aa r X i v : . [ m a t h . F A ] J un LOCALITY AND DOMINATION OF SEMIGROUPS
KHALID AKHLIL
Dedicated to O. El-Mennaoui and W. Arendt
Abstract.
We characterize all semigroups ( T ( t )) t ≥ on L (Ω) sandwichedbetween Dirichlet and Neumann ones, i.e.: e t ∆ D ≤ T ( t ) ≤ e t ∆ N , for all t ≥ T ( t )) t ≥ . Introduction
Let Ω be an open set of R d . We want to characterize all semigroups ( T ( t )) t ≥ on L (Ω) sandwiched between Dirichlet and Neumann ones, i.e.: e t ∆ D ≤ T ( t ) ≤ e t ∆ N , for all t ≥ a, D ( a )) associated with such semigroups is local then there exists a uniquepositive measure on the boundary Γ := ∂ Ω which charges no set of zero relativecapacity such that the form is given explicitly by a ( u, v ) = Z Ω ∇ u ∇ vdx + Z Γ e u e vdµ, D ( a µ ) = f H (Ω) ∩ L (Γ , dµ ) (1.2)The regularity of a is needed to ensure the uniqueness of the measure µ , otherwisethe form will be regular just on a subset X = Ω \ Γ ∞ , where Γ ∞ is the part ofΓ where µ is locally infinite everywhere, and then µ can not be unique becauseobtained by extension to the whole boundary.In this paper we focus on the hypothesis of locality and to simplify we consider a as regular Dirichlet form (if not we do the same as in [5] or [19]). The motivation ofour main results was an intuitive remark that one don’t need to suppose the localityof a , and that the locality is contained in the fact that a is sandwiched between twolocal forms. In fact and as mentioned in [10, Remark 6.12] one can remark thatthe semigroup considered in the Example 4.5 in [5] can not be positive(actuallyit is eventually positive). It follows that one can not make use of Ouahabaz’sCharacterization theorem of semigroup’s domination.This work is organized as follow: In section 2 we recall the framework and theresults of [5]. In section 3 we discuss the example 4.5 in [5] and we show why it is Date : June 5, 2020.2000
Mathematics Subject Classification.
Key words and phrases.
Robin boundary conditions; locality; Domination of Semigroups.The author was supported, for this work, by “Deutscher Akademischer Austausch Di-enst“(German Academic Exchange Service) Grant Number: A/11/97482. not positive. In section 4 we prove our main result using beurling-Deny and Lejan(BDL) formula. 2. Preliminaries
First of all, we recall the situation treated in [5]. Let Ω be an open set andΓ := ∂ Ω. Let µ : B → [0 , ∞ ] be a measure where B denotes the σ − algebra of Borelsets of Γ. We now consider the symmetric form a µ on L (Ω) given by a µ ( u, v ) = Z Ω ∇ u ∇ vdx + Z Γ u ( x ) v ( x ) dµ with domain D ( a µ ) = { u ∈ H (Ω) ∩ C c (Ω) : Z Γ | u | dµ < ∞} Let Γ µ = { z ∈ Γ : ∃ r > µ (Γ ∩ B ( z, r )) < ∞} be the part of Γ on which µ is locally finite.Assume now that Γ µ = ∅ , thus Γ µ is a locally compact space and µ is a regularBorel measure on Γ µ . We say that µ is admissible if for each Borel set A ⊂ Γ µ one has Cap Ω ( A ) = 0 ⇒ µ ( A ) = 0where Cap Ω ( A ) refer to the relative capacity.It is proved in [5] that the form a µ is closable if and only if µ is admissible, andthe closure of a µ is given by D ( a µ ) = { u ∈ e H (Ω) : ˜ u = 0 r.q.e. on Γ \ Γ µ , Z Γ µ | ˜ u | dµ < ∞} a µ ( u, v ) = Z Ω ∇ u ∇ vdx + Z Γ µ ˜ u ˜ vdµ Here ˜ u is the relatively quasi-continuous representative of u . Note that the form( a µ , D ( a µ ) is regular if and only if µ is a radon measure on Γ, which means thatΓ µ = Γ.Denote by ∆ µ the operator associated with a µ . Then it follows that ∆ µ is arealization of the Laplacian in L (Ω)( see [4] and [19] for more details and otherproperties).Next we give two trivial examples of measure µ : Example 2.1. (i) If µ = 0 then D ( a µ ) = e H (Ω) and a µ ( u, v ) = R Ω ∇ u ∇ vdx . Let∆ N be the operator associated with ( a , D ( a )). It is the Neumann Laplacian, andit coincides with the usual Neumann Laplacian when Ω is bounded with Lipschitzboundary.(ii) If Γ µ = ∅ , then D ( a µ ) = H (Ω) and a µ ( u, v ) = R Ω ∇ u ∇ vdx . Let ∆ D be theoperator associated with ( a ∞ , D ( a ∞ )). It is the Dirichlet Laplacian.We have the following domination result Theorem 2.2.
For each admissible measure µ , the semigroup ( e t ∆ µ ) t ≥ satisfies e t ∆ D ≤ e t ∆ µ ≤ e t ∆ N (2.1) for all t ≥ in the sense of positive operators. The proof is a simple application of the Ouhabaz criterion and is given in [4,Theorem 3.1.]. For a probabilistic proof see [1].In [4, Section 4], W. Arendt and M. Warma explored the converse of the aboveTheorem. More precisely, they answered the following question: Having a sand-wiched semigroup between Dirichlet and Neumann semigroups, can one write theassociated form with help of an admissible measure? The answer was affirmativeunder a central hypothesis of locality of the associated Dirichlet form.The first Theorem says ([4, Theorem 4.1])
Theorem 2.3.
Let Ω be an open subset of R d with boundary Γ . Let T be a sym-metric C − semigroup on L (Ω) associated with a positive closed form ( a, D ( a )) .Then the following assertions are equivalent(i) There exist an admissible measure µ such that a = a µ (ii) (a) One has e t ∆ D ≤ T ( t ) ≤ e t ∆ N , ( t ≥ .(b) a is local.(c) D ( a ) ∩ C c (Ω) is dense in ( D ( a ) , k . k a ) . In order to characterize those sandwiched semigroups which come from boundedmeasure we have the following Theorem ([4, Theorem 4.2.])
Theorem 2.4.
Let Ω be a bounded open set of R d . Let T be a symmetric C − semigroupon L (Ω) associated with a positive closed form ( a, D ( a )) . Then the following as-sertions are equivalent(i) There exist a bounded admissible measure µ on Γ such that a = a µ (ii) (a) One has e t ∆ D ≤ T ( t ) ≤ e t ∆ N , ( t ≥ .(b) a is local.(c) ∈ D ( a ) . To end this section we give the following fascinating Beurling-Deny and Lejanformula. The proof can be found for example in [11] or [7], another proof based oncontraction operators can be found in [2] or [3].
Theorem 2.5. (The Beurling-Deny and Lejan formula) Any regular Dirichlet form ( a, D ( a )) on L ( X ; m ) can be expressed for u, v ∈ D ( a ) ∩ C c ( X ) as follow: a ( u, v ) = a ( c ) ( u, v ) + Z X u ( x ) v ( x ) k ( dx )+ Z X × X \ d ( u ( x ) − u ( y ))( v ( x ) − v ( y )) J ( dx, dy ) where a ( c ) is a strongly local symmetric form with domain D ( a ( c ) ) = D ( a ) ∩ C c ( X ) , J is a symmetric positive Radon measure on X × X \ d ( d = diagonal), and k is a pos-itive Radon measure on X . In addition such a ( c ) , J and k are uniquely determinedby a . The Beurling-Deny and Lejan formula had catched a lot of attention last yearsand many generalization in other directions was explored. For example in the caseof semi-regular Dirichlet forms, or Regular but non-symmetric Dirichlet forms. Formore information we refer to [12], [13], [14] and [15] and references therein. Here,we will use the above “conventional” Beurling-Deny and Lejan formula that is theone for Regular Dirichlet forms, but there is no reason that the same arguments donot work also for the other cases.3.
About an example in [5]Let Ω ⊂ R d be a bounded open set with Lipschitz boundary Γ := ∂ Ω, σ is the( d − − dimensional Hausdorff measure on Γ and B a bounded operator on L (Γ).Define the bilinear form a B with domain H (Ω) on L (Ω) by a B ( u, v ) = Z Ω ∇ u ∇ vdx + Z Γ Bu | Γ v | Γ dσ The choice of the domain D ( a B ) = H (Ω) comes from the fact that B is boundedin L (Γ) and from the continuity of the embedding H (Ω) ֒ → L (Γ).It is clear that the billinear form ( a B , H (Ω)) is continuous and elliptic. Let − ∆ B be the associated operator of the form a B , and we note ( e t ∆ B ) t ≥ the associatedsemigroup. It is natural to expect that − ∆ B is a realization of the Laplacian in L (Ω), which means that if we let u, f ∈ L (Ω), then u ∈ D (∆ B ) and − ∆ B u = f if and only if u ∈ H (Ω), − ∆ u = f and ∂ ν u + Bu = 0 on Γ. The operator − ∆ B is then called the Laplacian with nonlocal Robin boundary conditions. Finally, by[16, C-II, Theorem 1.11. p:255] one can deduce easily that e t ∆ B ≥ t ≥ c such that B − cI ≤ L (Γ)[6]( I stands forthe identity operator on L (Γ)).Now, we consider the case where d = 1. Let Ω = (0 ,
1) then Γ = { , } and B = ( b ij ) i,j =1 , ∈ R × . In this simple situation, one can see easily when thesemigroup ( e t ∆ B ) t ≥ is positive or not. In fact, using the above characterization,one can conclude that e t ∆ B ≥ t ≥ b ij ≤ i = j , it sufficeto choose c = b ∨ b (see also [16, C-II, Example 1.13] for similar situations).The Example 4.5 of [5] corresponds to the choice B = (cid:18) (cid:19) and gives, a ( u, v ) = Z u ′ v ′ dx + u (0) v (0) + u (1) v (0) + u (0) v (1) + u (1) v (1)fot all u, v ∈ H (0 , e t ∆ B is sandwichedbetween Dirichlet semigroup and Neumann semigroup, but e t ∆ B is obviously notpositive, the situation where Ouhabaz’s domination criterion can not be used. Infact one can prove that any positive semigroup dominated by Neumann semigroupis automatically local. Consider then the situation of Theorem 4.1 in [4]. Let T be a symmetric C − semigroup on L (Ω) associated with a positive closed form ( a, D ( a )) such that0 ≤ T ( t ) ≤ e t ∆ N , ( t ≥ u ∈ D ( a ) that a ( u + , u − ) ≤ a ( u + , u − ) ≥ R Ω ∇ u + ∇ u − dx = 0, which means that a ( u + , u − ) = 0 for all u ∈ D ( a ) and thenthat a is local[6]. One can take also, instead of Neumann semigroup, any symmetric C − semigroup associated with a positive closed local form ( b, D ( b ). Remark . We established above that the semigroup in [4, Example 4.5] cannot be positive. This fact suggested to explore wether the semigroup is or is noteventual positive. The answer is affirmative, see [10, Example 6.12].4.
The sandwiched property using bdl formula
In this section we will prove that one can drop the assumption of locality inresults about sandwiched property. This can be made, for example, by using theBeurling-Deny and Lejan decomposition formula of regular Dirichlet forms. Onecan then obtain the following result:
Theorem 4.1.
Let Ω be a bounded open set of R d and ( T ( t )) t ≥ be a C − semigroupon L (Ω) associated with a regular Dirichlet form ( a, D ( a )) . Then the followingassertions are equivalent:(1) a = a µ for a unique Radon measure µ on ∂ Ω which charges no set of zerorelative capacity.(2) e t ∆ D ≤ T ( t ) ≤ e t ∆ N for all t ≥ .Proof. Let ( a, D ( a ) be the regular Dirichlet form associated with ( T ( t )) t ≥ . Fromthe formula of Beurling-Deny and Lejan we get for all u, v ∈ D ( a ) ∩ C c (Ω) a ( u, v ) = a ( c ) ( u, v ) + Z Ω u ( x ) v ( x ) k ( dx )+ Z Ω × Ω \ d ( u ( x ) − u ( y ))( v ( x ) − v ( y )) J ( dx, dy )where a ( c ) is a strongly local form, k a positive radon measure on Ω and J asymmetric positive radon measure on Ω × Ω \ d . To simplify calculations, we notefor all u, v ∈ D ( a ) ∩ C c (Ω) a k ( u, v ) = a ( c ) ( u, v ) + Z Ω u ( x ) v ( x ) k ( dx ) , and b ( u, v ) = a ( u, v ) − a k ( u, v ) , We have then for all u, v ∈ D ( a ) ∩ C c (Ω) such that supp[ u ] ∩ supp[ v ] = ∅ b ( u, v ) = Z Ω × Ω \ d ( u ( x ) − u ( y ))( v ( x ) − v ( y )) J ( dx, dy )= 2 Z Ω u ( x ) v ( x ) J ( dx, dy ) − Z Ω × Ω \ d u ( x ) v ( y ) J ( dx, dy ) (4.1)= − Z Ω × Ω \ d u ( x ) v ( y ) J ( dx, dy ) (4.2)From the fact that a k is local, we get for all u, v ∈ D ( a ) ∩ C c (Ω) such that supp[ u ] ∩ supp[ v ] = ∅ a ( u, v ) = − Z Ω × Ω \ d u ( x ) v ( y ) J ( dx, dy )From (1.1), and the Ouhabaz’s domination criterion[17, Th´eor`eme 3.1.7.], we havethat a N ( u, v ) ≤ a ( u, v ) for all u, v ∈ D ( a ) + , and then for all u, v ∈ D ( a ) + ∩ C c (Ω)such that supp[ u ] ∩ supp[ v ] = ∅ we have0 ≤ − Z Ω × Ω \ d u ( x ) v ( y ) J ( dx, dy ) , which means that − R Ω × Ω \ d u ( x ) v ( y ) J ( dx, dy ) = 0. Thus supp[ J ] ⊂ d and thenfor all u, v ∈ D ( a ) ∩ C c (Ω) Z Ω × Ω \ d u ( x ) v ( y ) J ( dx, dy ) = Z Ω u ( x ) v ( x ) J ( dx, dy )Then, we deduce that b ( u, v ) = 0 for all u, v ∈ D ( a ) ∩ C c (Ω) and thus the form a is immediately local and is reduced to the following a ( u, v ) = a ( c ) ( u, v ) + Z Ω u ( x ) v ( x ) k ( dx ) , for all u, v ∈ D ( a ) ∩ C c (Ω)We have C ∞ c (Ω) ⊂ D ( a ), and then for all u, v ∈ C ∞ c (Ω) we have from (1.1) Z Ω ∇ u ∇ vdx ≤ a ( c ) ( u, v ) + Z Ω u ( x ) v ( x ) k ( dx ) ≤ Z Ω ∇ u ∇ vdx It follows that for all u, v ∈ C ∞ c (Ω), we get a ( u, v ) = a ( c ) ( u, v ) + Z Ω u ( x ) v ( x ) k ( dx ) (4.3)= Z Ω ∇ u ∇ vdx (4.4)Thus supp[ k ] ⊂ ∂ Ω and then by putting µ = k | ∂ Ω , we have for all u, v ∈ D ( a ) ∩ C c (Ω) a ( u, v ) = Z Ω ∇ u ∇ vdx + Z ∂ Ω u ( x ) v ( x ) µ ( dx )For the rest of the proof, one can follow exactly the end of the proof of [19, Corollary3.4.23] (cid:3) One can see easily that the locality property was automatic in the above proofand that just the right hand side domination property was needed for it. We givethen the following Corollary.
Corollary 4.2.
Let Ω be a bounded open set of R d and ( T ( t )) t ≥ be a C − semigroupon L (Ω) associated with a regular Dirichlet form ( a, D ( a )) such that ≤ T ( t ) ≤ e t ∆ N for all t ≥ . Then the regular Dirichlet form ( a, D ( a )) is local.Proof. The corollary come directly from the proof of Theorem 4.1. (cid:3)
We deduce from the above discussion the following general theorem
Theorem 4.3.
Let X be locally compact separable metric space and m a positiveRadon measure on X such that supp[ m ] = X . Let ( T ( t )) t ≥ (resp. ( S ( t )) t ≥ ) be a C − semigroup on L ( X ; m ) associated with a regular Dirichlet form ( a, D ( a )) (resp.with a Dirichlet form ( b, D ( b )) ). Assume in addition that T ( t ) is subordinated by S ( t ) that is T ( t ) ≤ S ( t ) for all t ≥ in the positive operators sense. Then b is localimplies a is local.Proof. The proof is based on the same argumets as for Corollary 4.2. In fact, wehave that b ≤ a and b is local then for all u, v ∈ D ( a ) + ∩ C c ( X ) with disjointsupport we have a ( u, v ) ≥
0. Thus, by using Beurling-Deny and Lejan formula andthe positivity of u and v one obtain Z X × X \ d u ( x ) v ( y ) J ( dx, dy ) = 0for all u, v ∈ D ( a ) ∩ C c ( X ) with disjoint support. Thus supp[ J ] ⊂ d , and then thejump integral in the Beurling-Deny and Lejan decomposition of the form ( a, D ( a ))is null, which achieve the proof. (cid:3) Corollary 4.4.
Let Ω be a bounded open set of R d and ( T ( t )) t ≥ be a C − semigroupon L (Ω) associated with a Dirichlet form ( a, D ( a )) . Then the following assertionsare equivalent:(i) a = a µ for some positive measure µ on ∂ Ω which charges no set of zerorelative capacity on which it is locally finite.(ii) (a) e t ∆ D ≤ T ( t ) ≤ e t ∆ N for all t ≥ (b) D ( a ) ∩ C c (Ω) is dense in D ( a ) .Proof. Let Γ = { z ∈ Γ : ∃ u ∈ D ( a ) ∩ C c (Ω) , u ( z ) = 0 } From the Stone-Weierstrass theorem, one can see that the form ( a, D ( a )) is regularon L ( X ) where X = Ω ∪ Γ , one can see it exactly from the proof in [4, Theorem4.1] or the one in [19, Theorem 3.4.21]. Following the same procedure as in Theorem4.1, we obtain that there exist a unique positive Radon measure k on X such that a ( u, v ) = a ( c ) ( u, v ) + Z X u ( x ) v ( x ) k ( x )for all u, v ∈ D ( a ) ∩ C c ( X ).Now one can just again follow the same steps as in the proof of [4, Theorem 4.1]or the one in [19, Theorem 3.4.21] (cid:3) Proposition 4.5.
Let Ω ⊂ R d be a bounded open set of class C with boundary Γ . Let T be a symmetric C − semigroup associated with a regular Dirichlet form ( a, D ( a )) . Denote by A the generator of T . Assume thata) e t ∆ D ≤ T ( t ) ≤ e t ∆ N for all t ≥ , and thatb) there exists u ∈ D ( A ) ∩ C (Ω) such that u ( z ) > for all z ∈ Γ . Then thereexists a function β ∈ C (Γ) + such that A = − ∆ β .Proof. The proof is exactly the same as [4, Proposition 5.2], we have just omittedthe locality hypothesis in the proposition. (cid:3)
In what follow we give some complements and remarks about our approach andthe one of W. Arendt and M. Warma. • Apparently the both methods, ours and the one in [4] seem to be different,but in fact there is some connection between them. Seeing things more indetails in the proofs of [4, Theorem 4.1], and the proof of the formula ofBeurling-Deny and Lejan, one can see that they make use of [11, Lemma1.4.1](or more exactly its proof) as a central tool to find a measure. • The ideal property was not used in the sufficient implication of Theorem4.1. • One can expect to generalize our results without difficulty to the semi-regular Dirichlet forms and to regular but non-symmetric regular Dirichletforms. • In the context of p − Laplacian operator R. Chill and M. Warma proved aversion of the [4, Theorem 4.1. and Theorem 4.2] see [9]. Unfortunately,there is no version of the formula of Beurling-Deny and Lejan in L p ( X, m ),see [8] as a beginning of interest. • It is proved in an unpublished note of W. Arendt and M. Warma that alocality of the Dirichlet form implies the locality of the associated operator,but the converse is not true. We can see easily from our results that theoperator is local if and only if J | Ω × Ω \ d = 0, which means that there is anequivalence between the locality of the operator and the one of the formwhen there is no jump inside the domain. Acknowledgments
The author would like to thank Wolfgang Arendt Omar El-Mennaoui and JochenGl¨uck for many stimulating and helpful discussions. This work was achieved duringa research stay in Ulm, Germany.
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Akhlil KhalidApplied Analysis InstituteUniversity of UlmUlm, GermanyPolydisciplinary Faculty of OuarzazateIbn Zohr UniversityOuarzazate, Morocco.
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