Semigroups Generated by Elliptic Operators in Non-Divergence Form

SEMIGROUPS GENERATED BY ELLIPTIC OPERATORS IN NON-DIVERGENCE FORM ON C0 (Ω)

arXiv:1010.1703v1 [math.AP] 8 Oct 2010

¨ WOLFGANG ARENDT AND REINER SCHATZLE

Abstract. Let Ω ⊂ Rn be a bounded open set satisfying the uniform exterior cone condition. Let A be a uniformly elliptic operator given by Au =

n X

aij ∂ij u +

i,j=1

n X

bj ∂j u + cu

j=1

where ¯ and bj , c ∈ L∞ (Ω), c ≤ 0 . aij ∈ C(Ω) We show that the realization A0 of A in ¯ : u| C0 (Ω) := {u ∈ C(Ω) = 0} ∂Ω given by D(A0 )

:=

2,n (Ω) : Au ∈ C0 (Ω)} {u ∈ C0 (Ω) ∩ Wloc

A0 u

:=

Au

generates a bounded holomorphic C0 -semigroup on C0 (Ω). The result is in particular true if Ω is a Lipschitz domain. So far the best known result seems to be the case where Ω has C 2 -boundary [Lun95, Section 3.1.5]. We also study the elliptic problem −Au = f u|∂Ω = g .

0. Introduction The aim of this paper is to study elliptic and parabolic problems for operators in non-divergence form with continuous second order coefficients and to prove the existence (and uniqueness) of solutions which are continuous up to the boundary of the domain. Throughout this paper Ω is a bounded open set in Rn , n ≥ 2, with boundary ∂Ω. We consider the operator A given by

Au :=

n X

aij ∂ij u +

n X

bj ∂j u + cu

j=1

i,j=1

Date: September 30, 2010. 2010 AMS Subject Classification:. 35K20, 35J25, 47D06. Key words and phrases. holomorphic semigroups, elliptic operators in non-divergence form, Dirichlet problem, Wiener regular, Lipschitz domain, exterior cone property. 1

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with real-valued coefficients aij , bj , c satisfying bj ∈ L∞ (Ω) , j = 1, . . . , n , c ∈ L∞ (Ω) , c ≤ 0 ¯ , aij = aji , aij ∈ C(Ω) n P ¯ ξ ∈ Rn ) aij (x)ξi ξj ≥ Λ|ξ|2 (x ∈ Ω, i,j=1

where Λ > 0 is a fixed constant. Our best results are obtained under the hypothesis that Ω satisfies the uniform exterior cone condition (and thus in particular if Ω has Lipschitz boundary). Then ¯ ∩ we show that for each f ∈ Ln (Ω), g ∈ C(∂Ω) there exists a unique u ∈ C(Ω) 2,n Wloc (Ω) such that ( −Au = f (E) u|∂Ω = g . (Corollary 2.3). This result is proved with the help of Alexandrov’s maximum principle (which is responsible for the choice of p = n) and other standard results for elliptic second order differential operators (put together in the appendix). Our main concern is the parabolic problem   ut = Au  (P ) u(0, ·) = u0   u(t, x) = 0 x ∈ ∂Ω , t > 0 .

¯ : v| = 0}. Under with Dirichlet boundary conditions. Let C0 (Ω) := {v ∈ C(Ω) ∂Ω the uniform exterior cone condition, we show that the realization A0 of A in C0 (Ω) given by D(A0 ) A0 v

2,n := {v ∈ C0 (Ω) ∩ Wloc (Ω) : Av ∈ C0 (Ω)}

:= Av

generates a bounded, holomorphic C0 -semigroup on C0 (Ω). This improves the known results, which are presented in the monographie of Lunardi [Lun95, Corollary 3.1.21] for Ω of class C 2 (and bj , c uniformly continuous). If the second order coefficients are Lipschitz continuous, then the results mentioned so far hold if Ω is merely Wiener-regular. For elliptic operators in divergence form, this is proved in [GT98, Theorem 8.31] for the elliptic problem (E) and in [AB99, Corollary 4.7] for the parabolic problem (P ). Concerning the elliptic problem (E), and in particular the Dirichlet problem; i.e., the case f = 0 in (E), there is earlier work by Krylov [Kry67, Theorem 4], who shows well-posedness of the Dirichlet problem if Ω is merely Wiener regular and the second order coefficients are Dini-continuous. Krylov also obtains the well-posedness of the Dirichlet ¯ if Ω satisfies the uniform exterior cone condition [Kry67, problem for aij ∈ C(Ω) Theorem 5]. He uses different (partially probabilistic) methods, though. 1. The Poisson problem We consider the bounded open set Ω ⊂ Rn and the elliptic operator A from the Introduction. At first we consider the case where the second order conditions are

SEMIGROUPS GENERATED BY ELLIPTIC OPERATORS IN NON-DIVERGENCE FORM

3

Lipschitz continuous. Then we merely need a very mild regularity condition on Ω. We say that Ω is Wiener regular (or Dirichlet regular ) if for each g ∈ C(∂Ω) there ¯ of the Dirichlet problem exists a solution u ∈ C 2 (Ω) ∩ C(Ω) ∆u

=

0

u|∂Ω

=

g.

If Ω satisfies the exterior cone condition, then Ω is Dirichlet regular. Theorem 1.1. Assume that the second order coefficients aij are globally Lipschitz continuous. If Ω is Wiener-regular, then for each f ∈ Ln (Ω), there exists a unique 2,n u ∈ Wloc (Ω) ∩ C0 (Ω) such that −Au = f . The point is that for Lipschitz continuous aij the operator A may be written in divergence form. This is due to the following lemma. Lemma 1.2. Let h : Ω → R be Lipschitz continuous. Then h ∈ W 1,∞ (Ω). In particular, hu ∈ W 1,2 (Ω) for all u ∈ W 1,2 (Ω) and ∂j (hu) = (∂j h)u + h∂j u. Proof. One can extend h to a Lipschitz function on Rn (without increasing the Lipschitz constant, see [Min70]). Now the result follows from [Eva98, 5.8 Theorem 4]. Proof of Theorem 1.1. We assume that Ω is Dirichlet regular. Uniqueness follows from Aleksandrov’s maximum principle Theorem A.1. In order to solve the problem we replace A by an operator in divergence form in the following way. Let n ˜bj := bj − P ∂i aij , j = 1, . . . , n. Then ˜bj ∈ L∞ (Ω). Consider the elliptic operator i=1

Ad in divergence form given by Ad u =

n X

∂i (aij ∂j u) +

˜bj ∂j u + cu .

j=1

i,j=1 q

n X

a) Let f ∈ L (Ω) for q > n. By [GT98, Theorem 8.31] or [AB99, Corollary 4.6] 1,2 there exists a unique u ∈ C0 (Ω) ∩ Wloc (Ω) such that −Ad u = f weakly, i.e., Z Z d Z d Z X X ˜bj ∂j uv − cuv = f v aij ∂j u∂i v − i,j=1 Ω

j=1 Ω

Ω

Ω

for all v ∈ D(Ω) (the space of all test functions). We mention in passing that u ∈ W01,2 (Ω) by [AB99, Lemma 4.2]. For our purposes, it is important that u ∈ 2,2 Wloc (Ω) by Friedrich’s theorem [GT98, Theorem 8.8]. Here we use again that the aij are uniformly Lipschitz continuous but do not need any further hypothesis on 1,2 bj and c. It follows from Lemma 1.2 that aij ∂j u ∈ Wloc (Ω) and ∂i (aij ∂j u) = (∂i aij )∂j u + aij ∂ij u. Thus Ad u = Au. Now it follows from the interior Calderon2,q 2,n Zygmund estimate Theorem A.2 that u ∈ Wloc (Ω) ⊂ Wloc (Ω). This settles the q result if f ∈ L (Ω) for some q > n. b) Let f ∈ Ln (Ω). Choose fk ∈ L∞ (Ω) sucht that lim fk = f in Ln (Ω). Let k→∞


4

2,n uk ∈ Wloc ∩C0 (Ω) such that −Auk = fk (use case a)). By Aleksandrov’s maximum principle Thereom A.1, we have

kuk − uℓ kL∞ (Ω) ≤ ckfk − fℓ kLn (Ω) . Thus uk converge uniformly to a function u ∈ C0 (Ω) as k → ∞. By the CalderonZygmund estimate (Theorem A.2), kuk kW 2,n (B̺ ) ≤ c(kuk kLn (B2̺ ) + kfk kLn (B2̺ ) ) if B2̺ ⊂ Ω, where the constant c does not depend on k. Thus the sequence (uk )k∈N is bounded in W 2,n (B̺ ). It follows from reflexivity that u ∈ W 2,n (B̺ ) and uk ⇀ u in W 2,n (B̺ ) as k → ∞ after extraction of a subsequence. Conse2,n quently, u ∈ Wloc (Ω) ∩ C0 (Ω). Since −Auk = fk for all k ∈ N, it follows that −Au = f . ¯ and do no longer assume Now we return to the general assumption aij ∈ C(Ω) that the aij are Lipschitz continuous. We need the following lemma which we prove for convenience. Lemma 1.3. a) There exist a ˜ij ∈ C b (Rn ) such that a ˜ij = a ˜ji , a ˜ij (x) = aij (x) if x ∈ Ω and n X Λ aij (x)ξi ξj ≥ |ξ|2 2 i,j=1 for all ξ ∈ Rn , x ∈ Ω.

¯ such that ak = ak , b) There exist akij ∈ C ∞ (Ω) ij ji ¯ lim akij (x) = aij (x) uniformly on Ω.

k→∞

n P

i,j=1

akij (x)ξi ξj ≥

Λ 2 2 |ξ|

and

Proof. a) Let bij : Rn → R be a bounded, continuous extension of aij to Rn . b +b Replacing bij by ij 2 ji , we may assume that bij = bji . Since the function ϕ : n P bij (x)ξi ξj is continuous and S 1 := {ξ ∈ Rn : Rn × S 1 → R given by ϕ(x, ξ) := i,j=1

|ξ| = 1} is compact, the set Ω1 := {x ∈ Rn : ϕ(x, ξ) > Λ2 for all ξ ∈ S 1 } is open and ¯ Let 0 ≤ ϕ1 , ϕ2 ∈ C(Rn ) such that ϕ1 (x) + ϕ2 (x) = 1 for all x ∈ Rn contains Ω. ¯ Then a and ϕ2 (x) = 1 for x ∈ Rn \ Ω1 , ϕ1 (x) = 1 for x ∈ Ω. ˜ij := ϕ1 bij + Λ2 ϕ2 δij fulfills the requirements. b) Let (̺k )k∈N be a mollifier satisfying supp ̺k ⊂ B1/k (0). Then akij = a ˜ij ∗ ̺k ∈ ¯ If 1 < dist(∂Ω1 , Ω), C ∞ (Rn ) and lim akij (x) = a ˜ij (x) = aij (x) uniformly in x ∈ Ω. k k→∞

then for x ∈ Ω, ξ ∈ Rn n X

akij (x)ξi ξj

Z

=

i,j=1

|y| 0 such that (2.1)

kuk∞ ≤ 2c1 kµu − AukLn (Ω)

for all u ∈ D(A), µ ≥ 0. In order to show that A is closed, let uk ∈ D(A) such that uk → u in Ln (Ω) and Auk → f in Ln (Ω). It follows from (2.1) that u ∈ C0 (Ω) and lim uk = u in C0 (Ω). Let B2̺ be a ball of radius 2̺ such that B2̺ ⊂ Ω. By the

k→∞

Calderon-Zygmund estimate (Theorem A.2) kuk kW 2,n (B̺ ) ≤ c̺ (kuk kLn (B2̺ ) + kAuk kLn (B2̺ ) ) . It follows that (uk )k∈N is bounded in W 2,n (B̺ ). By passing to a subsequence we can assume that uk ⇀ u in W 2,n (B̺ ). Consequently Auk ⇀ Au in Ln (B̺ ). Thus Au = f on B̺ . Since the ball is arbitrary, it follows that u ∈ D(A) and Au = f . Now assume that µ1 − A is invertible for some µ1 ≥ 0. Let µ2 ≥ 0. Define B(t) = t(µ1 − A) + (1 − t)(µ2 − A). Since (µ1 − A), (µ2 − A) ∈ L(D(A), Ln (Ω)) where D(A) is considered as a Banach space with respect to the graph norm kukA := kukLn(Ω) + kAukLn(Ω) , since by (2.1) 2c1 kB(t)ukLn(Ω) ≥ kukC(Ω) ¯ ≥

1 kukLn(Ω) , |Ω|1/n

for all t ∈ [0, 1] and since B(1) is invertible, it follows from [GT98, Theorem 5.2] that B(0) is also invertible. 2,p We call a function u on Ω A-harmonic if u ∈ Wloc (Ω) for some p > 1 and T 2,q Au = 0. By [GT98, Theorem 9.16] each A-harmonic function u is in Wloc (Ω). q>1

Given g ∈ C(∂Ω), the Dirichlet problem consists in finding an A-harmonic function ¯ such that u| = g. We say that Ω is A-regular if for each g ∈ C(∂Ω) u ∈ C(Ω) ∂Ω there is a solution of the Dirichlet problem. Uniqueness follows from the maximum principle [GT98, Theorem 9.6] (2.2)

+ ∞ ∞ − ku− |∂Ω kL (∂Ω) ≤ u(x) ≤ ku|∂Ω kL (∂Ω)

¯ which holds for each A-harmonic function u ∈ C(Ω). ¯ In particular, for all x ∈ Ω, (2.3)

kukC(Ω) ¯ ≤ kukC(∂Ω) .

Theorem 2.2. The operator A is invertible if and only if Ω is A-regular.


7

Proof. a) Assume that A is invertible. ¯ First step: Let g ∈ C(∂Ω) be of the form g = G|∂Ω where G ∈ C 2 (Ω). Then n −1 AG ∈ L (Ω). Let v = A (AG), then u := G − v solves the Dirichlet problem for g. Second step: Let g ∈ C(∂Ω) be arbitrary. Extending g continuously and mollifying we find gk ∈ C(∂Ω) of the kind considered in the first step such that g = lim gk k→∞ ¯ be A-harmonic satisfying uk |∂Ω = gk . By (2.3) u := in C(∂Ω). Let uk ∈ C(Ω) ¯ lim uk exists in C(Ω). In particular, u|∂Ω = g. Let B2̺ ⊂ Ω. Then by the k→∞

Calderon-Zygmund estimate Theorem A.2 kuk kW 2,p (B̺ ) ≤ c̺ kuk kLp (B2̺ ) ≤ c̺ ckuk kC(Ω) ¯ (remember that Auk = 0). Thus (uk )k∈N is bounded in W 2,p (B̺ ). Passing to a subsequence, we can assume that uk ⇀ u in W 2,p (B̺ ). This implies that Au = 0 in B̺ . Since the ball is arbitrary, it follows that u is A-harmonic. Thus u is a solution of the Dirichlet problem (D). b) Conversely, assume that Ω is A-regular. Let f ∈ Ln (Ω). We want to find ¯ and extend f by 0 to B. u ∈ D(A) such that Au = f . Let B be a ball containig Ω 2,n ˜ = f . Here A˜ is Then by Theorem 1.4 we find v ∈ C0 (B) ∩ Wloc (B) such that Av an extension of A to the ball B according to Lemma 1.3a. Let g = v|∂Ω . Then by ¯ such that w| = g. our assumption there exists an A-harmonic function w ∈ C(Ω) ∂Ω 2,n Let u = v − w. Then u ∈ C0 (Ω) ∩ Wloc (Ω) and Au = Av = f ; i.e. u ∈ D(A) and Au = f . We have shown that A is surjective, which implies invertibility by Proposition 2.1.

Corollary 2.3. Assume that one of the following two conditions is satisfied: a) Ω is Wiener regular and the coefficients aij are globally Lipschitz continuous, or b) Ω satisfies the exterior cone condition. Then Ω is A-regular. More generally, for all f ∈ Ln (Ω), g ∈ C(∂Ω) there exists a ¯ ∩ W 2,n (Ω) satisfying unique u ∈ C(Ω) loc −Au u|∂Ω

= f = g.

Proof. Since A is closed by Proposition 2.1 it follows from Theorem 1.1 (in the case a)) and from Theorem 1.4 (in the case b)) that A is invertible. Thus Ω is A regular by Theorem 2.2. Let f ∈ Ln (Ω), g ∈ C(∂Ω). Since Ω is A-regular, there exists an ¯ such that u1 = g. Since A is invertible, there A-harmonic function u1 ∈ C(Ω) |∂Ω 2,n exists a function u0 ∈ C0 (Ω) ∩ Wloc (Ω) such that −Au0 = f . Let u := u0 + u1 . ¯ ∩ W 2,n (Ω), u| = g and −Au = f . Uniqueness follows from Then u ∈ C(Ω) ∂Ω loc Theorem A.1. For the Laplacian A = ∆, ∆-regularity is the usual regularity of Ω with respect to the classical Dirichlet problem, which is frequently called Wiener-regularity because of Wiener’s characterization via capacity [GT98, (2.37)]. It is a most interesting question how A-regularity and ∆-regularity are related. In general it is not true

8


that A-regularity implies Wiener regularity. In fact, K. Miller [Mil70] gives an example of an elliptic operator A with bj = c = 0 such that the pointed unit disc {x ∈ R2 : 0 < |x| < 1} is A-regular even though it is not ∆-regular. The other implication seems to be open. The fact that the uniform exterior cone property (which is much stronger than ∆-regularity) implies A-regularity (Corollary 2.3) had been proved before by Krylov [Kry67, Theorem 5] with the help of probabilistic methods. If Ω is merely ∆-regular, then it seems not to be known whether Ω is A-regular. Known results concerning this question are based on further restrictive conditions on the coefficients aij . In Theorem 1.1 we gave a proof for globally Lipschitz continuous aij . The best result seems to be [Kry67, Theorem 4] which goes in both directions: If the aij are Dini-continuous (in particular, if they are H¨ older-continuous), then Ω is ∆-regular if and only if Ω is A-regular. 3. Generation results An operator B on a complex Banach space X is said to generate a bounded holomorphic semigroup if (λ − B) is invertible for Re λ > 0 and sup kλ(λ − B)−1 k < ∞ . Re λ>0

Then there exist θ ∈ (0, π/2) and a holomorphic bounded function T : Σθ → L(X) satisfying T (z1 + z2 ) = T (z1 )T (z2 ) such that lim etBn = T (t) in L(X)

(3.1)

n→∞

for all t > 0, where Bn = nB(n − B)−1 ∈ L(X). Here Σθ is the sector Σθ := {reiα : r > 0, |α| < θ}. If B is an operator on a reel Banach space X we say that B generates a bounded holomorphic semigroup if its linear extension BC to the complexification XC of X generates a bounded holomorphic semigroup TC on XC . In that case TC (t)X ⊆ X (see [Lun95, Corollary 2.1.3]); in particular T (t) := TC (t)|X ∈ L(X). We call T = (T (t))t>0 the semigroup generated by B. It satisfies lim T (t)x = x for all t↓0

x ∈ X (i.e., it is a C0 -semigroup) if and only if D(B) = X. We refer to [Lun95, Chapter 2] and [ABHN01, Sec. 3.7] for these facts and further information. ¯ and C0 (Ω) as In this section we consider the parts Ac and A0 of A in C(Ω) follows: D(Ac ) := Ac u := D(A0 ) := A0 u :=

2,n ¯ {u ∈ C0 (Ω) ∩ Wloc (Ω) : Au ∈ C(Ω)}

Au

and

2,n {u ∈ C0 (Ω) ∩ Wloc (Ω) : Au ∈ C0 (Ω)}

Au .

¯ and A0 the part of Ac in C0 (Ω). Note that Thus Ac is the part of A in C(Ω) T 2,q D(A0 ) ⊆ D(Ac ) ⊆ Wloc by [GT98, Lemma 9.16]. The main result of this q>1

section is the following.


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Theorem 3.1. Assume that Ω is A-regular. Then Ac generates a bounded holo¯ The operator A0 generates a bounded holomorphic morphic semigroup T on C(Ω). C0 -semigroup T0 on C0 (Ω). Moreover, T (t)C0 (Ω) ⊆ C0 (Ω) and T0 (t) = T (t)|C0 (Ω) . Recall that Ω is A-regular if one of the following conditions is satisfied: (a) Ω satisfies the uniform exterior cone condition or (b) Ω is Wiener regular and the coefficients aij are Dini-continuous. In particular, Ω is A-regular if (a’) Ω is a Lipschitz-domain or (b’) Ω is Wiener-regular and the aij are H¨ older continuous. In the following complex maximum principle (Proposition 3.3) we extend A to 2,p the complex space Wloc (Ω) without changing the notation. We first proof a lemma. Lemma 3.2. Let B ⊆ Ω be a ball of center x0 and let u ∈ W 2,p (B), p > n, be a complex-valued function such that Au ∈ C(B). If |u(x0 )| ≥ |u(x)| for all x ∈ B, then i h Re u(x0 )(Au)(x0 ≤ 0 .

Proof. We may assume that x0 = h 0. If the claim i is wrong, then there exist ε > 0 and a ball B̺ ⊂ B such that Re u(x)(Au)(x) ≥ ε on B̺ .

u], and ∂ij (u¯ u) = (∂ij u)¯ u + ∂i u∂j u + Since ∂j |u|2 = (∂j u)¯ u + u∂j u = 2Re [∂j u¯ ∂j u∂i u + u∂ij u, and since by ellipticity X X aij ∂j u∂i u ≥ 0 , aij ∂i u∂j u ≥ 0 , Re Re i,j

i,j

it follows that A|u|2

≥

Re

X

aij (∂ij u)¯ u + Re

i,j

+

X

X

aij u∂ij u

i,j

bj 2Re [∂j u¯ u] + cu¯ u

j

≥

2Re (Au¯ u) ≥ 2ε on B̺ .

Let ψ(x) = |u|2 − τ |x|2 , τ > 0. Then A|ψ|2 ≥ 2ε − c1τ on B̺ for all τ > 0 and some c1 > 0. Choosing τ > 0 small enough, we have A|ψ|2 ≥ ε on B̺ . Since ψ ∈ W 2,p (B̺ )∩C(B̺ ), by Aleksandrov’s maximum principle [GT98, Theorem 9.1], see Theorem A.1, it follows that |u(0)|2 = |ψ(0)|2

≤

sup ψ ∂B̺ (0)

=

sup |u|2 − τ ̺2 ∂B̺ (0)

≤ |u(0)|2 − τ ̺2 < |u(0)|2 , a contradiction.

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¯ ∩ W 2,n (Ω) such Proposition 3.3. (complex maximum principle). Let u ∈ C(Ω) loc that λu − Au = 0 where Re λ > 0. If there exists x0 ∈ Ω such that |u(x)| ≤ |u(x0 )| for all x ∈ Ω, then u ≡ 0. Consequently, max |u(x)| = max |u(x)| . ¯ Ω

∂Ω

h i Proof. If |u(x)| ≤ |u(x0 )| for all x ∈ Ω, then by Lemma 3.2, Re u(x0 )(Au)(x0 ) ≤ 0. Since λu = Au, it follows that i h Re λ|u(x0 )|2 = Re u(x0 )(Au)(x0 ) ≤ 0 .

Hence u(x0 ) = 0.

Next, recall that an operator B on a real Banach space X is called m-dissipative if λ − B is invertible and λk(λ − B)−1 k ≤ 1 for all λ > 0 . Now we show that the operator Ac is m-dissipative and that the resolvent is positive (i.e., maps non-negative functions to non-negative functions). Proposition 3.4. Assume that Ω is A-regular. Then Ac is m-dissipative and (λ − Ac )−1 ≥ 0 for λ > 0. Proof. Let λ > 0. Since by Theorem 2.2 the operator (λ − A) is bijective, also (λ − Ac ) is bijective. ¯ f ≤ 0, u := (λ − Ac )−1 f . Assume a) We show that (λ − Ac )−1 ≥ 0. Let f ∈ C(Ω), + that u 6= 0. Since u ∈ C0 (Ω), there exists x0 ∈ Ω such that u(x0 ) = max u > 0. Ω

Then by Lemma 3.2, Au(x0 ) ≤ 0. Since λu − Au = f , it follows that λu(x0 ) ≤ f (x0 ) ≤ 0 a contradiction. ¯ u = (λ − Ac )−1 f . We show that kλukC(Ω) b) Let f ∈ C(Ω), ¯ ≤ kf kC(Ω) ¯ . Assume first that f ≥ 0, f 6= 0. Then u ≥ 0 by a) and u 6= 0. Let x0 ∈ Ω such that u(x0 ) = kukC(Ω) ¯ . Then (Ac u)(x0 ) ≤ 0 by Lemma 3.2. Hence λu(x0 ) ≤ λu(x0 ) − (Ac u)(x0 ) = f (x0 ) ≤ kf kC(Ω) ¯ . ¯ If f ∈ C(Ω) is arbitrary, then by a) |(λ − Ac )−1 f | ≤ (λ − Ac )−1 |f | and so kλ(λ − Ac )−1 f kC(Ω) ¯ ≤ kf kC(Ω) ¯ . Now we consider the complex extension of Ac (still denoted by Ac ) to the space ¯ which we still denote by C(Ω). ¯ Our aim is to of all complex-valued functions on Ω prove that for Re λ > 0 the operator (λ − Ac )−1 is invertible and k(λ − Ac )−1 k ≤

M , |λ|

where M is a constant. For that, we extend the coefficents aij to uniformly continuous bounded real-valued functions on Rn satisfying the strict ellipticity condition Re

d X

Λ aij (x)ξi ξ¯j ≥ |ξ|2 2 i,j=1

SEMIGROUPS GENERATED BY ELLIPTIC OPERATORS IN NON-DIVERGENCE FORM 11

(ξ ∈ Rn , x ∈ Rn ), keeping the some notation, see Lemma 1.3a. We extend bj , c to bounded measurable functions on Rn such that c ≤ 0 (keeping the same notation). Now we define the operator B∞ on L∞ (Rn ) by T 2,p D(B∞ ) := {u ∈ Wloc (Rn ) : u, Bu ∈ L∞ (Rn )} p>1

where

B∞ u

B∞ u

:= Bu , d d P P 2,p bj ∂j u + cu for u ∈ Wloc (Rn ) . aij ∂ij u + := i,j=1

j=1

The operator B∞ is sectorial. This is proved in [Lun95, Theorem 3.1.7] under the assumption that the coefficients bj , c are uniformly continuous. We give a perturbation argument to deduce the general case from the case bj = c = 0. The following lemma shows in particular that the domain of B∞ is independent of bj and c. Lemma 3.5. One has D(B∞ ) ⊂ W 1,∞ (Rn ). Moreover, for each ε > 0 there exists cε ≥ 0 such that kukW 1,∞ (Rn ) ≤ εkB∞ ukL∞ (Rn ) + cε kukL∞ (Rn ) for all u ∈ D(B∞ ). Proof. Consider an arbitrary ball B1 in Rn of radius 1 and the corresponding ball B2 of radius 2. Let p > n. Since the injection of W 2,p (B1 ) into C 1 (B¯1 ) is compact, for each ε > 0 there exists c′ε > 0 such that kukC 1 (B¯1 ) ≤ εkukW 2,p (B1 ) + c′ε kukL∞ (B1 ) . By the Calderon-Zygmund estimate this implies that kukC 1(B¯1 )

≤ εc1 (kB∞ ukL∞ (B2 ) + kukL∞(B2 ) ) +c′ε kukL∞ (B1 ) ≤ εc1 kB∞ ukL∞ (Rn ) + (εc1 + c′ε ) · kukL∞ (Rn ) .

Since kukL∞ (Rn ) = sup kukL∞ (B1 ) , where the supremum is taken over all balls of B1

radius 1 in Rn , the claim follows.

Theorem 3.6. There exist M ≥ 0, ω ∈ R such that (λ − B∞ ) is invertible and kλ(λ − B∞ )−1 k ≤ M

(Re λ > ω) .

0 Proof. Denote by B∞ the operator with the coefficients bj , c replaced by 0. Lemma 0 0 3.5 implies that D(B∞ ) = D(B∞ ) and (applied to B∞ ) that 0 0 k(B∞ − B∞ )ukL∞ (Rn ) ≤ εkB∞ ukL∞ (Rn ) + c′ε kukL∞ (Rn ) 0 0 for all u ∈ D(B∞ ), ε > 0 and some c′ε ≥ 0. Since B∞ is sectorial by [Lun95, Theorem 3.1.7] the claim follows from the usual holomorphic perturbation result [ABHN01, Theorem 3.7.23].


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Now we use the maximum principle, Lemma 3.2, to carry over the sectorial estimate from Rn to Ω. This is done in a very abstract framework by Lumer-Paquet [LP77], see [Are04, Section 2.5] for the Laplacian. Proof of Theorem 3.1. Let ω be the constant from Theorem 3.6 and let Re λ > ¯ u = (λ − Ac )−1 f . Then ω, f ∈ C(Ω), \ 2,p u ∈ C0 (Ω) ∩ Wloc (Ω) and λu − Au = f . p>1

n

Extend f by 0 to R and let v = (λ − B∞ )−1 f . Then λv − Av = f on Ω ¯ ∩ and kλvkL∞ (Ω) ≤ M kf kC(Ω) ¯ by Theorem 3.6. Moreover, w := v − u ∈ C(Ω) T 2,p Wloc (Ω), λw − Aw = 0 on Ω and w(z) = v(z) for all z ∈ ∂Ω. Then by the p≥1

complex maximum principle Proposition 3.3,

kwkC(Ω) ¯ = max |v(z)| ≤ z∈∂Ω

M kf kC(Ω) ¯ . |λ|

Consequently, kukC(Ω) ¯

= ku − v + vkC(Ω) ¯ ≤ kwkC(Ω) ¯ + kvkC(Ω) ¯ ≤

2M kf kC(Ω) ¯ . |λ|

This is the desired estimate which shows that Ac is sectorial. By [Lun95, Proposition 2.1.11] there exist a sector Σθ + ω := {ω + reiα : r > 0, |α| < θ} with θ ∈ ( π2 , π), ω ≥ 0, and a constant M1 > 0 such that (λ − Ac )−1 exists for λ ∈ Σθ + ω and kλ(λ − Ac )−1 k ≤ M1 . Thus there exists r > 0 such that (λ − Ac ) is invertible and kλ(λ − Ac )−1 k ≤ M whenever Re λ > 0 and |λ| > r. Since A is invertible by Theorem 2.2, it follows that Ac is bijective. Since the resolvent set of Ac is nonempty, Ac is closed. Thus Ac is invertible. Since by Proposition 3.4 Ac is resolvent positive, it follows from [ABHN01, Proposition 3.11.2] that there exists ε > 0 such that (λ−Ac ) is invertible whenever Re λ > −ε. As a consequence, sup kλ(λ − Ac )−1 k < ∞ . |λ|≤r Re λ>0

Together with the previous estimates this implies that kλ(λ − Ac )−1 k ≤ M2 whenever Re λ > 0 for some constant M2 . Thus Ac generates a bounded holomor¯ Since D(Ac ) ⊂ C0 (Ω) and D(Ω) ⊂ D(Ac ) it follows phic semigroup T on C(Ω). that D(Ac ) = C0 (Ω). The part of Ac in C0 (Ω) is A0 . So it follows from [Lun95, Remark 2.1.5, Proposition 2.1.4] that A0 generates a bounded, holomorphic C0 semigroup T0 on C0 (Ω) and T0 (t) = T (t)|C0 (Ω) on C0 (Ω). Finally we mention compactness and strict positivity.


Proposition 3.7. Assume that Ω satisfies the uniform exterior cone condition. Then (λ − Ac )−1 and T (t) are compact operators (λ > 0, t > 0). Proof. It follows from Theorem A.3 that D(Ac ) ⊂ C α (Ω). Since the embedding of ¯ is compact, it follows that the resolvent of Ac is compact. Since C α (Ω) into C(Ω) T is holomorphic, it follows that T (t) is compact for all t > 0. Proposition 3.8. Assume that Ω is A-regular. Let t > 0, 0 ≤ f ∈ C0 (Ω), f 6≡ 0. Then (T0 (t)f )(x) > 0 for all x ∈ Ω. Proof. a) We show that u := (λ− A0 )−1 f is strictly positive. Assume that u(x) ≤ 0 for some x ∈ Ω. Let v = −u. Then Av − λv = f ≥ 0. It follows from the maximum principle [GT98, Theorem 9.6] that v is constant. Since v ∈ C0 (Ω), it follows that v ≡ 0. Hence also f ≡ 0. b) It follows from a) that T0 is a positive, irreducible C0 -semigroup on C0 (Ω). Since the semigroup is holomorphic, the claim follows from [Na86, C-III.Theorem 3.2.(b)]. Appendix A. Results on elliptic partial differential equations In this section, we collect some results on elliptic partial differential equations, which can be found in text books, for example [GT98]. We consider the elliptic operator A from the Introduction and assume that the ellipticity constant Λ > 0 is so small that kaij kL∞ , kbj kL∞ , kckL∞ ≤ Λ1 .

Theorem A.1 (Aleksandrov’s maximum principle, [GT98, Theorem 9.1]). Let f ∈ ¯ ∩ W 2,n (Ω) such that Ln (Ω), u ∈ C(Ω) loc −Au ≤ f . Then sup u ≤ sup u+ + c1 kf + kLn (Ω) Ω

∂Ω

where the constant c1 depends merely on n, diam Ω and kbj kLn (Ω) , j = 1 . . . , n. Consequently, if u ∈ C0 (Ω) and −Au = f , then kukL∞ (Ω) ≤ 2c1 kf kLn(Ω)) and u ≤ 0 if f ≤ 0. Theorem A.2 (Interior Calderon-Zygmund estimate, [GT98, Theorem 9.11]). Let B2̺ be a ball of radius 2̺ such that B2̺ ⊂ Ω, and let u ∈ W 2,p (B2̺ ), where 1 < p < ∞. Then kukW 2,p (B̺ ) ≤ c̺ (kAukLp (B2̺ ) + kukLp(B2̺ ) ) where B̺ is the ball of radius ̺ concentric with B2̺ . The constant c merely depends on Λ, n, ̺, p and the continuity moduli of the aij .


14

Theorem A.3 (H¨older regularity, [GT98, Corollary 9.29]). Assume that Ω satisfies 2,n the uniform exterior cone condition. Let u ∈ C0 (Ω) ∩ Wloc (Ω) and f ∈ Ln (Ω) such that −Au = f . Then u ∈ C α (Ω) and kukC α (Ω) ≤ C(kf kLn(Ω) + kukL2 (Ω) ) where α > 0 and c > 0 depend merely on Ω, Λ and n. In [GT98, Corollary 9.29] it is supposed that u ∈ W 2,n (Ω). But an inspection of the proof and of the results preceding [GT98, Corollary 9.29] shows that u ∈ 2,n Wloc (Ω) suffices. The above H¨ older regularity also holds for solutions of equations in divergence form when the right-hand side f is in Lq (Ω) for some q > n2 , see [GT98, Theorem 8.29].

References [AB99]

Arendt, W., B´ enilan, Ph.: Wiener regularity and heat semigroups on spaces of continuous functions. Progress in Nonlinear Differential Equations and Their Applications Vol. 35 Birkh¨ auser Basel 1999, 29–49.

[ABHN01]

Arendt, W., Batty, C., Hieber, M., Neubrander, F.: Vector-valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics. Birkh¨ auser, Basel, (2001) ISBN 3-7643-6549-8.

[ADN59]

Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Communications on Pure and Applied Mathematics, 12 (1959), pp. 623-727.

[ADN64]

Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Communications on Pure and Applied Mathematics, 17 (1964), pp. 35-92.

[Are04]

Arendt, W. Semigroups and Evolution Equations: Functional Calculus, Regularity and Kernel Estimates. Handbook of Differential Equations. Evolutionary Equations, Vol. 1. C.M. Dafermos and E. Feireisl eds., Elsevier (2004), 1–85.

[GT98]

Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, Springer Verlag, 3.Auflage, Berlin (1998).

[Eva98]

Evans, L.C.: Partial Differential Equations. American Math. Soc., Providence, R. I. 1998.

[Kry67]

Krylov, N. V.: The first boundary value problem for elliptic equations of second order. Differencial’nye Uravnenja 3 (1967), 315–326.

[LP77]

Lumer, G., Paquet, L.: Semi-groupes holomorphes et ´ equations d’´ evolution, CR

[Lun95]

Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems,

Acad. Sc. Paris 284 S´ erie A (1977), pp. 237–240. Birkh¨ auser Basel, (1995). [Mil70]

Miller, K.: Nonequivalence of regular boundary points for the Laplace and nondivergence equations, even with continuous coefficients. Ann. Scuola Norm. Sup. Pisa 3 24 (1970), 159–163.

[Min70]

Minty, G. J.: On the extension of Lipschitz, Lipschitz-H¨ older continuous, and

[Na86]

Nagel, R. (ed.): One-parameter Semigroups of Positive Operators. Springer LN

monotone functions. Bull. Amer. Math. Soc. 76 (1970), 334–339. 1184, (1986) Berlin.


Institute of Applied Analysis, University of Ulm, D - 89069 Ulm, Germany E-mail address: [email protected] ¨ bingen, D-72076 Tu ¨ bingen, Institute of Mathematics, Eberhard-Karls-University of Tu Germany E-mail address: [email protected]