aa r X i v : . [ m a t h . A P ] A p r From forms to semigroups
Wolfgang Arendt and A. F. M. ter Elst
Introduction
Form methods give a very efficient tool to solve evolutionary prob-lems on Hilbert space. They were developed by T. Kato [Kat] and, inslightly different language by J.L. Lions. In this expository article wegive an introduction based on [AE2]. The main point in our approachis that the notion of closability is not needed anymore. The new settingis particularly efficient for the Dirichlet-to-Neumann operator and de-generate equations. Besides this we give several other examples. Thispresentation starts by an introduction to holomorphic semigroups. In-stead of the contour argument found in the literature, we give a moredirect argument based on the Hille–Yosida theorem.1.
The Hille–Yosida Theorem A C -semigroup on a Banach space X is a mapping T : (0 , ∞ ) →L ( X ) satisfying T ( t + s ) = T ( t ) T ( s )lim t ↓ T ( t ) x = x ( x ∈ X ) . The generator A of such a C -semigroup is defined by D ( A ) := { x ∈ X : lim t ↓ T ( t ) x − xt exists } Ax := lim t ↓ T ( t ) x − xt ( x ∈ D ( A )) . Thus the domain D ( A ) of A is a subspace of X and A : D ( A ) → X islinear. One can show that D ( A ) is dense in X . The main interest insemigroups lies in the associated Cauchy problem (CP) ( ˙ u ( t ) = Au ( t ) ( t > u (0) = x . Indeed, if A is the generator of a C -semigroup, then given x ∈ X , thefunction u ( t ) := T ( t ) x is the unique mild solution of ( CP ) ; i.e. u ∈ C ([0 , ∞ ); X ) , t Z u ( s ) ds ∈ D ( A ) for all t > and u ( t ) = x + A t Z u ( s ) dsu (0) = x . If x ∈ D ( A ) , then u is a classical solution ; i.e. u ∈ C ([0 , ∞ ); X ) , u ( t ) ∈ D ( A ) for all t ≥ and ˙ u ( t ) = Au ( t ) for all t > . Conversely,if for each x ∈ X there exists a unique mild solution of ( CP ) , then A generates a C -semigroup [ABHN, Theorem 3.1.12]. In view of thischaracterization of well-posedness, it is of big interest to decide whethera given operator generates a C -semigroup. A positive answer is givenby the famous Hille–Yosida Theorem. Theorem 1.1. (Hille–Yosida (1948)) . Let A be an operator on X . Thefollowing are equivalent.(i) A generates a contractive C -semigroup;(ii) the domain of A is dense, λ − A is invertible for some ( all ) λ > and k λ ( λ − A ) − k ≤ . Here we call a semigroup T contractive if k T ( t ) k ≤ for all t > .By λ − A we mean the operator with domain D ( A ) given by ( λ − A ) x := λx − Ax ( x ∈ D ( A )) . So the condition in (ii) means that λ − A : D ( A ) → X is bijective and k λ ( λ − A ) − x k ≤ k x k for all λ > and x ∈ X . If X is reflexive, then this existence of the resolvent ( λ − A ) − and the contractivity k λ ( λ − A ) − k ≤ imply already thatthe domain is dense [ABHN, Theorem 3.3.8].Yosida’s proof is based on the Yosida-approximation: Assuming (ii),one easily sees that lim λ →∞ λ ( λ − A ) − x = x ( x ∈ D ( A )) , ROM FORMS TO SEMIGROUPS 3 i.e. λ ( λ − A ) − converges strongly to the identity as λ → ∞ . Thisimplies that A λ := λA ( λ − A ) − = λ ( λ − A ) − − λ approximates A as λ → ∞ in the sense that lim λ →∞ A λ x = Ax ( x ∈ D ( A )) . The operator A λ is bounded, so one may define e tA λ := ∞ X n =0 t n n ! A nλ by the power series. Note that k λ ( λ − A ) − k ≤ λ . Since e tA λ = e − λt e tλ ( λ − A ) − , it follows that k e tA λ k ≤ e − λt e t k λ ( λ − A ) − k ≤ . The key element in Yosida’s proof consists in showing that for all x ∈ X the family ( e tA λ x ) λ> is a Cauchy net as λ → ∞ . Then the C -semigroup generated by A is given by T ( t ) x := lim λ →∞ e tA λ x ( t > for all x ∈ X . We will come back to this formula when we talk aboutholomorphic semigroups. Remark . Hille’s independent proof is based on Euler’s formula forthe exponential function. Note that putting t = λ one has λ ( λ − A ) − = ( I − tA ) − . Hille showed that T ( t ) x := lim n →∞ ( I − tn A ) − n x exists for all x ∈ X , see [Kat, Section IX.1.2]. WOLFGANG ARENDT AND A. F. M. TER ELST Holomorphic semigroups A C -semigroup is defined on the real half-line (0 , ∞ ) with values in L ( X ) . It is useful to study when extensions to a sector Σ θ := { re iα : r > , | α | < θ } for some θ ∈ (0 , π/ exist. In this section X is a complex Banachspace. Definition 2.1. A C -semigroup T is called holomorphic if there exist θ ∈ (0 , π/ and a holomorphic extension e T : Σ θ → L ( X ) of T which is locally bounded; i.e. sup z ∈ Σ θ | z |≤ k e T ( z ) k < ∞ . If k e T ( z ) k ≤ for all z ∈ Σ θ , then we call T a sectorially contractiveholomorphic C -semigroup ( of angle θ , if we want to make precise theangle).The holomorphic extension e T automatically has the semigroup prop-erty e T ( z + z ) = e T ( z ) e T ( z ) ( z , z ∈ Σ θ ) . Because of the boundedness assumption it follows that lim z → z ∈ Σ θ e T ( z ) x = x ( x ∈ X ) . These properties are easy to see. Moreover, e T can be extended contin-uously (for the strong operator topology) to the closure of Σ θ , keepingthese two properties. In fact, if x = T ( t ) y for some t > and some y ∈ X , then lim w → z T ( w ) x = lim w → z T ( w + t ) y = T ( z + t ) y exists. Since the set { T ( t ) y : t ∈ (0 , ∞ ) , y ∈ X } is dense the claimfollows. In the sequel we will omit the tilde and denote the extension e T simply by T . We should add a remark on vector-valued holomorphicfunctions. ROM FORMS TO SEMIGROUPS 5
Remark . If Y is a Banach space, Ω ⊂ C open, then a function f : Ω → Y is called holomorphic if f ′ ( z ) = lim h → f ( z + h ) − f ( z ) h exists in the norm of Y for all z ∈ Ω and f ′ : Ω → Y is continuous.It follows as in the scalar case that f is analytic. It is remarkablethat holomorphy is the same as weak holomorphy (first observed byGrothendieck): A function f : Ω → Y is holomorphic if and only if y ′ ◦ f : Ω → C is holomorphic for all y ′ ∈ Y ′ . In our context the space Y is L ( X ) , thespace of all bounded linear operators on X with the operator norm.If the function f is bounded it suffices to test holomorphy with fewfunctionals. We say that a subspace W ⊂ Y ′ separates points if for all x ∈ Y , h y ′ , x i = 0 for all y ′ ∈ W implies x = 0 . Assume that f : Ω → Y is bounded such that y ′ ◦ f is holomorphic for all y ′ ∈ W where W is a separating subspace of Y ′ . Then f is holomorphic.This result is due to [AN], see also [ABHN, Theorem A7]. In particular,if Y = L ( X ) , then a bounded function f : Ω → L ( X ) is holomorphicif and only if h x ′ , f ( · ) x i is holomorphic for all x in a dense subspace of X and all x ′ in a separating subspace of X ′ .We recall a special form of Vitali’s Theorem (see [AN], [ABHN, The-orem A5]). Theorem 2.3. (Vitali) . Suppose Ω ⊂ C is connected. For all n ∈ N let f n : Ω → L ( X ) be holomorphic, let M ∈ R and suppose thata) k f n ( z ) k ≤ M for all z ∈ Ω and n ∈ N , and;b) Ω := { z ∈ Ω : lim n →∞ f n ( z ) x exists for all x ∈ X } has a limitpoint in Ω , i.e. there exist a sequence ( z k ) k ∈ N in Ω and z ∈ Ω such that z k = z for all k ∈ N and lim k →∞ z k = z .Then f ( z ) x := lim n →∞ f n ( z ) x exists for all x ∈ X and z ∈ Ω , and f : Ω → L ( X ) is holomorphic. WOLFGANG ARENDT AND A. F. M. TER ELST
Now we want to give a simple characterization of holomorphic sec-torially contractive semigroups. Assume that A is a densely definedoperator on X such that ( λ − A ) − exists and k λ ( λ − A ) − k ≤ λ ∈ Σ θ ) , where < θ ≤ π/ . Let z ∈ Σ θ . Then for all λ > , ( zA ) λ = zA λz is holomorphic in z . For each z ∈ Σ θ , the operator zA satisfies Condi-tion (ii) of Theorem 1.1. By the Hille–Yosida Theorem T ( z ) x := lim λ →∞ e ( zA ) λ x exists for all x ∈ X and z ∈ Σ θ . Since z e ( zA ) λ = e zA λ/z is holo-morphic, T : Σ θ → L ( X ) is holomorphic by Vitali’s Theorem. If t > ,then T ( t ) = lim λ →∞ e tA λ/t = T A ( t ) where T A is the semigroup generated by A . Since T A ( t + s ) = T A ( t ) T A ( s ) ,it follows from analytic continuation that T ( z + z ) = T ( z ) T ( z ) ( z , z ∈ Σ θ ) . Thus A generates a sectorially contractive holomorphic C -semigroupof angle θ on X . One sees as above that T zA ( t ) = T ( zt ) for all t > and z ∈ Σ θ . We have shown the following. Theorem 2.4.
Let A be a densely defined operator on X and θ ∈ (0 , π/ . The following are equivalent.(i) A generates a sectorially contractive holomorphic C -semigroupof angle θ ;(ii) ( λ − A ) − exists for all λ ∈ Σ θ and k λ ( λ − A ) − k ≤ λ ∈ Σ θ ) . We refer to [AEH] for a similar approach to possibly noncontractiveholomorphic semigroups.
ROM FORMS TO SEMIGROUPS 7 The Lumer–Phillips Theorem
Let H be a Hilbert space over K = R or C . An operator A on H iscalled accretive or monotone if Re( Ax | x ) ≥ x ∈ D ( A )) . Based on this notion the following very convenient characterization isan easy consequence of the Hille–Yosida Theorem.
Theorem 3.1. (Lumer–Phillips) . Let A be an operator on H . Thefollowing are equivalent.(i) − A generates a contraction semigroup;(ii) A is accretive and I + A is surjective. For a proof, see [ABHN, Theorem 3.4.5]. Accretivity of A can bereformulated by the condition k ( λ + A ) x k ≥ k λx k ( λ > , x ∈ D ( A )) . Thus if λ + A is surjective, then λ + A is invertible and k λ ( λ + A ) − k ≤ .We also say that A is m -accretive if Condition (ii) is satisfied. If A is m -accretive and K = C , then one can easily see that λ + A is invertiblefor all λ ∈ C satisfying Re λ > and k ( λ + A ) − k ≤ λ . Due to the reflexivity of Hilbert spaces, each m -accretive operator A is densely defined (see [ABHN, Proposition 3.3.8]). Now we want toreformulate the Lumer–Phillips Theorem for generators of semigroupswhich are contractive on a sector. Theorem 3.2. (generators of sectorially contractive semigroups). Let A be an operator on a complex Hilbert space H and let θ ∈ (0 , π ) . Thefollowing are equivalent.(i) − A generates a holomorphic C -semigroup which is contractiveon the sector Σ θ ;(ii) e ± iθ A is accretive and I + A is surjective.Proof. ( ii ) ⇒ ( i ) . Since e ± iθ A is accretive the operator zA is accretivefor all z ∈ Σ θ . Since ( I + A ) is surjective, the operator A is m -accretive.Thus ( λ + A ) is invertible whenever Re λ > . Consequently ( I + zA ) = WOLFGANG ARENDT AND A. F. M. TER ELST z ( z − + A ) is invertible for all z ∈ Σ θ . Thus zA is m -accretive for all z ∈ Σ θ . Now (i) follows from Theorem 2.4. ( i ) ⇒ ( ii ) . If − A generates a holomorphic semigroup which is con-tractive on Σ θ , then e iα A generates a contraction semigroup for all α with | α | ≤ θ . Hence e iα A is m -accretive whenever | α | ≤ θ . (cid:3) Forms: the complete case
We recall one of our most efficient tool to solve equations, the Lax–Milgram lemma, which is just a non-symmetric generalization of theRiesz–Fréchet representation theorem from 1905.
Lemma 4.1. (Lax–Milgram (1954)) . Let V be a Hilbert space over K , where K = R or K = C , and let a : V × V → K be sesquilinear,continuous and coercive, i.e. Re a ( u ) ≥ α k u k V ( u ∈ V ) for some α > . Let ϕ : V → K be a continuous anti-linear form, i.e. ϕ is continuous and satisfies ϕ ( u + v ) = ϕ ( u ) + ϕ ( v ) and ϕ ( λu ) = λϕ ( u ) for all u, v ∈ V and λ ∈ K . Then there is a unique u ∈ V such that a ( u, v ) = ϕ ( v ) ( v ∈ V ) . Of course, to say that a is continuous means that | a ( u, v ) | ≤ M k u k V k v k V ( u, v ∈ V ) for some constant M . We let a ( u ) := a ( u, u ) for all u ∈ V .In general, the range condition in the Hille–Yosida Theorem is diffi-cult to prove. However, if we look at operators associated with a form,the Lax–Milgram Lemma implies automatically the range condition.We describe now our general setting in the complete case. Given isa Hilbert space V over K with K = R or K = C , and a continuous,coercive sesquilinear form a : V × V → K . Moreover, we assume that H is another Hilbert space over K and j : V → H is a continuous linear mapping with dense image. Nowwe associate an operator A on H with the pair ( a, j ) in the following ROM FORMS TO SEMIGROUPS 9 way. Given x, y ∈ H we say that x ∈ D ( A ) and Ax = y if there existsa u ∈ V such that j ( u ) = x and a ( u, v ) = ( y | j ( v )) H for all v ∈ V .
We first show that A is well-defined. Assume that there exist u , u ∈ V and y , y ∈ H such that j ( u ) = j ( u ) ,a ( u , v ) = ( y | j ( v )) H ( v ∈ V ) , and, a ( u , v ) = ( y | j ( v ) H ( v ∈ V ) . Then a ( u − u , v ) = ( y − y | j ( v )) H for all v ∈ V . Since j ( u − u ) = 0 ,taking v := u − u gives a ( u − u , u − u ) = 0 . Since a is coercive,it follows that u = u . It follows that ( y | j ( v )) H = ( y | j ( v )) H for all v ∈ V . Since j has dense image, it follows that y = y .It is clear from the definition that A : D ( A ) → H is linear. Ourmain result is the following generation theorem. We first assume that K = C . Theorem 4.2. (generation theorem in the complete case) . The opera-tor − A generates a sectorially contractive holomorphic C -semigroup T .If a is symmetric, then A is selfadjoint.Proof. Letting M ≥ be the constant of continuity and α > theconstant of coerciveness as before, we have | Im a ( v ) | Re a ( v ) ≤ M k v k V α k v k V = Mα for all v ∈ V \ { } . Thus there exists a θ ′ ∈ (0 , π ) such that a ( v ) ∈ Σ θ ′ ( v ∈ V ) . Let x ∈ D ( A ) . There exists a u ∈ V such that x = j ( u ) and a ( u, v ) =( Ax | j ( v )) H for all v ∈ V . In particular, ( Ax | x ) H = a ( u ) ∈ Σ θ ′ . Itfollows that e ± iθ A is accretive where θ = π − θ ′ . In order to prove therange condition, let y ∈ H . Consider the form b : V × V → C given by b ( u, v ) = a ( u, v ) + ( j ( u ) | j ( v )) H . Then b is continuous and coercive. Let y ∈ H . Then ϕ ( v ) := ( y | j ( v )) H defines a continuous anti-linear form ϕ on V . By the Lax–Milgram Lemma 4.1 there exists a unique u ∈ V such that b ( u, v ) = ϕ ( v ) ( v ∈ V ) . Hence ( y | j ( v )) H = a ( u, v ) + ( j ( u ) | j ( v )) H ; i.e. a ( u, v ) = ( y − j ( u ) | j ( v )) H for all v ∈ V . This means that x := j ( u ) ∈ D ( A ) and Ax = y − x . (cid:3) The result is also valid in real Banach spaces. If T is a C -semigroupon a real Banach space X , then the C -linear extension T C of T onthe complexification X C := X ⊕ iX of X is a C -semigroup givenby T C ( t )( x + iy ) := T ( t ) x + iT ( t ) y . We call T holomorphic if T C is holomorphic. The generation theorem above remains true on realHilbert spaces.In order to formulate a final result we want also allow a rescaling.Let X be a Banach space over K and T be a C -semigroup on X withgenerator A . Then for all ω ∈ K and t > define T ω ( t ) := e ωt T ( t ) . Then T ω is a C -semigroup whose generator is A + ω . Using this weobtain now the following general generation theorem in the completecase.Let V, H be Hilbert spaces over K and j : V → H linear with denseimage. Let a : V × V → K be sesquilinear and continuous. We call theform a j - elliptic if there exist ω ∈ R and α > such that(4.1) Re a ( u ) + ω k j ( u ) k H ≥ α k u k V ( u ∈ V ) Then we define the operator A associated with ( a, j ) as follow. Given x, y ∈ H we say that x ∈ D ( A ) and Ax = y if there exists a u ∈ V such that j ( u ) = x and a ( u, v ) = ( y | j ( v )) H for all v ∈ V .
Theorem 4.3.
The operator defined in this way is well-defined. More-over, − A generates a holomorphic C -semigroup on H .Remark . The form a satisfies Condition (4.1) if and only if the form a ω given by a ω ( u, v ) = a ( u, v ) + ω ( j ( u ) | j ( v )) H ROM FORMS TO SEMIGROUPS 11 is coercive. If T ω denotes the semigroup associated with ( a ω , j ) and T the semigroup associated with ( a, j ) , then T ω ( t ) = e − ωt T ( t ) ( t > as is easy to see. 5. The Stokes Operator
In this section we show as an example that the Stokes operator isselfadjoint and generates a holomorphic C -semigroup. The followingapproach is due to Monniaux [Mon]. Let Ω ⊂ R d be a bounded openset. We first discuss the Dirichlet Laplacian. Theorem 5.1. (Dirichlet Laplacian) . Let H = L (Ω) and define theoperator ∆ D on L (Ω) by D (∆ D ) = { u ∈ H (Ω) : ∆ u ∈ L (Ω) } ∆ D u := ∆ u . Then ∆ D is selfadjoint and generates a holomorphic C -semigroup on L (Ω) .Proof. Define a : H (Ω) × H (Ω) → R by a ( u, v ) = R Ω ∇ u ∇ v . Then a is clearly continuous. Poincaré’s inequality says that a is coercive.Consider the injection j of H (Ω) into L (Ω) . Let A be the operatorassociated with ( a, j ) . We show that A = − ∆ D . In fact, let u ∈ D ( A ) and write f = Au . Then R Ω ∇ u ∇ v = R Ω f v for all v ∈ H (Ω) . Taking inparticular v ∈ C ∞ c (Ω) we see that − ∆ u = f . Conversely, let u ∈ H (Ω) be such that f := − ∆ u ∈ L (Ω) . Then R Ω f ϕ = R Ω ∇ u ∇ ϕ = a ( u, ϕ ) for all ϕ ∈ C ∞ c (Ω) . This is just the definition of the weak partialderivatives in H (Ω) . Since C ∞ c (Ω) is dense in H (Ω) , it follows that R Ω f v = a ( u, v ) for all v ∈ H (Ω) . Thus u ∈ D ( A ) and Au = f . (cid:3) For our treatment of the Stokes operator it will be useful to considerthe Dirichlet Laplacian also in L (Ω) d = L (Ω) ⊕ . . . ⊕ L (Ω) . Theorem 5.2.
Define the symmetric form a : H (Ω) d × H (Ω) d → R by a ( u, v ) = Z Ω ∇ u ∇ v := d X j =1 Z Ω ∇ u j ∇ v j , where u = ( u , . . . , u d ) . Then a is continuous and coercive. Moreover,let j : H (Ω) d → L (Ω) d be the identity. The operator A associatedwith ( a, j ) on L (Ω) d is given by D ( A ) = { u ∈ H (Ω) d : ∆ u j ∈ L (Ω) for all j ∈ { , . . . , d }} ,Au = ( − ∆ u , . . . , − ∆ u d ) =: − ∆ u . We call ∆ D := − A the Dirichlet Laplacian on L (Ω) d . In order to define the Stokes operator we need some preparation.Let D (Ω) := C ∞ c (Ω) d and let D (Ω) := { ϕ ∈ D (Ω) : div ϕ = 0 } , where div ϕ = ∂ ϕ + . . . + ∂ d ϕ d and ϕ = ( ϕ , . . . , ϕ d ) . By D (Ω) ′ we denote thedual space of D (Ω) (with the usual topology). Each element S of D (Ω) ′ can be written in a unique way as S = ( S , . . . , S d ) with S j ∈ C ∞ c (Ω) ′ so that h S, ϕ i = d X j =1 h S j , ϕ j i for all ϕ = ( ϕ , . . . , ϕ d ) ∈ D (Ω) .We say that S ∈ H − (Ω) if there exists a constant c ≥ such that |h S, ϕ i| ≤ c ( Z |∇ ϕ | ) ( ϕ ∈ D (Ω)) where |∇ ϕ | = |∇ ϕ | + . . . + |∇ ϕ d | . For the remainder of this sectionwe assume that Ω has Lipschitz boundary. We need the following result(see [Tem, Remark 1.9, p. 14]). Theorem 5.3.
Let T ∈ H − (Ω) . The following are equivalent.(i) h T, ϕ i = 0 for all ϕ ∈ D (Ω) ;(ii) there exists a p ∈ L (Ω) such that T = ∇ p . Note that Condition (ii) means that h T, ϕ i = d X j =1 h ∂ j p, ϕ j i = − d X j =1 h p, ∂ j ϕ j i = −h p, div ϕ i . Now the implication (ii) ⇒ (i) is obvious. We omit the other implica-tion.Consider the real Hilbert space L (Ω) d with scalar product ( f | g ) = d X j =1 ( f j | g j ) L (Ω) = d X j =1 Z Ω f j g j . ROM FORMS TO SEMIGROUPS 13
We denote by H := D (Ω) ⊥⊥ = D (Ω) the closure of D (Ω) in L (Ω) d . We call H the space of all divergencefree vectors in L (Ω) d . The orthogonal projection P from L (Ω) d onto H is called the Helmholtz projection . Now let V be the closure of D (Ω) in H (Ω) d . Thus V ⊂ H (Ω) d and div u = 0 for all u ∈ V . One canactually show that V = { u ∈ H (Ω) d : div v = 0 } . We define the form a : V × V → R by a ( u, v ) = d X j =1 ( ∇ u j |∇ v j ) L (Ω) ( u = ( u , . . . , u d ) , v = ( v , . . . , v d ) ∈ V ) . Then a is continuous and coercive. The space V is dense in H sinceit contains D (Ω) . We consider the identity j : V → H . Let A be theoperator associated with ( a, j ) . Then A is selfadjoint and − A generatesa holomorphic C -semigroup. The operator can be described as follows. Theorem 5.4.
The operator A has the domain D ( A ) = { u ∈ V : ∃ π ∈ L (Ω) such that − ∆ u + ∇ π ∈ H } and is given by Au = − ∆ u + ∇ π , where π ∈ L (Ω) is such that − ∆ u + ∇ π ∈ H . If u ∈ H (Ω) d , then ∆ u ∈ H − (Ω) . In fact, for all ϕ ∈ D (Ω) , |h− ∆ u, ϕ i| = |−h u, ∆ ϕ i| = | d X j =1 Z Ω ∇ u j ∇ ϕ j | ≤ k u k H (Ω) d k ϕ k H (Ω) d . Proof of Theorem 5.4.
Let u ∈ D ( A ) and write f = Au . Then f ∈ H , u ∈ V and a ( u, v ) = ( f | v ) H for all v ∈ V . Thus, the distribution − ∆ u ∈ H − (Ω) coincides with f on D (Ω) . By Theorem 5.3 thereexists a π ∈ L (Ω) such that − ∆ u + ∇ π = f . Conversely, let u ∈ V , f ∈ H , π ∈ L (Ω) and suppose that − ∆ u + ∇ π = f in D (Ω) ′ . Thenfor all ϕ ∈ D (Ω) , a ( u, ϕ ) = Z Ω ∇ u ∇ ϕ = Z Ω ∇ u ∇ ϕ + h∇ π, ϕ i = ( f | ϕ ) L (Ω) d . Since D (Ω) is dense in V , it follows that a ( u, ϕ ) = ( f | ϕ ) L (Ω) d for all ϕ ∈ V . Thus, u ∈ D ( A ) and Au = f . (cid:3) The operator A is called the Stokes operator . We refer to [Mon] forthis approach and further results on the Navier–Stokes equation. Weconclude this section by giving an example where j is not injective.Further examples will be seen in the sequel. Proposition 5.5.
Let e H be a Hilbert space and H ⊂ e H a closed sub-space. Denote by P the orthogonal projection onto H . Let e V be aHilbert space which is continuously and densely embedded into e H andlet a : e V × e V → R be a continuous, coercive form. Denote by A theoperator on e H associated with ( a, j ) where j is the injection of e V into e H and let B be the operator on H associated with ( a, P ◦ j ) . Then D ( B ) = { P w : w ∈ D ( A ) and Aw ∈ H } ,BP w = Aw ( w ∈ D ( A ) , Aw ∈ H ) . This is easy to see. In the context considered in this section weobtain the following example.
Example 5.6.
Let e H = L (Ω) d , H = D (Ω) and e V := H (Ω) d . Define a : e V × e V → R by a ( u, v ) = Z Ω ∇ u ∇ v . Moreover, define j : e V → e H by j ( u ) = u . Then the operator associatedwith ( a, j ) is A = − ∆ D as we have seen in Theorem 5.2. Now let P bethe Helmholtz projection and B the operator associated with ( a, P ) .Then D ( B ) = { u ∈ H : ∃ π ∈ L (Ω) such that u + ∇ π ∈ D (∆ D ) and ∆( u + ∇ π ) ∈ H } and Bu = − ∆( u + ∇ π ) , if π ∈ L (Ω) is such that u + ∇ π ∈ D (∆ D ) and ∆( u + ∇ π ) ∈ H .This follows directly from Proposition 5.5 and Theorem 5.3. Thus, theoperator B is selfadjoint and generates a holomorphic semigroup. ROM FORMS TO SEMIGROUPS 15 From forms to semigroups: the incomplete case
In the preceding sections we considered forms which were defined ona Hilbert space V . Now we want to study a purely algebraic conditionconsidering forms whose domain is an arbitrary vector space. At firstwe consider the complex case. Let H be a complex Hilbert space. A sectorial form on H is a sesquilinear form a : D ( a ) × D ( a ) → C , where D ( a ) is a vector space, together with a linear mapping j : D ( a ) → H with dense image such that there exist ω ≥ and θ ∈ (0 , π/ suchthat a ( u ) + ω k j ( u ) k H ∈ Σ θ ( u ∈ D ( a )) . If ω = 0 , then we call the form -sectorial . To a sectorial form, weassociate an operator A on H by defining for all x, y ∈ H that x ∈ D ( A ) and Ax = y : ⇔ there exists a sequence ( u n ) n ∈ N in D ( a ) such thata) lim n →∞ j ( u n ) = x in H ;b) sup n ∈ N Re a ( u n ) < ∞ , and;c) lim n →∞ a ( u n , v ) = ( y | j ( v )) H for all v ∈ D ( a ) .It is part of the next theorem that the operator A is well-defined(i.e. that y depends only on x and not on the choice of the sequencesatisfying a), b) and c)). We only consider single-valued operators inthis article. Theorem 6.1.
The operator A associated with a sectorial form ( a, j ) is well-defined and − A generates a holomorphic C -semigroup on H . The proof of the theorem consists in a reduction to the complete caseby considering an appropriate completion of D ( a ) . Here it is importantthat in Theorem 4.2 a non-injective mapping j is allowed. For a proofwe refer to [AE2, Theorem 3.2].If C ⊂ H is a closed convex set, we say that C is invariant under asemigroup T if T ( t ) C ⊂ C ( t > . Invariant sets are important to study positivity, L ∞ -contractivity, andmany more properties. If the semigroup is associated with a form, thenthe following criterion, [AE2, Proposition 3.9], is convenient. Theorem 6.2. (invariance) . Let C ⊂ H be a closed convex set and let P be the orthogonal projection onto C . Then the semigroup T associ-ated with a sectorial form ( a, j ) on H leaves C invariant if and only iffor each u ∈ D ( a ) there exists a sequence ( w n ) n ∈ N in D ( a ) such thata) lim n →∞ j ( w n ) = P j ( u ) in H ;b) lim sup n →∞ Re a ( w n , u − w n ) ≥ , and;c) sup n ∈ N Re a ( w n ) < ∞ . Corollary 6.3.
Assume that for each u ∈ D ( a ) , there exists a w ∈ D ( a ) such that j ( w ) = P j ( u ) and Re a ( w, u − w ) ≥ . Then T ( t ) C ⊂ C for all t > . In this section we want to use the invariance criterion to prove ageneration theorem in the incomplete case which is valid in real Hilbertspaces. Let H be a real Hilbert space. A sectorial form on H is abilinear mapping a : D ( a ) × D ( a ) → R , where D ( a ) is a real vector space, together with a linear mapping j : D ( a ) → H with dense image such that there are α, ω ≥ suchthat | a ( u, v ) − a ( v, u ) | ≤ α ( a ( u ) + a ( v )) + ω ( k j ( u ) k H + k j ( v ) k H )( u, v ∈ D ( a )) . It is easy to see that the form a is sectorial on the real space H ifand only if the sesquilinear extenion a C of a to the complexification of D ( a ) together with the C -linear extension of j is sectorial in the senseformulated in the beginning of this section.To such a sectorial form ( a, j ) we associate an operator A on H bydefining for all x, y ∈ H that x ∈ D ( A ) and Ax = y : ⇔ there exists asequence ( u n ) in D ( a ) satisfyinga) lim n →∞ j ( u n ) = x in H ;b) sup n ∈ N a ( u n ) < ∞ , and;c) lim n →∞ a ( u n , v ) = ( y | j ( v )) H for all v ∈ D ( a ) . ROM FORMS TO SEMIGROUPS 17
Then the following holds.
Theorem 6.4.
The operator A is well-defined and − A generates aholomorphic C -semigroup on H .Proof. Consider the complexifications H C = H ⊕ iH and D ( a C ) := D ( a ) + iD ( a ) . Letting a C ( u, v ) := a (Re u, Re v )+ a (Im u, Im v )+ i ( a (Re u, Im v )+ a (Im u, Re v )) for all u = Re u + i Im u, v = Re v + i Im v ∈ D ( a C ) . Then a C is asesquilinear form. Let J : D ( a C ) → H C be the C -linear extension of j .Let b ( u, v ) = a C ( u, v ) + ω ( J ( u ) | J ( v )) H C ( u, v ∈ D ( a C )) . Then Im b ( u ) = a (Im u, Re u ) − a (Re u, Im u ) , Re b ( u ) = a (Re u ) + a (Im u ) + ω ( k j (Re u ) k H + k j (Im u ) k H ) . The assumption implies that there is a c > such that | Im b ( u ) | ≤ c Re b ( u ) for all u ∈ D ( a C ) . Consequently, b ( u ) ∈ Σ θ , where θ =arctan c . Thus the operator B associated with b generates a C -semi-group S C on H C . It follows from Corollary 6.3 that H is invariant. Thepart A ω of B in H is the generator of S , where S ( t ) := S C ( t ) | H . It iseasy to see that A ω − ω = A . (cid:3) Remark . It is remarkable, and important for some applications,that Condition b) in Theorem 6.1 as well as in Theorem 6.4 may bereplaced by b ′ ) lim n,m →∞ a ( u n − u m ) = 0 . For later purposes we carry over the invariance criterion Theorem5.3 to the real case.
Corollary 6.6.
Let H be a real Hilbert space and ( a, j ) a sectorialform on H with associated semigroup T . Let C ⊂ H be a closed convexset and P the orthogonal projection onto C . Assume that for each u ∈ D ( a ) there exists a w ∈ D ( a ) such that j ( w ) = P j ( u ) and a ( w, u − w ) ≥ . Then T ( t ) C ⊂ C for all t > . We want to formulate a special case of invariance. An operator S ona space L p (Ω) is called positive if (cid:16) f ≥ a.e. implies Sf ≥ a.e. (cid:17) and submarkovian if (cid:16) f ≤ a.e. implies Sf ≤ a.e. (cid:17) .Thus, an operator S is submarkovian if and only if it is positive and k Sf k ∞ ≤ k f k ∞ for all f ∈ L ∩ L ∞ . Proposition 6.7.
Consider the real space H = L (Ω) and a sectorialform a on H . Assume that for each u ∈ D ( a ) one has u ∧ ∈ D ( a ) and a ( u ∧ , ( u − + ) ≥ . Then the semigroup T associated with a is submarkovian.Proof. The set C := { u ∈ L (Ω) : u ≤ a.e. } is closed and convex.The orthogonal projection P onto C is given by P u = u ∧ . Thus u − P u = ( u − + and the result follows from Corollary 6.3. (cid:3) We conclude this section by some references to the literature. Inmany text books, for example [Dav], [Kat], [MR], [Ouh], [Tan] one findsthe notion of a sectorial form a on a complex Hilbert space H . By thisone understands a sesquilinear form a : D ( a ) × D ( a ) → C where D ( a ) is a dense subspace of H such that there are θ ∈ (0 , π/ and ω ≥ such that a ( u ) + ω k u k H ∈ Σ θ for all u ∈ D ( a ) . Then k u k a := (Re a ( u ) + ( ω + 1) k u k H ) / defines a norm on D ( a ) . The form is called closed if D ( a ) is completefor this norm. This corresponds to our complete case with V = D ( a ) and j the identity. If the form is not closed, then one may consider thecompletion V of D ( a ) . Since the injection D ( a ) → H is continuous forthe norm k k a , it has a continuous extension j : V → H . This extensionmay be injective or not. The form is called closable if j is injective. Inthe literature only for closable forms generation theorems are given, see[AE2] for precise references. The results above show that the notion ofclosability is not needed.There is a unique correspondence between sectorially quasi contrac-tive holomorphic semigroups and closed sectorial forms (see [Kat, The-orem VI.2.7]). One looses uniqueness if one considers forms which aremerely closable or in our general setting if one allows arbitrary maps ROM FORMS TO SEMIGROUPS 19 j : D ( a ) → H with dense image. However, examples show that in manycases a natural operator is obtained by this general framework.7. Degenerate diffusion
In this section we use our tools to show that degenerate elliptic op-erators generate holomorphic semigroups on the real space L (Ω) . Westart with a -dimensional example. Example 7.1. (degenerate diffusion in dimension 1). Consider thereal Hilbert space H = L ( a, b ) , where −∞ ≤ a < b ≤ ∞ , and let α, β, γ ∈ L ∞ loc ( a, b ) be real coefficients. We assume that there is a c ≥ such that γ − ∈ L ∞ ( a, b ) and β ( x ) ≤ c · α ( x ) ( x ∈ ( a, b )) . We define the bilinear form a on L ( a, b ) by a ( u, v ) = b Z a (cid:16) α ( x ) u ′ ( x ) v ′ ( x ) + β ( x ) u ′ ( x ) v ( x ) + γ ( x ) u ( x ) v ( x ) (cid:17) dx with domain D ( a ) = H c ( a, b ) . We choose j : H c ( a, b ) → L ( a, b ) to be the identity map. Then theform a is sectorial , i.e. there exist constants c, ω ≥ , such that | a ( u, v ) − a ( v, u ) | ≤ c ( a ( u ) + a ( v )) + ω ( k u k L + k v k L )( u, v ∈ D ( a )) . Proof.
We use Young’s inequality | xy | ≤ εx + 14 ε y twice. Let u, v ∈ D ( a ) . On one hand we have for all δ > , | a ( u, v ) − a ( v, u ) | = | b Z a β ( u ′ v − uv ′ ) |≤ b Z a δβ ( u ′ + v ′ ) + 14 δ ( u + v ) . On the other hand, for all c, ω, ε > one has c ( a ( u ) + a ( v )) + ω ( k u k H + k v k H )= b Z a cα ( u ′ + v ′ ) + cβ ( u ′ u + v ′ v ) + ( cγ + ω )( u + v ) ≥ b Z a ( cα − εβ )( u ′ + v ′ ) − c ε ( u + v ) + ( cγ + ω )( u + v ) ≥ b Z a ( cα − εβ )( u ′ + v ′ ) + ( ω − c k γ − k L ∞ − c ε )( u + v ) . Therefore (7.1) is valid if ( cα − εβ ) ≥ δβ and ( ω − c k γ − k L ∞ − c ε ) ≥ δ . Since β ≤ c α one can find δ, ε, c, ω such that the conditions aresatisfied. (cid:3) As a consequence, letting A be the operator associated with ( a, j ) ,we know that − A generates a holomorphic C -semigroup T on L (Ω) .Moreover, T is submarkovian.The condition β ≤ c α shows in particular that { x ∈ ( a, b ) : α ( x ) =0 } ⊂ { x ∈ ( a, b ) : β ( x ) = 0 } . This inclusion is a natural hypothe-sis, since in general an operator of the form βu ′ does not generate aholomorphic semigroup.A special case is the Black–Scholes Equation u t + σ x u xx + rxu x − ru = 0 . This one obtains by choosing H = L (0 , ∞ ) , a ( u, v ) = ∞ Z ( σ x u ′ v ′ + ( σ − r ) xu ′ v + ruv ) and D ( a ) = H c (0 , ∞ ) .It is not difficult to extend the example above to higher dimensions. Example 7.2.
Let Ω ⊂ R d be open and for all i, j ∈ { , . . . , d } let a ij , b j , c ∈ L ∞ loc (Ω) be real coefficients. Assume c − ∈ L ∞ (Ω) , a ij = a ji and there exists a c > such that c A ( x ) − B ( x ) is positive semidefinite ROM FORMS TO SEMIGROUPS 21 for almost all x ∈ Ω , where A ( x ) = ( a ij ( x )) and B ( x ) = diag( b ( x ) , . . . , b d ( x )) . Define the form a on L (Ω) by a ( u, v ) = Z Ω (cid:16) d X i,j =1 a ij ( ∂ i u )( ∂ j v ) + d X j =1 b j ( ∂ j u ) v + cuv (cid:17) with domain D ( a ) = H c (Ω) . Then a is sectorial. The associated semigroup T on L (Ω) is submarko-vian.This and the previous example incorporate Dirichlet boundary con-ditions. In the next one we consider a degenerate elliptic operator withNeumann boundary conditions. Example 7.3.
Let Ω ⊂ R d be an open, possibly unbounded subset of R d . For all i, j ∈ { , . . . , d } let a ij ∈ L ∞ (Ω) be real coefficients andassume that there exists a θ ∈ (0 , π/ such that d X i,j =1 a ij ( x ) ξ i ξ j ∈ Σ θ ( ξ ∈ C d , x ∈ Ω) . Consider the form a on L (Ω) given by a ( u, v ) = Z Ω d X i,j =1 a ij ( ∂ i u )( ∂ j v ) with domain D ( a ) = H (Ω) . Then a is sectorial. Let T be the associ-ated semigroup. Our criteria show right away that T is submarkovian.It is remarkable that even T ∞ ( t )11 Ω = 11 Ω ( t > . For bounded Ω this is easy to prove, but otherwise more sophisticatedtools are needed (see [AE2, Corollar 4.9]). Note that T extends consis-tently to semigroups T p on L p (Ω) for all p ∈ [1 , ∞ ] , where T p is stronglycontinuous for all p < ∞ and T ∞ is the adjoint of a strongly continuoussemigroup on L (Ω) . We want to add an abstract result which shows that our solutionsare some kind of viscosity solutions . This is illustrated particularly wellin the situation of Example 7.3.
Proposition 7.4. ([AE2, Corollary 3.9]) . Let
V, H be real Hilbertspaces such that
V ֒ → d H . Let j : V → H be the identity map. Let a : V × V → R be continuous and sectorial. Assume that a ( u ) ≥ forall u ∈ V . Let b : V × V → R be continuous and coercive. Then foreach n ∈ N the form a + 1 n b : V × V → R is continuous and coercive. Let A n be the operator associated with ( a + n b, j ) and A with ( a, j ) . Then lim n →∞ ( A n + λ ) − f = ( A + λ ) − f in H for all f ∈ H and λ > . Moreover, denoting by T n and T the semi-group generated by − A n and by − A one has lim n →∞ T n ( t ) f = T ( t ) f in H for all f ∈ H . The point in the result is that the form a is merely sectorial and maybe degenerate. For instance, in Example 7.3 a ij ( x ) = 0 is allowed. Ifwe perturb by the Laplacian, we obtain a coercive form a n : H (Ω) × H (Ω) → R given by a n ( u, v ) = a ( u, v ) + 1 n Z Ω ∇ u ∇ v . Then Proposition 7.4 says that in the situation of Example 7.3 for thisperturbation one has lim n →∞ ( A n + λ ) − f = ( A + λ ) − f in L (Ω) forall f ∈ L (Ω) .8. The Dirichlet-to-Neumann operator
The following example shows how the general setting involving non-injective j can be used. It is taken from [AE1] where also the interplaybetween trace properties and the semigroup generated by the Dirichlet-to-Neumann operator is studied. Let Ω ⊂ R d be a bounded open set ROM FORMS TO SEMIGROUPS 23 with boundary ∂ Ω . Our point is that we do not need any regularityassumption on Ω , except that we assume that ∂ Ω has a finite ( d − -dimensional Hausdorff measure. Still we are able to define the Dirichlet-to-Neumann operator on L ( ∂ Ω) and to show that it is selfadjoint andgenerates a submarkovian semigroup on L (Ω) . Formally, the Dirichlet-to-Neumann operator D is defined as follows. Given ϕ ∈ L (Γ) , onesolves the Dirichlet problem ( ∆ u = 0 in Ω u | ∂ Ω = ϕ and defines D ϕ = ∂u∂ν . We will give a precise definition using weakderivatives. We consider the space L ( ∂ Ω) := L ( ∂ Ω , H d − ) with the ( d − -dimensional Hausdorff measure H d − . Integrals over ∂ Ω arealways taken with respect to H d − , those over Ω always with respect tothe Lebesgue measure. Throughout this section we only assume that H d − ( ∂ Ω) < ∞ and that Ω is bounded. Definition 8.1. (normal derivative). Let u ∈ H (Ω) be such that ∆ u ∈ L (Ω) . We say that ∂u∂ν ∈ L ( ∂ Ω) if there exists a g ∈ L ( ∂ Ω) such that Z Ω (∆ u ) v + Z Ω ∇ u ∇ v = Z ∂ Ω gv for all v ∈ H (Ω) ∩ C (Ω) . This determines g uniquely and we let ∂u∂ν := g .Recall that for all u ∈ L (Ω) the Laplacian ∆ u is defined in thesense of distributions. If ∆ u = 0 , then u ∈ C ∞ (Ω) by elliptic regularity.Next we define traces of a function u ∈ H (Ω) . Definition 8.2. (traces). Let u ∈ H (Ω) . We let tr( u ) = { g ∈ L (Ω) : ∃ ( u n ) n ∈ N in H (Ω) ∩ C (Ω) such that lim n →∞ u n = u in H (Ω) and lim n →∞ u n | ∂ Ω = g in L ( ∂ Ω) } . For arbitrary open sets and u ∈ H (Ω) the set tr( u ) might be empty,or contain more than one element. However, if Ω is a Lipschitz domain,then for each u ∈ H (Ω) the set tr( u ) contains precisely one element,which we denote by u | ∂ Ω ∈ L ( ∂ Ω) . Now we are in the position todefine the Dirichlet-to-Neumann operator D . Its domain is given by D ( D ) := { ϕ ∈ L ( ∂ Ω) : ∃ u ∈ H (Ω) such that ∆ u = 0 , ϕ ∈ tr( u ) and ∂u∂ν ∈ L ( ∂ Ω) } and we define D ϕ = ∂u∂ν where u ∈ H (Ω) is such that ∆ u = 0 , ∂u∂ν ∈ L ( ∂ Ω) and ϕ ∈ tr( u ) . Itis part of our result that this operator is well-defined. Theorem 8.3.
The operator D is selfadjoint and − D generates asubmarkovian semigroup on L ( ∂ Ω) . In the proof we use Theorem 6.4. Here a non-injective mapping j isneeded. We also need Maz’ya’s inequality. Let q = dd − . There exists aconstant c M > such that (cid:16) Z Ω | u | q (cid:17) /q ≤ c M (cid:16) Z Ω |∇ u | + Z ∂ Ω | u | (cid:17) for all u ∈ H (Ω) ∩ C (Ω) . (See [Maz, Example 3.6.2/1 and Theorem3.6.3] and [AW, (19)].) Proof of Theorem 8.3.
We consider real spaces. Our Hilbert space is L ( ∂ Ω) . Let D ( a ) = H (Ω) ∩ C (Ω) , a ( u, v ) = R Ω ∇ u ∇ v and define j : D ( a ) → L ( ∂ Ω) by j ( u ) = u | ∂ Ω ∈ L ( ∂ Ω) . Then a is symmetricand a ( u ) ≥ for all u ∈ D ( a ) . Thus the sectoriality condition beforeTheorem 6.4 is trivially satisfied. Denote by A the operator on L ( ∂ Ω) associated with ( a, j ) . Let ϕ, ψ ∈ L ( ∂ Ω) . Then ϕ ∈ D ( A ) and Aϕ = ψ if and only if there exists a sequence ( u n ) n ∈ N in H (Ω) ∩ C (Ω) such that lim n →∞ u n | ∂ Ω = ϕ in L ( ∂ Ω) , lim n →∞ a ( u n , v ) = R ∂ Ω ψv | ∂ Ω for all v ∈ D ( a ) and lim n,m →∞ R Ω |∇ ( u n − u m ) | = 0 (here we use Remark 6.5). Now Maz’ya’sinequality implies that ( u n ) n ∈ N is a Cauchy sequence in H (Ω) . Thus lim n →∞ u n = u exists in H (Ω) , and so ϕ ∈ tr( u ) . Moreover R ∂ Ω ψv = ROM FORMS TO SEMIGROUPS 25 lim n →∞ R Ω ∇ u n ∇ v = R Ω ∇ u ∇ v for all v ∈ H (Ω) ∩ C (Ω) . Taking as v testfunctions, we see that ∆ u = 0 . Thus Z Ω ∇ u ∇ v + Z Ω (∆ u ) v = Z ∂ Ω ψv for all v ∈ H (Ω) . Consequently, ∂u∂ν = ψ . We have shown that A ⊂ D .Conversely, let ϕ ∈ D ( D ) , D ϕ = ψ . Then there exists a u ∈ H (Ω) such that ∆ u = 0 , ϕ ∈ tr( u ) and ∂u∂v = ψ . Since ϕ ∈ tr( u ) there existsa sequence ( u n ) n ∈ N in H (Ω) ∩ C (Ω) such that u n → u in H (Ω) and u n | ∂ Ω → ϕ in L ( ∂ Ω) . It follows that j ( u n ) = u n | ∂ Ω → ϕ in L ( ∂ Ω) ,the sequence ( a ( u n )) n ∈ N is bounded and a ( u n , v ) = Z Ω ∇ u n ∇ v → Z Ω ∇ u ∇ v = Z Ω ∇ u ∇ v + Z Ω (∆ u ) v = Z ∂ Ω ψv for all v ∈ H (Ω) ∩ C (Ω) . Thus, ϕ ∈ D ( A ) and Aϕ = ψ by thedefinition of the associated operator. Since a is symmetric, the operator A is selfadjoint. Now the claim follows from Theorem 6.4.Our criteria easily apply and show that semigroup generated by − D is submarkovian. (cid:3) References [ABHN]
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Institute of Applied Analysis, University of Ulm, D - 89069 Ulm,Germany
E-mail address : [email protected] Department of Mathematics, University of Auckland, Private Bag92019, Auckland 1142, New Zealand
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