k ′ → µ c = 0 and µ c = 0 is quite similar to that for parabolic equationsconsidered 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 renormalizedsolution of (1.1), then u is a weak solution in the sense of [4].(ii) Let α ∈ (0 , E ⊂ R d be a bounded domain, and let L be the fractional Laplacian∆ α/ on E with zero boundary conditions. By [11, Remark 4.13], if u is a renormalizedsolution 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 theform satisfies (ACT). Lemma 4.3.
Assume that ν ∈ M ∩ S + , µ ∈ M + b . If Rν ≤ Rµ m -a.e. then ν ∈ M +0 ,b .In fact, k ν k T V ≤ k µ k T V .Proof.
Set g n = n (1 − nR n Rg n = nR n ≤ , n ≥ . Since by [3, Chapter II, Proposition (2.2)] the constant function 1 is excessive relativeto M , g n ≥ Rg n ր
1. Since the resolvent7ensity r is symmetric, applying Fubini’s theorem we get h µ, Rg n i = Z E (cid:16) Z E r ( x, y ) g n ( y ) dy (cid:17) µ ( dx )= Z E (cid:16) Z E r ( y, x ) µ ( dx ) (cid:17) g n ( y ) dy = h g n , Rµ i . Likewise, h ν, Rg n i = h g n , Rν i . Since Rν ≤ Rµ m -a.e., it follows from the above that h µ, Rg n i ≥ h ν, Rg n i , n ≥ . Therefore k ν k T V = lim n →∞ h Rg n , ν i ≤ lim n →∞ h Rg n , µ i = k µ k T V , which proves the lemma. Theorem 4.4.
Let µ ∈ M b . (i) If u is a probabilistic solution of (1.1) and f u ∈ L ( E ; m ) then u is a renormalizedsolution of (1.1) . (ii) If u is a renormalized solution of (1.1) then u is a probabilistic solution of (1.1) .Proof. (i) Let Y t = u ( X t ), t ≥
0. By (3.2), for q.e. x ∈ E , Y t = Y − Z t f u ( X s ) ds − Z t dA µ d s + Z t dM s , t ≥ , P x -a.s. (4.2)By Itˆo’s formula for convex functions (see, e.g., [22, Theorem IV.66]), u + ( X t ) − u + ( X ) = Z t { Y s − > } dY s + A t , t ≥ , (4.3) u − ( X t ) − u − ( X ) = − Z t { Y s − ≤ } dY s + A t , t ≥ A , A . 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 R t ∧ σ n { Y s − ≤ } dM s = 0, t ≥ n ≥
1. Therefore, by (4.2) and (4.3), E x A τ k ∧ σ n = E x u + ( X τ k ∧ σ n ) − u + ( x ) + E x Z τ k ∧ σ n { Y s − > } ( f u ( X s ) ds + dA µ d s )for all k, n ≥
1. Letting n → ∞ we get E x A τ k = E x u + ( X τ k ) − u + ( x ) + E x Z τ k { Y s − > } ( f u ( X s ) ds + dA µ d s ) . Similarly, by (4.2) and (4.4), E x A τ k = E x u − ( X τ k ) − u − ( x ) − E x Z τ k { Y s − ≤ } ( f u ( X s ) ds + dA µ d s ) . k → ∞ in the above two equalities and using (3.4) shows that for q.e. x ∈ E , E x A ζ ≤ Rµ + c ( x ) + E x Z ζ ( | f u ( X t ) | ds + dA | µ d | t ) = Rµ + c ( x ) + R ( | f u | · m + | µ d | )( x ) ,E x A ζ ≤ Rµ − c ( x ) + E x Z ζ ( | f u ( X t ) | ds + dA | µ d | t ) = Rµ − c ( x ) + R ( | f u | · m + | µ d | )( x ) . By this and Proposition 2.1, E x ( A ζ + A ζ ) < + ∞ for q.e. x ∈ E . Therefore by [9,Theorem A.3.16] there exists positive AFs of B , B of M such that B i , i = 1 ,
2, isa compensator of A i under P x for q.e. x ∈ E . The processes B , B are increasing,because A and A are increasing. Since by [9, Theorem A.3.2] the process X has nopredictable jumps, it follows from [9, Theorem A.3.5] that B , B are continuous. Thus B , B are increasing continuous AFs of M such that A i − B i , i = 1 ,
2, is a martingaleunder 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), Rb i ( x ) = E x B iζ = E x A iζ < + ∞ , i = 1 , , for q.e. x ∈ E . From this and Lemma 4.3 it follows that b , b ∈ M ,b . By Itˆo’sformula, for k > u + ∧ k )( X t ) − ( u + ∧ k )( X ) = Z t { u + ( X s − ) ≤ k } du + ( X s ) − A ,kt , t ≥ , (4.5)( u − ∧ k )( X t ) − ( u − ∧ k )( X ) = Z t { u − ( X s − ) ≤ k } du − ( X s ) − A ,kt , t ≥ , (4.6)for some increasing processes A ,k , A ,k . By (4.3) and (4.5), E x A ,kt ≤ u + ( x ) ∧ k + E x Z t { u + ( X s − ) ≤ k } { Y s − > } dY s + E x Z t { u + ( X s − ) ≤ k } dA s whereas by (4.4) and (4.6), E x A ,kt ≤ u − ( x ) ∧ k − E x Z t { u − ( X s − ) ≤ k } { Y s − ≤ } dY s + E x Z t { u − ( X s − ) ≤ k } dA s . By the above two inequalities, E x ( A ,kζ + A ,kζ ) ≤ u + ( x ) ∧ k + u − ( x ) ∧ k + R ( | f u | · m + | µ d | )( x ) + R ( b + b )( x ) . Hence E x ( A ,kζ + A ,kζ ) < + ∞ for q.e. x ∈ E . Let B ,k , B ,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 incase of B , B , we show that B ,k , B ,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 , B i,k in the Revuz sense. Then R ( b ,k + b ,k )( x ) = E x ( A ,kζ + A ,kζ ) < + ∞ for q.e. x ∈ E , and hence, by Lemma 4.3, that b ,k , b ,k ∈ M ,b .Let Y kt = T k u ( X t ). Since T k u = ( u + ∧ k ) − ( u − ∧ k ), from (4.2)–(4.6) we get Y kt − Y k = − Z t {− k ≤ Y s − ≤ k } ( f u ( X s ) ds + dA µ d s ) − B ,kt + Z t { u + ( X s ) ≤ k } dB s + B ,kt − Z t { u − ( X s ) ≤ k } dB s + M kt , (4.7)9here M kt = Z t {− k ≤ Y s − ≤ k } dM s − ( A ,kt − B ,kt ) + ( A ,kt − B ,kt )+ Z t { u + ( X s − ) ≤ k } d ( A s − B s ) − Z t { u − ( X s − ) ≤ k } d ( A s − B s ) . 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 , T k u ( x ) = E x T k ( X t ) + E x Z t {− k ≤ Y s − ≤ k } ( f u ( X s ) ds + dA µ d s )+ E x B ,kt − E x Z t { u + ( X s ) ≤ k } dB s − E x B ,kt + E x Z t { u − ( X s ) ≤ k } dB s . Since T k u ( X t ) → P x -a.s. as t → ∞ , E x T k u ( X t ) → T k u ( x ) = R ( {− k ≤ u ≤ k } ( f u · m + µ d )) + R ( b ,k − { u + ≤ k } b ) − R ( b ,k − { u − ≤ k } b ) . Set ν k = { u/ ∈ [ − k,k ] } ( f u · m + µ d ) + b ,k − { u + ≤ k } b − b ,k + { u − ≤ k } b . Then ν k ∈ M ,b and for q.e. x ∈ E , T k u ( x ) = R ( f u · m + µ d )( x ) + Rν k ( x ) . (4.8)On the other hand, by Proposition 3.4, u ( x ) = R ( f u · m + µ d )( x ) + Rµ c ( x ) for q.e. x ∈ E . Hence Rν k ( x ) → Rµ c ( x ) for q.e. x ∈ E . By Remark 3.5(ii), T k u ∈ D e ( E ).Finally, since T k u = Rλ k with λ k = f u · m + µ d + ν k ∈ M ,b , repeating step by stepthe reasoning following [13, (3.14)] shows that T k u satisfies (4.1), which completes theproof of (i).(ii) Assume that u is a renormalized solution of (1.1). Then T k u is a solution in thesense of duality of the linear equation − L ( T k u ) = f u + µ d + ν k , and hence T k u is a probabilistic solution of the above equation (see the arguments in[13, p. 1924]). Hence T k u ( x ) = E x (cid:16) Z ζ ( f u ( X t ) dt + dA µ d t ) + Z ζ dA ν k t (cid:17) = R ( f u · m + µ d )( x ) + Rν k ( x )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 Definition3.1. 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, Propo-sition 4.12] we may replace “probabilistic solution” by “solutions in the sense of Stam-pacchia”.By Theorem 4.4, a probabilistic solution u is a renormalized solution once we knowthat f u ∈ L ( E ; m ). We close this section with describing some interesting situationsin which this condition holds true. 10 roposition 4.5. Let µ ∈ M b and let f : E × R → R be a measurable function suchthat f ( · , ∈ L ( E ; m ) and for every x ∈ E the mapping R ∋ y f ( x, y ) is continuousand nonincreasing. If u is a probabilistic solution of (1.1) then f u ∈ L ( E ; m ) .Proof. See [11, Proposition 4.8].Following [4, 11] we call µ ∈ M a good measure (relative to L and f ) if there existsa 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 ≥ , E ( T k u, T k u ) ≤ k ( k µ k T V + k f u k L ( E ; m ) ) , (4.9) k f u k L ( E ; m ) ≤ k f ( · , k L ( E ; m ) + k µ k T V . (4.10) Proof.
The existence of a solution follows immediately from Theorem 4.4(i) and Propo-sition 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 inPropositions 4.5 and 4.6 can be relaxed in case µ is nonnegative. Remark 4.7. (i) Assume that µ ∈ M is nonnegative and f satisfies the following “signcondition”: for every x ∈ E , yf ( x, y ) ≤ , y ∈ R . (4.11)Then if u is a probabilistic solution of (1.1), then u ≥ { τ k } for u . Then by (4.2), (4.4) and Itˆo’s formula for convexfunctions (see [22, Theorem IV.66]), for q.e. x ∈ E we have u − ( x ) = E x u − ( X τ k ) − Z τ k { Y s − ≤ } f ( X s , Y s ) ds − Z τ k { Y s − ≤ } dA µ d s − E x A τ k . Since µ ≥ µ d ≥ µ c ≥
0. In particular, A µ d is increasing. Since A is alsoincreasing and f satisfies (4.11), it follows that u − ( x ) ≤ E x u − ( X τ k ). By this and (3.4), u ( x ) ≤ lim sup k →∞ E x u − ( X τ k ) = Rµ − c ( x ) = 0 for q.e. x ∈ E .(ii) Obviously (4.11) is satisfied if f ( x,
0) = 0 and f is nonincreasing. Therefore if µ in Proposition 4.5 is nonnegative, then without loss of generality we may assume that f ( · , y ) = 0 for y ≤
0, i.e. f satisfies the condition imposed on f in [4] (see [4, Remark1]) and in [11, Section 5].(iii) If f satisfies (4.11) and µ ∈ M + b is good (relative to L and f ), then f u ∈ L ( E ; m ),and hence there exists a renormalized solution of (1.1). Indeed, if µ ≥ u ≥ Rf u + Rµ ≥ f u ≤
0. Hence 0 ≤ R ( − f u ) = − Rf u ≤ Rµ q.e. By this and Lemma 4.3, − f u · m ∈ M + b , so f u ∈ L ( E ; m ).The problem of existence of solutions of (1.1) for f satisfying the assumptions ofProposition 4.5 (or more general “sign condition” (4.11)) and the related problem ofcharacterizing the set of good measures are very subtle, and are beyond the scope ofthe present paper. For many positive results in this direction in the case where A is the11aplace operator we defer the reader to [4, 21]. Interesting existence and uniquenessresults for equations involving the fractional Laplace operator are to be found in [11, 5]. Acknowledgements
This work was supported by the Polish National Science Centre under Grant2012/07/B/ST1/03508.
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