Hölder kernel estimates for Robin operators and Dirichlet-to-Neumann operators
aa r X i v : . [ m a t h . A P ] O c t H¨older kernel estimates for Robin operatorsand Dirichlet-to-Neumann operators
A.F.M. ter Elst and M.F. Wong
Abstract
Consider the elliptic operator A = − d X k,l =1 ∂ k c kl ∂ l + d X k =1 a k ∂ k − d X k =1 ∂ k b k + a on a bounded connected open set Ω ⊂ R d with Lipschitz boundaryconditions, where c kl ∈ L ∞ (Ω , R ) and a k , b k , a ∈ L ∞ (Ω , C ), subject toRobin boundary conditions ∂ ν u + β Tr u = 0, where β ∈ L ∞ ( ∂ Ω , C )is complex valued. Then we show that the kernel of the semigroupgenerated by − A satisfies Gaussian estimates and H¨older Gaussian es-timates. If all coefficients and the function β are real valued, then weprove Gaussian lower bounds.Finally, if Ω is of class C κ with κ > c kl = c lk is H¨oldercontinuous, a k = b k = 0 and a is real valued, then we show that thekernel of the semigroup associated to the Dirichlet-to-Neumann operatorcorresponding to A has H¨older Poisson bounds.AMS Subject Classification: 35K05, 35B45, 35J25.Keywords: Robin boundary conditions, Dirichlet-to-Neumann operator, heat kernel esti-mates. Home institution:
Department of MathematicsUniversity of AucklandPrivate bag 92019AucklandNew Zealand
Introduction
Second-order strongly elliptic operators in divergence form with real measurable boundedcoefficients, subject to Dirichlet, Neumann or mixed boundary conditions, are well stud-ied on a bounded Lipschitz domain Ω. The kernel of the associated semigroup satisfiesGaussian kernel bounds [Aro], [Dav], [AE1], [Dan1]. For Dirichlet and Neumann boundaryconditions it was proved that the kernel is even H¨older continuous with the appropriateH¨older Gaussian bounds [AT]. Recently, also H¨older Gaussian kernel bounds have beenproved for operators with mixed boundary conditions [ERe]. The situation is different ifthe operator has Robin boundary conditions ∂ ν u + β u = 0, where β ∈ L ∞ ( ∂ Ω). If β ≥ β ≥ β ∈ L ∞ ( ∂ Ω , R ) in [Dan2] Theorem 2.2and Lemma 3.2. No H¨older Gaussian kernel bounds are known if β = 0. In this paper weshow that the kernel has both Gaussian kernel bounds and H¨older Gaussian bounds for any β ∈ L ∞ ( ∂ Ω), even complex valued. Also the lower-order coefficients can be complex, butwe still require that the principal coefficients are real valued (and measurable, althoughthey do not have to be symmetric).The first main theorem of this paper is as follows.
Theorem 1.1.
Let Ω ⊂ R d be a bounded Lipschitz domain. For all k, l ∈ { , . . . , d } let c kl : Ω → R and a k , b k , a : Ω → C be bounded and measurable. Let β : ∂ Ω → C be boundedand measurable. Suppose there exists a µ > such that Re P dk,l =1 c kl ( x ) ξ k ξ l ≥ µ | ξ | forall x ∈ Ω and ξ ∈ C d . Define the sesquilinear form a β : W , (Ω) × W , (Ω) → C by a β ( u, v ) = Z Ω d X k,l =1 c kl ( ∂ k u ) ∂ l v + Z Ω d X k =1 a k ( ∂ k u ) v + Z Ω d X k =1 b k u ∂ k v + Z Ω a u v + Z ∂ Ω β (Tr u ) Tr v. Let A be the operator associated with the closed sectorial form a β . Then the semigroupgenerated by − A has a kernel K . Moreover, for all τ > and τ ′ ∈ (0 , there exist κ ∈ (0 , and b, c, ω > such that | K t ( x, y ) | ≤ c t − d/ e − b | x − y | t e ωt and | K t ( x, y ) − K t ( x ′ , y ′ ) | ≤ c t − d/ (cid:16) | x − x ′ | + | y − y ′ | t / + | x − y | (cid:17) κ e − b | x − y | t e ωt for all x, x ′ , y, y ′ ∈ Ω and t > with | x − x ′ | + | y − y ′ | ≤ τ t / + τ ′ | x − y | . For the proof we use a modification of the technique of Auscher [Aus] to use Morreyand Campanato spaces to deduce H¨older Gaussian kernel bounds. Here we will use apointwise version of Morrey and Campanato seminorms as in [ERe]. Estimates are obtainedseparately for the interior and regions near to the boundary. Then the Gaussian boundsfollow from a similar result in [EO1]. We also prove that the constants in Theorem 1.12an be chosen uniformly with respect to the ellipticity constant and the L ∞ -norm of thecoefficients and β .If the lower order coefficients are real valued and the function β is real valued, thenthe kernel K is real valued. If in addition the operator is self-adjoint, then K satisfiesGaussian lower bounds. Theorem 1.2.
Adopt the notation and assumptions of Theorem 1.1. Assume in additionthat the a k , b k , a and β are real valued. Moreover, assume that the operator A is self-adjoint. Then there are b, c, ω > such that K t ( x, y ) ≥ c t − d/ e − b | x − y | t e − ωt for all x, y ∈ Ω and t > . These lower bounds have been proved before by [CK] for C , -domains and the parabolicproblem associated to operators in non-divergence form, possibly non-autonomous, andwith Neumann boundary conditions.The main regularity proposition that is used in the proof of Theorem 1.1 can also beused for the Dirichlet-to-Neumann operator N , which we describe next.Let Ω ⊂ R d be a bounded Lipschitz domain with boundary Γ. Let ϕ ∈ L (Γ). Then wesay that ϕ ∈ D ( N ) if there exists a u ∈ W , (Ω) such that ∆ u = 0 weakly on Ω, Tr u = ϕ and with normal derivative ∂ ν u ∈ L (Γ). Then N ϕ = ∂ ν u . The Dirichlet-to-Neumannoperator N is a positive self-adjoint operator. Let T be the semigroup on L (Γ) generatedby −N . If there exists a κ > C κ domain, then ter Elst–Ouhabaz[EO2] proved that T has a kernel satisfying Poisson bounds. The last main theorem of thispaper is that the kernel of T is H¨older continuous and satisfies H¨older continuous Poissonbounds. Theorem 1.3.
Suppose d ≥ . Let Ω ⊂ R d be a bounded domain with C κ -boundary forsome κ > . Let K be the kernel of the semigroup on L (Γ) generated by −N , where N isthe Dirichlet-to-Neumann operator. Then for all ε ∈ (0 , and τ > there exist c, κ ′ > such that | K t ( x, y ) − K t ( x ′ , y ′ ) | ≤ c ( t ∧ − ( d − (cid:16) | x − x ′ | + | y − y ′ | t (cid:17) κ ′ (cid:16) | x − y | t (cid:17) d − ε for all x, y, x ′ , y ′ ∈ Γ with | x − x ′ | + | y − y ′ | ≤ τ t . In Theorem 5.11 we will prove an extension of Theorem 1.3, where the Laplacian isreplaced by a pure second-order strongly elliptic operator in divergence form with realsymmetric H¨older continuous coefficients. This theorem will be a consequence of the pre-viously mentioned Poisson kernel bounds in [EO2] and a new theorem, Theorem 5.5, whichprovides optimal bounds for the semigroup from L (Γ) into the space of H¨older continuousfunctions on Γ. The latter theorem is valid even for operators in divergence form with realmeasurable principal coefficients and complex lower-order terms.The outline of the paper is as follows. In Section 2 we introduce the notation and classesof coefficients that we need in this paper. In Section 3 H¨older continuity of the semigroup3ill be proved, together with a uniform version of Theorem 1.1. For this we use pointwiseversions of Morrey and Campanato spaces and a regularity proposition to obtain ellipticregularity. For the convenience of the reader we repeat in Section 2 the pointwise versions ofMorrey and Campanato spaces as introduced in [ERe]. The regularity proposition containsa new boundary term to handle the Robin operator. The same regularity proposition isthen also used in Section 5 to prove an extension of Theorem 1.3. In Section 4 we provethe lower kernel bounds of Theorem 1.2. In the proof we use that a bounded connectedLipschitz domain satisfies the chain condition. We prove this fact in the appendix. In this section we introduce the classes of operators that we use throughout this paper.Since our proofs also involve Morrey and Campanato seminorms, we include those defini-tions as well.Let Ω ⊂ R d be open bounded with Lipschitz boundary Γ. Let µ, M >
0. Define E p (Ω , µ, M ) to be the set of all measurable C : Ω → R d × d such that k C ( x ) k ≤ M for all x ∈ Ω and satisfy the ellipticity condition d X k,l =1 c kl ( x ) ξ k ξ l ≥ µ | ξ | for all ξ ∈ R d and x ∈ Ω. Here k C ( x ) k is the ℓ -norm of C ( x ) in C d . Let E p (Ω) = S µ,M> E p (Ω , µ, M ).Let E (Ω , µ, M ) be the set of all tuples ( C, a, b, a ), where C ∈ E p (Ω , µ, M ), a, b : Ω → C d measurable and a : Ω → C measurable with k a ( x ) k , k b ( x ) k , | a ( x ) | ≤ M for all x ∈ Ω. Let E (Ω) = S µ,M> E (Ω , µ, M ).For all ( C, a, b, a ) ∈ E (Ω) define the closed sectorial forms a p , a : W , (Ω) × W , (Ω) → C by a p ( u, v ) = Z Ω d X k,l =1 c kl ( ∂ k u ) ∂ l v and a ( u, v ) = Z Ω d X k,l =1 c kl ( ∂ k u ) ∂ l v + Z Ω d X k =1 a k ( ∂ k u ) v + Z Ω d X l =1 b l u ∂ l v + Z Ω a u v. Let β : Γ → C be bounded measurable. Define the closed sectorial form a β : W , (Ω) × W , (Ω) → C by a β ( u, v ) = a ( u, v ) + Z Γ β (Tr u ) Tr v It will be clear from the context what are
C, a, b, a and β . Let A be the m-sectorialoperator associated to a β . We denote by S the semigroup generated by − A . We next showthat A is an elliptic operator with Robin boundary conditions. In order to describe thedomain of A , we need the notion of a weak co-normal derivative.Define the operator A : W , (Ω) → ( W , (Ω)) ∗ by hA u, v i ( W , (Ω)) ∗ × W , (Ω) = a ( u, v ) . u ∈ W , (Ω) and suppose that A u ∈ L (Ω). Let ψ ∈ L (Γ). Then we say that ψ is a weak co-normal derivative of u if a ( u, v ) − ( A u, v ) L (Ω) = ( ψ, Tr v ) L (Γ) for all v ∈ W , (Ω). Then ψ is unique by the Stone–Weierstraß theorem and we write ∂ ν u = ψ . If u and Ω are smooth enough, then ∂ ν u = P dk,l =1 ν l c kl ∂ k u + P dk =1 ν k b k u . Lemma 2.1. dom( A β ) = { u ∈ W , (Ω) : A u ∈ L (Ω) and ∂ ν u + Tr u = 0 } . If u ∈ dom( A β ) , then A β u = A u . Proof . The easy proof is left to the reader.Let κ ∈ (0 , C κ (Ω) is the space of all H¨older continuous functions of order κ on Ω with semi-norm ||| u ||| C κ (Ω) = sup { | u ( x ) − u ( y ) || x − y | κ : x, y ∈ Ω , < | x − y | ≤ } . It is a Banach space with the norm k u k C κ (Ω) = ||| u ||| C κ (Ω) + k u k L ∞ (Ω) .Finally we introduce the pointwise Morrey and Campanato semi-norms as in [ERe].Let Ω ⊂ R d be open. For all x ∈ R d and r > x, r ) = Ω ∩ B ( x, r ). For all γ ∈ [0 , d ], R e ∈ (0 ,
1] and x ∈ Ω define k · k
M,γ,x, Ω ,R e : L (Ω) → [0 , ∞ ] by k u k M,γ,x, Ω ,R e = sup r ∈ (0 ,R e ] (cid:16) r − γ Z Ω( x,r ) | u | (cid:17) / . Next, for all γ ∈ [0 , d + 2], R e ∈ (0 ,
1] and x ∈ Ω define ||| · ||| M ,γ,x, Ω ,R e : L (Ω) → [0 , ∞ ] by ||| u ||| M ,γ,x, Ω ,R e = sup r ∈ (0 ,R e ] (cid:16) r − γ Z Ω( x,r ) | u − h u i Ω( x,r ) | (cid:17) / , where for an L -function v we denote by h v i D = | D | R D v the average of v over a boundedmeasurable subset D of the domain of v with | D | > E = { x = (˜ x, x d ) : − < x d < k ˜ x k R d − < } the open cylinder in R d , the lower half by E − = { x ∈ E : x d < } and its mid plate by P = E ∩ { x ∈ R d : x d = 0 } . We emphasise that the field is the complex numbers and all our functions are complexvalued, except when explicitly stated otherwise.
In this section we aim to prove a uniform version of Theorem 1.1. The main tool is thefollowing regularity result. For this section one may choose ˜ γ = γ + δ . In Section 5 we willuse the same proposition, but then the choice ˜ γ = γ + δ does not work in the case d = 2and d = 3. In order not to repeat the major part of the proof, we prove a bit more in thenext proposition. 5 roposition 3.1. Let Ω ⊂ R d be open bounded with Lipschitz boundary Γ . Let U ⊂ R d be an open set and Φ a bi-Lipschitz map from an open neighbourhood of U onto an opensubset of R d such that Φ( U ) = E and Φ(Ω ∩ U ) = E − . Then for all µ, M > there exists a κ ∈ (0 , such that for all γ, ˜ γ ∈ [0 , d ) and δ ∈ (0 , with γ + δ < d − κ and γ + δ ≤ ˜ γ there exists a c > such that the following is valid. Let C ∈ E p (Ω , µ, M ) , u, g ∈ W , (Ω) , β ∈ L ∞ (Γ) and f, f , . . . , f d ∈ L (Ω) . Suppose that a p ( u, v ) = ( f, v ) L (Ω) + d X i =1 ( f i , ∂ i v ) L (Ω) + Z Γ β Tr g Tr v (1) for all v ∈ W , (Ω) . Then k∇ ( u ◦ Φ − ) k M,γ + δ,x,E − , ≤ c (cid:16) ε − δ k f ◦ Φ − k M,γ,x,E − , + d X i =1 k f i ◦ Φ − k M,γ + δ,x,E − , + ε − ( γ + δ ) k∇ u k L (Ω) + ε − δ k β k L ∞ (Γ) k∇ ( g ◦ Φ − ) k M,γ,x,E − , + ε ˜ γ − γ − δ k β k L ∞ (Γ) k g ◦ Φ − k M, ˜ γ,x,E − , (cid:17) for all x ∈ E − and ε ∈ (0 , . Proof . The proof is a modification of the proof of [ERe] Proposition 6.5. We indicate thedifferences and use the notation as in [ERe].Let K ∈ [1 , ∞ ) be larger than the Lipschitz constant of Φ | Ω ∩ U and Φ − | E − . The tracemap is continuous from W , (Ω) into L (Γ) by [Neˇc] Theorem 2.4.2. Hence there exists a c ≥ k Tr v k L (Γ) ≤ c k v k W , (Ω) for all v ∈ W , (Ω).After composition with Φ the equation (1) transforms to an equation on E − with theaid of [ERe] Proposition 4.3 with a form with measurable coefficients ˜ c kl . For all x ∈ E − and 0 < R ≤ define P ( x, R ) = P ∩ B ( x, R ). Recall that E − ( x, R ) = E − ∩ B ( x, R ).Let W , P ( x,R ) ( E − ( x, R )) be the closure in W , ( E − ( x, R )) of the space { w | E − ( x,R ) : w ∈ C ∞ c ( R d ) and supp w ∩ ( ∂ ( E − ( x, R )) \ P ( x, R )) = ∅} . By the De Giorgi estimates of [ERe]Lemma 5.1 there exist κ ∈ (0 ,
1) and c DG > Z E − ( x,r ) |∇ w | ≤ c DG (cid:16) rR (cid:17) d − κ Z E − ( x,R ) |∇ w | for all x ∈ E − , r, R ∈ (0 ,
1] and w ∈ W , ( E − ( x, R )) such that r ≤ R and d X k,l =1 Z E − ( x,R ) ˜ c kl ( ∂ k w ) ∂ l v = 0for all v ∈ W , x,R ) (Ω( x, R )).Let x ∈ E − and 0 < R ≤ . By the Dirichlet-type Poincar´e inequality of [ERe]Lemma 6.1(b) and the Lax–Milgram theorem there exists a unique ˜ v ∈ W , P ( x,R ) ( E − ( x, R ))such that d X k,l =1 Z E − ( x,R ) ˜ c kl ( ∂ k ˜ v ) ∂ l τ = d X k,l =1 Z E − ( x,R ) ˜ c kl ( ∂ k ( u ◦ Φ − )) ∂ l τ τ ∈ W , P ( x,R ) ( E − ( x, R )). Define v : Ω → C by v ( y ) = ( ˜ v (Φ( y )) if y ∈ Φ − ( E − ( x, R )) , y ∈ Ω \ Φ − ( E − ( x, R )) . Then v ∈ W , (Ω) by in [ERe] Lemma 6.4. Moreover, d X k,l =1 Z E − ( x,R ) ˜ c kl ( ∂ k ˜ v ) ∂ l ˜ v = d X k,l =1 Z E − ( x,R ) ˜ c kl ( ∂ k ( u ◦ Φ − )) ∂ l ˜ v = d X k,l =1 Z Ω ∩ U c kl ( ∂ k u ) ∂ l v = a p ( u, v ) = ( f, v ) L (Ω) + d X i =1 ( f i , ∂ i v ) L (Ω) + Z Γ β Tr g Tr v, where the last term in the last step is new. Using ellipticity and the Cauchy–Schwartzinequality, one obtains( d ! K d +2 ) − µ Z E − ( x,R ) |∇ ˜ v | ≤ d ! K d (cid:16) Z E − ( x,R ) | f ◦ Φ − | (cid:17) / (cid:16) Z E − ( x,R ) | ˜ v | (cid:17) / + d ! K d +1 d X i =1 (cid:16) Z E − ( x,R ) | f i ◦ Φ − | (cid:17) / (cid:16) Z E − ( x,R ) | ∂ i ˜ v | (cid:17) / + (cid:12)(cid:12)(cid:12) Z Γ β Tr g Tr v (cid:12)(cid:12)(cid:12) ≤ d ! K d k f ◦ Φ − k M,γ,x,E − , R ( γ +2) / (cid:16) Z E − ( x,R ) |∇ ˜ v | (cid:17) / + d ! K d +1 d X i =1 k f i ◦ Φ − k M,γ + δ,x,E − , R ( γ + δ ) / (cid:16) Z E − ( x,R ) |∇ ˜ v | (cid:17) / + (cid:12)(cid:12)(cid:12) Z Γ β Tr g Tr v (cid:12)(cid:12)(cid:12) , where we used the Dirichlet-type Poincar´e inequality of [ERe] Lemma 6.1(b) in the laststep. We next estimate the boundary integral. The boundedness of the trace gives (cid:12)(cid:12)(cid:12) Z Γ β Tr g Tr v (cid:12)(cid:12)(cid:12) ≤ c k β k L ∞ (Γ) k g v k W , (Ω) ≤ c k β k L ∞ (Γ) Z Ω (cid:16) | g | | v | + |∇ g | | v | + | g | |∇ v | (cid:17) ≤ c d ! d K d +1 k β k L ∞ (Γ) Z E − ( x,R ) (cid:16) | g ◦ Φ − | | ˜ v | + |∇ ( g ◦ Φ − ) | | ˜ v | + | g ◦ Φ − | |∇ ˜ v | (cid:17) ≤ c d ! d K d +1 k β k L ∞ (Γ) (cid:16) R ˜ γ/ k g ◦ Φ − k M, ˜ γ,x,E − , · R + R γ/ k∇ ( g ◦ Φ − ) k M,γ,x,E − , · R + R ˜ γ/ k g ◦ Φ − k M, ˜ γ,x,E − , (cid:17)(cid:16) Z E − ( x,R ) |∇ ˜ v | (cid:17) / c d ! d K d +1 k β k L ∞ (Γ) (cid:16) R (˜ γ − γ − δ ) / k g ◦ Φ − k M, ˜ γ,x,E − , + 2 R (2 − δ ) / k∇ ( g ◦ Φ − ) k M,γ,x,E − , (cid:17) · R ( γ + δ ) / (cid:16) Z E − ( x,R ) |∇ ˜ v | (cid:17) / , where we used again the Dirichlet-type Poincar´e inequality of [ERe] Lemma 6.1(b). So Z E − ( x,R ) |∇ ˜ v | ≤ (3 c d ! d K d +3 ) µ − (cid:16) R (2 − δ ) / k f ◦ Φ − k M,γ,x,E − , + d X i =1 k f i ◦ Φ − k M,γ + δ,x,E − , + R (˜ γ − γ − δ ) / k β k L ∞ (Γ) k g ◦ Φ − k M, ˜ γ,x,E − , + R (2 − δ ) / k β k L ∞ (Γ) k∇ ( g ◦ Φ − ) k M,γ,x,E − , (cid:17) R γ + δ . Next let r ∈ (0 , R ]. We apply the De Giorgi estimates to the function w = u − v . Then Z E − ( x,r ) |∇ ( u ◦ Φ − ) | ≤ Z E − ( x,r ) |∇ ( w ◦ Φ − ) | + 2 Z E − ( x,r ) |∇ ˜ v | ≤ c DG (cid:16) rR (cid:17) d − κ Z E − ( x,R ) |∇ ( w ◦ Φ − ) | + 2 Z E − ( x,r ) |∇ ˜ v | ≤ c DG (cid:16) rR (cid:17) d − κ Z E − ( x,R ) |∇ ( u ◦ Φ − ) | + (2 + 4 c DG ) Z E − ( x,R ) |∇ ˜ v | ≤ c DG (cid:16) rR (cid:17) d − κ Z E − ( x,R ) |∇ ( u ◦ Φ − ) | + c (cid:16) R (2 − δ ) / k f ◦ Φ − k M,γ,x,E − , + d X i =1 k f i ◦ Φ − k M,γ + δ,x,E − , + R (˜ γ − γ − δ ) / k β k L ∞ (Γ) k g ◦ Φ − k M, ˜ γ,x,E − , + R (2 − δ ) / k β k L ∞ (Γ) k∇ ( g ◦ Φ − ) k M,γ,x,E − , (cid:17) R γ + δ , where c = (2 + 4 c DG )(3 c d ! d K d +3 M ) µ − . Note that these bounds are uniform for all x ∈ E − and 0 < r ≤ R ≤ . Moreover, γ + δ < d − κ . Hence they can be improvedby use of Lemma III.2.1 of [Gia] and one deduces that there exists a c >
0, dependingonly on c DG , γ + δ and d − κ , such that Z E − ( x,r ) |∇ ( u ◦ Φ − ) | ≤ c (cid:16) rR (cid:17) γ + δ Z E − ( x,R ) |∇ ( u ◦ Φ − ) | + c c (cid:16) ε − δ k f ◦ Φ − k M,γ,x,E − , + d X i =1 k f i ◦ Φ − k M,γ + δ,x,E − , + ε ˜ γ − γ − δ k β k L ∞ (Γ) k g ◦ Φ − k M, ˜ γ,x,E − , + ε − δ k β k L ∞ (Γ) k∇ ( g ◦ Φ − ) k M,γ,x,E − , (cid:17) r γ + δ , x ∈ E − , ε ∈ (0 ,
1] and 0 < r ≤ R ≤ ε . Choosing R = ε gives Z E − ( x,r ) |∇ ( u ◦ Φ − ) | ≤ γ + δ c ( ε − ( γ + δ ) k∇ ( u ◦ Φ − ) k L ( E − ) ) r γ + δ + c c (cid:16) ε − δ k f ◦ Φ − k M,γ,x,E − , + d X i =1 k f i ◦ Φ − k M,γ + δ,x,E − , + ε ˜ γ − γ − δ k β k L ∞ (Γ) k g ◦ Φ − k M, ˜ γ,x,E − , + ε − δ k β k L ∞ (Γ) k∇ ( g ◦ Φ − ) k M,γ,x,E − , (cid:17) r γ + δ , for all x ∈ E − and 0 < r ≤ ε .The rest of the proof is similarly to the proof of [ERe] Proposition 6.5, which is amodification of the proof of Proposition 3.2 in [ERe].We also need the Davies perturbation. Let D = { ψ ∈ C ∞ c ( R d , R ) : k∇ ψ k ∞ ≤ } . For all ρ ∈ R and ψ ∈ D define the multiplication operator U ρ by U ρ u = e − ρ ψ u . Note that U ρ u ∈ W , (Ω) for all u ∈ W , (Ω). Let S ρt = U ρ S t U − ρ be the Davies perturbation for all t >
0. Let − A ( ρ ) the generator of S ρ . Then A ( ρ ) is the operator associated with the form a ( ρ ) β with form domain D ( a ( ρ ) β ) = W , (Ω) and a ( ρ ) β ( u, v ) = a p ( u, v ) + Z Ω d X k =1 (cid:16) a ( ρ ) k ( ∂ k u ) v + b ( ρ ) k u ( ∂ k v ) (cid:17) + Z Ω a ( ρ )0 u v + Z Γ β (Tr u ) Tr v (2)with a ( ρ ) k = a k − ρ d X l =1 c kl ∂ l ψ , b ( ρ ) k = b k + ρ d X l =1 c lk ∂ l ψ and a ( ρ )0 = a − ρ d X k,l =1 c kl ( ∂ k ψ ) ∂ l ψ + ρ d X k =1 a k ∂ k ψ − ρ d X k =1 b k ∂ k ψ. Lemma 3.2.
Let Ω ⊂ R d be open bounded with Lipschitz boundary Γ . Then for all µ, M > , there exist c , ω > such that k S ρt u k L (Ω) ≤ e ω (1+ ρ ) t k u k L (Ω) k∇ S ρt u k L (Ω) ≤ c t − / e ω (1+ ρ ) t k u k L (Ω) k A ( ρ ) S ρt u k L (Ω) ≤ c t − e ω (1+ ρ ) t k u k L (Ω) for all ( C, a, b, a ) ∈ E (Ω , µ, M ) , β ∈ L ∞ (Γ) , t > , ρ ∈ R and ψ ∈ D with k β k L ∞ (Γ) ≤ M ,where S ρ is the semigroup generated by − A ( ρ ) . roof . Without lost of generality we may assume that µ ≤
1. By [Neˇc] Theorem 2.4.2there exists a c > k Tr v k L (Γ) ≤ c k v k W , (Ω) for all v ∈ W , (Ω). Let u ∈ L (Ω). Then the boundary term can be estimated by | Z Γ β (Tr S ρt u ) Tr S ρt u | ≤ c k β k L ∞ (Γ) k ( S ρt u ) S ρt u k W , (Ω) ≤ c ( k S ρt u k L (Ω) + 2 k∇ S ρt u k L (Ω) k S ρt u k L (Ω) ) , where c = c M . Now by ellipticity µ k∇ S ρt u k L (Ω) ≤ Re a p ( S ρt u ) ≤ Re a ( ρ ) β ( S ρt u ) + 2 d M (1 + | ρ | ) k∇ S ρt u k L (Ω) k S ρt u k L (Ω) + M (1 + | ρ | ) k S ρt u k L (Ω) + c ( k S ρt u k + 2 k∇ S ρt u k L (Ω) k S ρt u k L (Ω) ) ≤ Re a ( ρ ) β ( S ρt u ) + 2(1 + | ρ | )( d M + c ) k∇ S ρt u k L (Ω) k S ρt u k L (Ω) + (1 + | ρ | ) ( M + c ) k S ρt u k L (Ω) ≤ Re a ( ρ ) β ( S ρt u ) + µ k∇ S ρt u k L (Ω) + ω (1 + ρ ) k S ρt u k L (Ω) for all t >
0, where ω = µ ( d M + c ) + 2( M + c ). Therefore12 µ k∇ S ρt u k L (Ω) ≤ Re a ( ρ ) β ( S ρt u ) + ω (1 + ρ ) k S ρt u k L (Ω) . Differentiating gives ddt k S ρt u k L (Ω) = − A ( ρ ) S ρt u, S ρt u ) L (Ω) = − a ( ρ ) β ( S ρt u ) ≤ ω (1 + ρ ) k S ρt u k L (Ω) . Hence by Gronwall’s lemma k S ρt u k L (Ω) ≤ e ω (1+ ρ ) t k u k L (Ω) . The estimates for k A ( ρ ) S t u k L (Ω) and k∇ S t u k L (Ω) follows from [ERe] Lemma 7.1.We next consider the L → L ∞ and H¨older estimates for the semigroup near Γ. Proposition 3.3.
Let Ω ⊂ R d be open bounded with Lipschitz boundary Γ . Let U ⊂ R d and Φ be a bi-Lipschitz map from an open neighbourhood of U onto an open subset of R d such that Φ( U ) = E and Φ(Ω ∩ U ) = E − . Then for all µ, M > there exist κ ∈ (0 , and c, ω > such that the following is valid. Let ( C, a, b, a ) ∈ E (Ω , µ, M ) and β ∈ L ∞ (Γ) with k β k L ∞ (Γ) ≤ M . Let S be the semigroup generated by − A . Then k S ρt u k L ∞ (Φ − ( E − )) ≤ c t − d/ e ω (1+ ρ ) t k u k L (Ω) and | ( S ρt u )( x ) − ( S ρt u )( y ) | ≤ c t − d/ t − κ/ e ω (1+ ρ ) t k u k L (Ω) | x − y | κ for all t > , u ∈ L (Ω) , ρ ∈ R , ψ ∈ D and x, y ∈ Φ − ( E − ) with | x − y | ≤ K , where K > is larger than the Lipschitz constant of Φ | Ω ∩ U and Φ − | E − . roof . Let µ, M > κ ∈ (0 ,
1) be as in Proposition 3.1. For all γ ∈ [0 , d − κ )let P ( γ ) be the hypothesisThere exist c, ω >
0, depending only on K , µ , M , κ and c DG , suchthat k ( S ρt u ) ◦ Φ − k M,γ,x,E − , ≤ c t − γ/ e ω (1+ ρ ) t k u k L (Ω) (3)and k∇ (( S ρt u ) ◦ Φ − ) k M,γ,x,E − , ≤ c t − γ/ t − / e ω (1+ ρ ) t k u k L (Ω) (4)for all t > u ∈ L (Ω), ρ ∈ R , ψ ∈ D and x ∈ E − .Clearly P (0) is valid by Lemma 3.2. Lemma 3.4.
Let γ ∈ [0 , d − κ ) and suppose that P ( γ ) is valid. Let δ ∈ (0 , andsuppose that γ + δ < d − κ . Then P ( γ + δ ) is valid. Proof . Let c , ω > t > u ∈ L (Ω), ρ ∈ R , ψ ∈ D and x ∈ E − . Note that k ( S ρt u ) ◦ Φ − k L ( E − ) ≤ d ! K d k S ρt u k L (Ω) ≤ d ! K d e ω (1+ ρ ) t k u k L (Ω) , (5)by Lemma 3.2.We first prove the bounds (3) for P ( γ + δ ). Choose ε = t / e − t ∈ (0 , c be as in[ERe] Lemma 6.2. Then it follows from [ERe] Lemma 6.2, (4) and (5) that k ( S ρt u ) ◦ Φ − k M ,γ + δ,x,E − , ≤ c ( ε − δ k∇ (( S ρt u ) ◦ Φ − ) k M,γ,x,E − , + ε − ( γ + δ ) k ( S ρt u ) ◦ Φ − k L ( E − ) ) ≤ c ( ε − δ c t − γ/ t − / e ω (1+ ρ ) t + ε − ( γ + δ ) d ! K d e ω (1+ ρ ) t ) k u k L (Ω) ≤ c ′ t − ( γ + δ ) / e ω ′ (1+ ρ ) t k u k L (Ω) where c ′ = c ( c + d ! K d ) and ω ′ = ω + ω + γ + δ . By [ERe] Lemma 3.1(a) there exist c , c > k v k M,γ + δ,x,E − , ≤ c k v k M ,γ + δ,x,E − , + c k v k L ( E − ) for all x ∈ E − and v ∈ L ( E − ). Hence k ( S ρt u ) ◦ Φ − k M,γ + δ,x,E − , ≤ c c ′ t − ( γ + δ ) / e ω ′ (1+ ρ ) t k u k L (Ω) + c d ! K d e ω (1+ ρ ) t k u k L (Ω) ≤ c ′′ t − ( γ + δ ) / e ω ′′ (1+ ρ ) t k u k L (Ω) , (6)where c ′′ = c ′ c + c d ! K d and ω ′′ = ω + ω ′ + d + 2. This gives the bound (3) for P ( γ + δ ).In order to obtain (4), we use Proposition 3.1. Note that a ( ρ ) β ( S ρt u, v ) = ( S ρt/ A ( ρ ) S ρt/ u, v ) L (Ω) v ∈ W , (Ω). It follows from (2) that a p ( S ρt u, v ) = ( f, v ) L (Ω) − d X i =1 ( f i , ∂ i v ) L (Ω) − Z Γ β (Tr S ρt u ) Tr v for all v ∈ W , (Ω), where f i = b ( ρ ) i S t u and f = S ρt/ A ( ρ ) S ρt/ u − a ( ρ )0 S ρt u − d X i =1 a ( ρ ) i ∂ i S ρt u. Apply Proposition 3.1 with ε = t / e − t ∈ (0 , f are approximatedseparately using Lemma 3.2 with ˜ γ = γ + δ . First, ε − δ k ( S ρt/ A ( ρ ) S ρt/ u ) ◦ Φ − k M,γ,x,E − , ≤ t (2 − δ ) / c ( t/ − γ/ e ω (1+ ρ ) t/ k A ( ρ ) S ρt/ u k L (Ω) ≤ c ( t/ − e ω (1+ ρ ) t/ t (2 − δ ) / c ( t/ − γ/ e ω (1+ ρ ) t/ k u k L (Ω) ≤ γ/ c c t − ( γ + δ ) / t − / e ( ω + ω )(1+ ρ ) t k u k L (Ω) . Secondly, ε − δ k ( a ( ρ )0 S ρt u ) ◦ Φ − k M,γ,x,E − , ≤ t (2 − δ ) / M (1 + ρ ) c t − γ/ e ω (1+ ρ ) t k u k L (Ω) ≤ c M t − ( γ + δ ) / t − / e ( ω +1)(1+ ρ ) t k u k L (Ω) . Thirdly, ε − δ k ( d X i =1 a ( ρ ) i ∂ i S t u ) ◦ Φ − k M,γ,x,E − , ≤ t (2 − δ ) / M (1 + | ρ | ) k ( ∇ S ρt u ) ◦ Φ − k M,γ,x,E − , ≤ t (2 − δ ) / M t − / e (1+ ρ ) t K k∇ (( S ρt u ) ◦ Φ − ) k M,γ,x,E − , ≤ c K M t − ( γ + δ ) / t − / e ( ω +1)(1+ ρ ) t k u k L (Ω) . The terms with f i in Proposition 3.1 can be estimated by d X i =1 k ( b ( ρ ) i S t u ) ◦ Φ − k M,γ + δ,x,E − , ≤ d M (1 + | ρ | ) k ( S ρt u ) ◦ Φ − k M,γ + δ,x,E − , ≤ d M c ′′ t − ( γ + δ ) / t − / e (1+ ρ ) t e ω ′′ (1+ ρ ) t k u k L (Ω) , where we used (6) in the last step. Next, ε − ( γ + δ ) k∇ S ρt u k L (Ω) ≤ t − ( γ + δ ) / e ( γ + δ ) t c t − / e ω (1+ ρ ) t k u k L (Ω) ≤ c t − ( γ + δ ) / t − / e ( ω + d +2)(1+ ρ ) t k u k L (Ω) . ε − δ k∇ (( S ρt u ) ◦ Φ − ) k M,γ,x,E − , ≤ t (2 − δ ) / c t − γ/ t − / e ω (1+ ρ ) t k u k L (Ω) ≤ c t ( γ + δ ) / t − / e ( ω +1)(1+ ρ ) t k u k L (Ω) and k ( S ρt u ) ◦ Φ − k M, ˜ γ,x,E − , ≤ c ′′ t − ( γ + δ ) / e ω ′′ (1+ ρ ) t k u k L (Ω) ≤ c ′′ t − ( γ + δ ) / t − / e ( ω ′′ +1)(1+ ρ ) t k u k L (Ω) , where (4) and (6) are used. Now (4) for P ( γ + δ ) follows from Proposition 3.1. End of proof of Proposition 3.3.
This follows as at the end of the proof of Proposi-tion 7.2 in [ERe].The part of Ω away from Γ can be estimated by interior regularity.
Proposition 3.5.
Let Ω ⊂ R d be open bounded with Lipschitz boundary Γ . Let ζ > .Then for all µ, M > there exist κ ∈ (0 , and c, ω > such that the following is valid.Let ( C, a, b, a ) ∈ E (Ω , µ, M ) and β ∈ L ∞ (Γ) with k β k L ∞ (Γ) ≤ M . Then k S ρt u k L ∞ (Ω ζ ) ≤ c t − d/ e ω (1+ ρ ) t k u k L (Ω) and | ( S ρt u )( x ) − ( S ρt u )( y ) | ≤ c t − d/ t − κ/ e ω (1+ ρ ) t k u k L (Ω) | x − y | κ for all t > , u ∈ L (Ω) , ρ ∈ R , ψ ∈ D and x, y ∈ Ω ζ with | x − y | ≤ , where Ω ζ = { x ∈ Ω : d ( x, Γ) > ζ } . Proof . This follows similarly to the proof of Proposition 3.3 using [ERe] Proposition 3.2instead of Proposition 3.1.We can now prove Gaussian H¨older kernel bounds for second-order operators withcomplex lower-order coefficients and complex Robin boundary conditions.
Theorem 3.6.
Let Ω ⊂ R d be open bounded with Lipschitz boundary Γ . Then for all µ, M, τ > and τ ′ ∈ (0 , there exist κ ∈ (0 , and b, c, ω > such that the followingis valid. Let ( C, a, b, a ) ∈ E (Ω , µ, M ) and β ∈ L ∞ (Γ) with k β k L ∞ (Γ) ≤ M . Let S be thesemigroup generated by − A . Then S has a kernel K . Moreover, | K t ( x, y ) | ≤ c t − d/ e − b | x − y | t e ωt and | K t ( x, y ) − K t ( x ′ , y ′ ) | ≤ c t − d/ (cid:16) | x − x ′ | + | y − y ′ | t / + | x − y | (cid:17) κ e − b | x − y | t e ωt for all x, x ′ , y, y ′ ∈ Ω and t > with | x − x ′ | + | y − y ′ | ≤ τ t / + τ ′ | x − y | . roof . By a compactness argument it follows from Propositions 3.3 and 3.5 that thereexist δ, κ ∈ (0 ,
1) and c, ω > | ( S ρt u )( x ) | ≤ c t − d/ e ω (1+ ρ ) t k u k L (Ω) and | ( S ρt u )( x ) − ( S ρt u )( y ) | ≤ c t − d/ t − κ/ e ω (1+ ρ ) t k u k L (Ω) | x − y | κ for all u ∈ L (Ω), t > ρ ∈ R and ψ ∈ D with | x − y | < δ . Then the H¨older Gaussiankernel bounds follow as in the proof of Lemma A.1 in [EO1]. Corollary 3.7.
For all t > let T t : C (Ω) → C (Ω) be the restriction of S t to C (Ω) . Then T is a holomorphic C -semigroup. Proof . This follows as in the proof of Theorem 4.3 in [Nit], with the use of Theorem 3.6.(Note that there is a gap in the proof of Theorem 4.3 in [Nit] in case the condition β ≥ S is irreducible in the following sense. Proposition 3.8.
Suppose that Ω is connected. Let Ω ⊂ Ω be measurable. Suppose that S t L (Ω ) ⊂ L (Ω ) . Then | Ω | = 0 or | Ω \ Ω | = 0 . Proof . It follows from [Ouh] Theorem 2.2 that Ω u ∈ W , (Ω) for all u ∈ W , (Ω). Thenthe proposition follows by the discussion on page 106 in [Ouh].Also the semigroup on C (Ω) is irreducible. Proposition 3.9.
Suppose that Ω is connected. Let T be the C -semigroup on C (Ω) asin Corollary 3.7. Let F ⊂ Ω be closed and suppose that T t I ⊂ I for all t > , where I = { u ∈ C (Ω) : u | F = 0 } . Then F = ∅ or F = Ω . Proof . Suppose that F = ∅ and F = Ω. Define f ∈ C (Ω) by f ( x ) = d ( x, F ). Let t > x ∈ F . If τ ∈ C (Ω), then f τ ∈ I , so 0 = ( T t ( f τ ))( x ) = R Ω K t ( x, y ) f ( y ) τ ( y ) dy .Hence K t ( x, y ) f ( y ) = 0 for almost every y ∈ Ω and by continuity for all y ∈ Ω. Therefore K t ( x, y ) = 0 for all y ∈ Ω \ F and by continuity for all y ∈ Ω \ F , where the closure is in R d . Let F ◦ denote the interior of F in R d . It is elementary to show that Ω \ ( F ◦ ) ⊂ Ω \ F .Hence we proved that K t ( x, y ) = 0 for all x ∈ F , y ∈ Ω \ ( F ◦ ) and t > J = { u ∈ L (Ω) : u | F = 0 a.e. } . If u ∈ J , t > x ∈ F , then( S t u )( x ) = Z Ω K t ( x, y ) u ( y ) dy = Z Ω \ F K t ( x, y ) u ( y ) dy = 0 . So S t J ⊂ J for all t >
0. Since S is irreducible by Proposition 3.8, it follows that | F | = 0or | Ω \ F | = 0. Since F = Ω there exists an x ∈ Ω and r > B ( x, r ) ⊂ R d \ F .Then 0 < | Ω( x, r ) | ≤ | Ω \ F | . Hence | F | = 0. Then also | F ◦ | = 0 and consequently F ◦ = ∅ .Therefore Ω \ ( F ◦ ) = Ω. It follows that K t ( x, y ) = 0 for all t > x ∈ F and y ∈ Ω, andthen by continuity for all y ∈ Ω. Then 1 = lim t ↓ ( T t Ω )( x ) = lim t ↓ R Ω K t ( x, y ) dy = 0 forall x ∈ F . This is a contradiction since F = ∅ .14 Lower kernel bounds
In this short section we prove the Gaussian lower bounds of Theorem 1.2. The generaloutline is standard. We first show on-diagonal lower bounds for small time. Secondly weuse the H¨older Gaussian upper bounds to obtain lower bounds close to the diagonal forsmall time. Finally we use the semigroup property together with the chain condition toprove Gaussian lower bounds.Adopt the notation and assumption of Theorem 1.2. Let T be the C -semigroup in C (Ω) as in Corollary 3.7. Then lim t ↓ k T t Ω − Ω k C (Ω) = 0. Hencelim t ↓ sup x ∈ Ω (cid:12)(cid:12)(cid:12) − Z Ω K t ( x, y ) dy (cid:12)(cid:12)(cid:12) = 0 . It follows from [ERo] Theorem 2.1 that there are c , c , t > K t ( x, y ) ≥ c t − d/ for all x, y ∈ Ω and t ∈ (0 , t ] with | x − y | ≤ c t / . Without loss of generality we mayassume that t ≤
1. By Proposition A.1 in Appendix A the set Ω satisfies the chaincondition. That is, there exists a c > x, y ∈ Ω and n ∈ N there exist x , . . . , x n ∈ Ω such that x = x , x n = y and | x k +1 − x k | ≤ c | x − y | n for all k ∈ { , . . . , n − } .Since Ω is bounded and Lipschitz, there exists a c > | Ω( x, r ) | ≥ c r d for all x ∈ Ω and r ∈ (0 , x, y ∈ Ω and t >
0. Let n ∈ N be the smallest natural number such that4 c | x − y | c t ≤ n and tt ≤ n. Then n − ≤ c | x − y | c t + tt . (7)By the chain condition there exist x , . . . , x n ∈ Ω such that x = x , x n = y and | x k +1 − x k | ≤ c n | x − y | for all k ∈ { , . . . , n − } . Then the semigroup property gives K t ( x, y ) = Z Ω . . . Z Ω K tn ( x, z ) K tn ( z , z ) . . . K tn ( z n − , z n − ) K tn ( z n − , y ) dz . . . dz n − ≥ Z B ( x , c √ t √ n ) . . . Z B ( x n − , c √ t √ n ) K tn ( x, z ) K tn ( z , z ) . . . K tn ( z n − , z n − ) K tn ( z n − , y ) dz . . . dz n − . If z k ∈ B ( x k , c √ t √ n ) for all k ∈ { , . . . , n − } and we set z = x and z n = x n , then | z k − z k +1 | ≤ | x k − x k +1 | + 2 c √ t √ n ≤ c n | x − y | + c √ t √ n ≤ c n c √ n √ t c + c √ t √ n = c (cid:16) tn (cid:17) / for all k ∈ { , . . . , n − } and tn ≤ t . Hence K tn ( z k , z k +1 ) ≥ c n d/ t − d/ and K t ( x, y ) ≥ (cid:16) c (cid:16) c √ t √ n (cid:17) d (cid:17) n − (cid:16) c n d/ t − d/ (cid:17) n = c ( c c d c ) n − n d/ t − d/ ≥ c ( c c d c ) n − t − d/ . M ∈ [1 , ∞ ) be such that M ≤ c c d c . Then( c c d c ) n − ≥ (cid:16) M (cid:17) n − = e − ( n −
1) log M ≥ e − (log M ) (cid:16) c | x − y | c t + tt (cid:17) , where we used (7). Then Theorem 1.2 follows. In this section we prove uniform estimates and H¨older continuity estimates for the ker-nel of the semigroup generated by minus the Dirichlet-to-Neumann operator on a domainwith Lipschitz boundary. The Dirichlet-to-Neumann operator can be associated to a gen-eral second-order elliptic differential operator in divergence form with real principal co-efficients and complex lower-order coefficients. Combining these estimates with Poissonkernel bounds for operators on C κ -domains and H¨older continuous principal coefficientswe can prove an extension of Theorem 1.3.We first introduce the Dirichlet-to-Neumann operator which is associated with a generalsecond-order elliptic operator.Let Ω ⊂ R d be a bounded Lipschitz domain. For all k, l ∈ { , . . . , d } let c kl : Ω → R and a k , b k , a : Ω → C be bounded and measurable. Suppose there exists a µ > P dk,l =1 c kl ( x ) ξ k ξ l ≥ µ | ξ | for all x ∈ Ω and ξ ∈ C d . As in Section 2 define the form a : W , (Ω) × W , (Ω) → C by a ( u, v ) = Z Ω d X k,l =1 c kl ( ∂ k u ) ∂ l v + Z Ω d X k =1 a k ( ∂ k u ) v + Z Ω d X l =1 b l u ∂ l v + Z Ω a u v. Let A D be the m-sectorial operator in L (Ω) associated with a | W , (Ω) × W , (Ω) . Then A D is an elliptic operator with Dirichlet boundary conditions. Throughout this section weassume that 0 σ ( A D ).Under the above assumptions one can solve the Dirichlet problem. Proposition 5.1.
Let ϕ ∈ Tr W , (Ω) . Then there exists a unique u ∈ W , (Ω) such that A u = 0 and Tr u = ϕ . Proof . See [AE3] Lemma 2.1 (or [BE] Lemma 3.2(a)).We are now able to define the Dirichlet-to-Neumann operator N . Let ϕ, ψ ∈ L (Γ).Then we say that ϕ ∈ dom N and N ϕ = ψ if there exists a u ∈ W , (Ω) such that Tr u = ϕ , A u = 0 and ∂ ν u = ψ . The operator N can be characterised by the form a . Proposition 5.2.
Let ϕ, ψ ∈ L (Γ) . Then the following are equivalent. (i) ϕ ∈ dom N and N ϕ = ψ . (ii) There exists a u ∈ W , (Ω) such that Tr u = ϕ and a ( u, v ) = ( ψ, Tr v ) L (Γ) for all v ∈ W , (Ω) . The proof is left to the reader.If the form a is symmetric, then the operator N is self-adjoint by [AEKS] Theorem 4.5.The non-symmetric extension is as follows. 16 roposition 5.3. The operator N is m-sectorial. Proof . There exist µ , ω > a ( u ) ≥ µ k u k W , (Ω) − ω k u k L (Ω) for all u ∈ W , (Ω). By Proposition 5.1 we can define the map γ D : H / (Γ) → W , (Ω)by γ D ( ϕ ) = u , where u ∈ W , (Ω) is such that A u = 0 and Tr u = ϕ . As in [AE2]Section 2 define V ( a ) = { u ∈ W , (Ω) : a ( u, v ) = 0 for all v ∈ W , (Ω) } . Then V ( a ) isclosed in W , (Ω). If u ∈ W , (Ω), then u = γ D (Tr u ) + ( u − γ D (Tr u )) ∈ V ( a ) + W , (Ω).Therefore W , (Ω) = V ( a ) + W , (Ω). Also V ( a ) ∩ W , (Ω) = { } since 0 ∈ ρ ( A D ). SoTr | V ( a ) : V ( a ) → L (Γ) is injective. By Ehrling’s lemma there exists a c > k u k L (Ω) ≤ µ ω k u k W , (Ω) + c k Tr u k L (Ω) for all u ∈ V ( a ). Then Re a ( u ) ≥ µ k u k W , (Ω) − c ω k Tr u k L (Γ) (8)for all u ∈ V ( a ). Now the statement follows from [AE2] Corollary 2.2 and Proposition 5.2. Remark 5.4.
Propositions 5.1, 5.2 and 5.3 remain valid if the principle coefficients c kl arecomplex valued. The proofs are word-by-word the same.Let T be the semigroup generated by the operator −N . Recall that we assume that0 σ ( A D ). The main result of this section is the following theorem. Theorem 5.5.
Suppose d ≥ . Then there exist κ ∈ (0 , and c, ω > such that T t L (Γ) ⊂ C κ (Γ) , k T t k L (Γ) → C κ (Γ) ≤ c t − d − t − κ e ωt (9) and k T t k L (Γ) → L ∞ (Γ) ≤ c t − d − e ωt (10) for all t > . The bounds (10) easily follows by interpolation of the bounds (9) and the bounds k T t k L (Γ) → L (Γ) ≤ c ′ e ω ′ t . So it remains to prove the H¨older bounds (9).In case d = 2, then we can also prove H¨older bounds, but unfortunately the singularityin t in (9) is not optimal, as we loose an ε . Since the proof is almost the same, we considerthe case d ≥ t ∈ (0 , ∞ ) and ϕ ∈ L (Γ), then T t ϕ ∈ dom( N ). Hence there exists a unique u t,ϕ ∈ W , (Ω) such that Tr u t,ϕ = T t ϕ and a ( u t,ϕ , v ) = ( N T t ϕ, Tr v ) L (Γ) for all v ∈ W , (Ω). The key idea for the proof of (9) is to estimate u t,ϕ .17 emma 5.6. There exist ˜ c , ˜ ω > such that k u t,ϕ k L (Ω) ≤ ˜ c t − / e ˜ ω t k ϕ k L (Γ) and k∇ u t,ϕ k L (Ω) ≤ ˜ c t − / e ˜ ω t k ϕ k L (Γ) for all t > and ϕ ∈ L (Γ) . Proof . As in [AE2] Section 2 define V ( a ) = { u ∈ W , (Ω) : a ( u, v ) = 0 for all v ∈ W , (Ω) } . Let c, µ , ω > µ k u k W , (Ω) ≤ Re a ( u ) + c ω k Tr u k L (Γ) for all u ∈ V ( a ). In particular µ k u t,ϕ k W , (Ω) ≤ Re( N T t ϕ, T t ϕ ) L (Γ) + c ω k T t ϕ k L (Γ) and the lemma follows from the analyticity of the semigroup T .By a compactness argument Theorem 5.5 is a consequence of the next proposition. Proposition 5.7.
Let U ⊂ R d be an open set and Φ a bi-Lipschitz map from an openneighbourhood of U onto an open subset of R d such that Φ( U ) = E and Φ(Ω ∩ U ) = E − . (a) If d ≥ , then there exist c, δ , ω > and κ ∈ (0 , such that | ( T t ϕ )( x ) − ( T t ϕ )( y ) | ≤ c t − d − t − κ e ωt k ϕ k L (Γ) | x − y | κ for all t > , ϕ ∈ L (Γ) and x, y ∈ Γ ∩ Φ − ( E ) with | x − y | ≤ δ . (b) If d = 2 , then for all ε > there exist c, δ , ω > and κ ∈ (0 , such that | ( T t ϕ )( x ) − ( T t ϕ )( y ) | ≤ c t − d − t − κ t − ε e ωt k ϕ k L (Γ) | x − y | κ for all t > , ϕ ∈ L (Γ) and x, y ∈ Γ ∩ Φ − ( E ) with | x − y | ≤ δ . Proof . There exists an
M >
C, a, b, a ) ∈ E (Ω , µ, M ). Let κ ∈ (0 ,
1) be asin Proposition 3.1. Let K ∈ [1 , ∞ ) be larger than the Lipschitz constant of Φ | Ω ∩ U andΦ − | E − . For all γ ∈ [0 , d − κ ) let P ( γ ) be the hypothesisThere exist c γ , ω γ > k∇ ( u t,ϕ ◦ Φ − ) k M,γ,x,E − , ≤ c γ t − γ +12 e ω γ t k ϕ k L (Γ) for all t > ϕ ∈ L (Γ) and x ∈ E − .Clearly P (0) is valid by Lemma 5.6.We need three lemmas. Lemma 5.8.
There exists a c ′ > such that k u k M ,γ + δ,x,E − , ≤ c ′ (cid:16) ε − δ k∇ u k M,γ,x,E − , + ε − ( γ + δ − k∇ u k L ( E − ) (cid:17) for all γ ∈ [0 , d ) and δ ∈ [0 , , ε ∈ (0 , , u ∈ W , ( E − ) and x ∈ E − with γ + δ ≥ . roof . By the Neumann type Poincar´e inequality of [ERe] Lemma 6.1(a) there exists a c > Z E − ( x ,R ) | u − h u i E − ( x ,R ) | ≤ c R Z E − ( x ,R ) |∇ u | for all x ∈ E − , R ∈ (0 , ] and u ∈ W , ( E − ).Now we prove the lemma. If r ∈ (0 , ε ], then r − ( γ + δ ) Z E − ( x,r ) | u − h u i E − ( x,r ) | ≤ c r − δ r − γ Z E − ( x,r ) |∇ u | ≤ c ε − δ ) k∇ u k M,γ,x,E − , Alternatively, Z E − ( x,r ) | u − h u i E − ( x,r ) | ≤ c r Z E − ( x,r ) |∇ u | ≤ γ + δ − ε − γ + δ − k∇ u k L ( E − ) r γ + δ if r ∈ [ ε , ], from which the lemma follows. Lemma 5.9.
Adopt the assumptions and notation of Proposition 5.7. (a) If γ ∈ [0 , ∩ [0 , d ) , then there exist c ′ , ω ′ > such that k u t,ϕ ◦ Φ − k M,γ,x,E − , ≤ c ′ t − / e ω ′ t k ϕ k L (Γ) for all t > , ϕ ∈ L (Γ) and x ∈ E − . (b) Let γ ∈ [0 , d ) and δ ∈ [0 , with γ + δ < d . Suppose that P ( γ ) is valid. Then thereexist c ′ , ω ′ > such that k u t,ϕ ◦ Φ − k M,γ + δ,x,E − , ≤ c ′ t − ∨ ( γ + δ − e ω ′ t k ϕ k L (Γ) for all t > , ϕ ∈ L (Γ) and x ∈ E − . Proof . ‘(a)’. By Lemma 5.6 we may assume that γ >
0. By the second part of [ERe]Lemma 6.2 there exists a c ′ > k u t,ϕ ◦ Φ − k M ,γ,x,E − , ≤ c ′ (cid:16) k∇ ( u t,ϕ ◦ Φ − ) k M, ,x,E − , + k u t,ϕ ◦ Φ − k L ( E − ) (cid:17) ≤ c ′ d ! K d +1 ˜ c t − / e ˜ ω t k ϕ k L (Γ) for all t > ϕ ∈ L (Γ), where ˜ c , ˜ ω > c ′′ , c ′′′ > k u t,ϕ ◦ Φ − k M,γ,x,E − , ≤ c ′′ ( k u t,ϕ ◦ Φ − k M ,γ,x,E − , + k u t,ϕ ◦ Φ − k L ( E − ) ) ≤ c ′′′ t − / e ˜ ω t k ϕ k L (Γ) for all t > ϕ ∈ L (Γ) and x ∈ E − .‘(b)’. By Statement (a) we may assume that γ + δ ≥
2. Let c ′ > ε = t / e − t ∈ (0 , k u t,ϕ ◦ Φ − k M ,γ + δ,x,E − , ≤ c ′ (cid:16) ε − δ k∇ ( u t,ϕ ◦ Φ − ) k M,γ,x,E − , + ε − ( γ + δ − k∇ ( u t,ϕ ◦ Φ − ) k L ( E − ) (cid:17) ≤ c ′ (cid:16) t − δ e − (2 − δ ) t c γ t − γ +12 e ω γ t k ϕ k L (Γ) + t − γ + δ − e ( γ + δ ) t d ! K d +1 ˜ c t − e ˜ ω t k ϕ k L (Γ) (cid:17) ≤ c ′′ t − γ + δ − e ω ′ t k ϕ k L (Γ) t > ϕ ∈ L (Γ) and x ∈ E − , with suitable c ′′ , ω ′ > c ′′′ , c ′′′′ , ω ′′ > k u t,ϕ ◦ Φ − k M,γ + δ,x,E − , ≤ c ′′′ ( k u t,ϕ ◦ Φ − k M ,γ + δ,x,E − , + k u t,ϕ ◦ Φ − k L ( E − ) ) ≤ c ′′′′ t − γ + δ − e ω ′′ t k ϕ k L (Γ) for all t > ϕ ∈ L (Γ) and x ∈ E − , and the lemma follows. Lemma 5.10.
Let γ ∈ [0 , d − κ ) and suppose that P ( γ ) is valid. Let δ ∈ (0 , andsuppose that γ + δ < d − κ . Then one has the following. (a) If d ≥ , then P ( γ + δ ) is valid. (b) If d = 2 , then for all η > there exist c ′ , ω ′ > such that k∇ ( u t,ϕ ◦ Φ − ) k M,γ + δ,x,E − , ≤ c ′ t − γ + δ +12 t − η e ω ′ t k ϕ k L (Γ) for all t > , ϕ ∈ L (Γ) and x ∈ E − . Proof . Without loss of generality we may assume in case d = 2 that 2 κ ≤ − η andhence γ + δ ≤ − η . Define ˜ γ ∈ [ γ + δ, d ) by˜ γ = γ + δ if γ + δ ≥ , γ + δ < d ≥ , − η if d = 2 . Note that ˜ γ ≥ d ≥
3. Let c > β = Γ . Byanalyticity of T there exist ˜ c, ˜ ω > kN T t ϕ k L (Γ) ≤ ˜ c t − e ˜ ωt k ϕ k L (Γ) for all t > ϕ ∈ L (Γ). By Lemma 5.9(b) there exist ˆ c, ˆ ω > k u t,ϕ ◦ Φ − k M,γ,x,E − , ≤ ˆ c t − ∨ ( γ − e ˆ ωt k ϕ k L (Γ) , k u t,ϕ ◦ Φ − k M,γ + δ,x,E − , ≤ ˆ c t − ∨ ( γ + δ − e ˆ ωt k ϕ k L (Γ) and k u t,ϕ ◦ Φ − k M, ˜ γ,x,E − , ≤ ˆ c t − ∨ (˜ γ − e ˆ ωt k ϕ k L (Γ) for all t > ϕ ∈ L (Γ) and x ∈ E − .Let t > ϕ ∈ L (Γ) and x ∈ E − . Since N T t ϕ = T t N T t ϕ it follows that a p ( u t,ϕ , v ) = ( f, v ) L (Ω) + d X k =1 ( f k , ∂ k v ) L (Ω) + Z Γ (Tr u t, N T t ϕ ) Tr v for all v ∈ W , (Ω), where f = − a u t,ϕ − P dk =1 a k ∂ k u t,ϕ and f k = − b k u t,ϕ . HenceProposition 3.1 with the choice ε = t / e − t ∈ (0 ,
1] gives k∇ ( u t,ϕ ◦ Φ − ) k M,γ + δ,x,E − , ≤ c (cid:16) ε − δ k ( a u t,ϕ ) ◦ Φ − k M,γ,x,E − , + ε − δ d X k =1 k ( a k ∂ k u t,ϕ ) ◦ Φ − k M,γ,x,E − , + d X k =1 k ( b k u t,ϕ ) ◦ Φ − k M,γ + δ,x,E − , + ε − ( γ + δ ) k∇ u t,ϕ k L (Ω) + ε − δ k∇ ( u t, N T t ϕ ◦ Φ − ) k M,γ,x,E − , + ε ˜ γ − γ − δ k u t, N T t ϕ ◦ Φ − k M, ˜ γ,x,E − , (cid:17) .
20e estimate the terms.First ε − δ k ( a u t,ϕ ) ◦ Φ − k M,γ,x,E − , ≤ M t − δ ˆ c (2 t ) − ∨ ( γ − e ωt k ϕ k L (Γ) ≤ ˆ c M t − δ t − ∨ ( γ − e ωt k ϕ k L (Γ) = ˆ c M t − γ + δ +12 t ∧ γ )2 e ωt k ϕ k L (Γ) ≤ ˆ c M t − γ + δ +12 e (2ˆ ω +1+ γ ) t k ϕ k L (Γ) . Secondly, the induction hypothesis P ( γ ) gives ε − δ d X k =1 k ( a k ∂ k u t,ϕ ) ◦ Φ − k M,γ,x,E − , ≤ t − δ d M k ( ∇ u t,ϕ ) ◦ Φ − k M,γ,x,E − , ≤ d M t − δ K k∇ ( u t,ϕ ◦ Φ − ) k M,γ,x,E − , ≤ d K M t − δ c γ (2 t ) − γ +12 e ω γ t k ϕ k L (Γ) = c γ d K M t − γ + δ +12 t e ω γ t k ϕ k L (Γ) ≤ c γ d K M t − γ + δ +12 e (2 ω γ +1) t k ϕ k L (Γ) . Thirdly, d X k =1 k ( b k u t,ϕ ) ◦ Φ − k M,γ + δ,x,E − , ≤ d M k u t,ϕ ◦ Φ − k M,γ + δ,x,E − , ≤ d M ˆ c (2 t ) − ∨ ( γ + δ − e ωt k ϕ k L (Γ) ≤ ˆ c d M t − γ + δ +12 t ∧ ( γ + δ )2 e ωt k ϕ k L (Γ) ≤ ˆ c d M t − γ + δ +12 e (2ˆ ω +1+ γ + δ ) t k ϕ k L (Γ) . Fourthly, ε − ( γ + δ ) k∇ u t,ϕ k L (Ω) ≤ t − γ + δ e ( γ + δ ) t ˜ c (2 t ) − / e ω t k ϕ k L (Γ) ≤ ˜ c t − γ + δ +12 e (2˜ ω + γ + δ ) t k ϕ k L (Γ) , where ˜ c , ˜ ω > ε − δ k∇ ( u t, N T t ϕ ◦ Φ − ) k M,γ,x,E − , ≤ t − δ M c γ t − γ +12 e ω γ t kN T t ϕ k L (Γ) ≤ t − δ M c γ t − γ +12 e ω γ t ˜ c t − e ˜ ωt k ϕ k L (Γ) = ˜ c c γ M t − γ + δ +12 e ( ω γ +˜ ω ) t k ϕ k L (Γ) . ε ˜ γ − γ − δ k u t, N T t ϕ ◦ Φ − k M, ˜ γ,x,E − , ≤ t ˜ γ − γ − δ M ˆ c t − ∨ (˜ γ − e ˆ ωt kN T t ϕ k L (Γ) ≤ t ˜ γ − γ − δ M ˆ c t − ∨ (˜ γ − e ˆ ωt ˜ c t − e ˜ ωt k ϕ k L (Γ) = ˜ c ˆ c M t − γ + δ +12 t − (2 − ˜ γ ) ∨ e (ˆ ω +˜ ω ) t k ϕ k L (Γ) . Note that t − (2 − ˜ γ ) ∨ = 1 if d ≥
3, since then ˜ γ ≥ End of proof of Proposition 5.7. ‘(a)’. (Suppose that d ≥ P (0) is valid. Then it follows by inductionfrom Lemma 5.10(a) that P ( γ ) is valid for all γ ∈ [0 , d − κ ). In particular P ( d − κ )is valid. Hence using Lemma 5.8 with δ = 2 and ε = t / e − t one deduces that there are c, ω > k u t,ϕ ◦ Φ − k M ,d + κ,x,E − , ≤ c t − d − κ e ωt k ϕ k L (Γ) for all t > ϕ ∈ L (Γ) and x ∈ E − . Therefore the function ( u t,ϕ ◦ Φ − ) | E − has acontinuous representative, which is H¨older continuous and it extends continuously to theclosure of E − . By [ERe] Lemma 3.1(c) there exists a c ′ > | ( u t,ϕ ◦ Φ − )( x ) − ( u t,ϕ ◦ Φ − )( y ) | ≤ c ′ t − d − κ e ωt k ϕ k L (Γ) | x − y | κ/ for all t > ϕ ∈ L (Γ) and x, y ∈ E − with | x − y | < . The latter estimates extendto all x, y ∈ E − with | x − y | ≤ . Since Φ is bi-Lipschitz with Lipschitz constants K , itfollows that | u t,ϕ ( x ) − u t,ϕ ( y ) | ≤ c ′ K κ/ t − d − κ e ωt k ϕ k L (Γ) | x − y | κ/ for all t > u ∈ L (Γ) and x, y ∈ Γ ∩ Φ − ( E ) with | x − y | ≤ K and Statement (a)follows.The prove of Statement (b) is similar by using Lemma 5.10(b).The uniform H¨older bounds of Theorem 5.5 can be combined with the Poisson kernelbounds of [EO2] to obtain H¨older Poisson kernel bounds in case the domain Ω is of class C κ for some κ > Theorem 5.11.
Assume d ≥ . Suppose there exists a κ > such that Ω is of class C κ .Assume that c kl = c lk is H¨older continuous for all k, l ∈ { , . . . , d } , the function a is realvalued and a k = b k = 0 for all k ∈ { , . . . , d } . Suppose that σ ( A D ) . Let K be thekernel of the semigroup on L (Γ) generated by −N , where N is the Dirichlet-to-Neumannoperator. Then for all ε, τ ′ ∈ (0 , and τ > there exist c, ν > such that | K t ( x, y ) − K t ( x ′ , y ′ ) | ≤ c ( t ∧ − ( d − (cid:16) | x − x ′ | + | y − y ′ | t + | x − y | (cid:17) ν (cid:16) | x − y | t (cid:17) d − ε (1 + t ) ν e − λ t for all x, y, x ′ , y ′ ∈ Γ and t > with | x − x ′ | + | y − y ′ | ≤ τ t + τ ′ | x − y | , where λ = min σ ( N ) . roof . It follows from Theorem 5.5 and [EO2] Theorem 1.1 that there exist c, ω > ν ′ ∈ (0 ,
1) such that | K t ( x, y ) | ≤ c t − ( d − (cid:16) | x − y | t (cid:17) d e ωt and | K t ( x, y ) − K t ( x ′ , y ) | ≤ c t − ( d − (cid:16) | x − x ′ | t (cid:17) ν ′ e ωt (11)for all x, y, x ′ ∈ Γ and t > | x − x ′ | ≤
1. By duality, we obtain similarly, without lossof generality, that | K t ( x, y ) − K t ( x, y ′ ) | ≤ c t − ( d − (cid:16) | y − y ′ | t (cid:17) ν ′ e ωt (12)for all x, y, y ′ ∈ Γ and t > | y − y ′ | ≤ x, y, x ′ , y ′ ∈ Γ and suppose that | x − x ′ | + | y − y ′ | ≤ τ t + τ ′ | x − y | . Then | x − y | ≤ | x ′ − y ′ | + τ t + τ ′ | x − y | , so | x − y | ≤ − τ ′ | x ′ − y ′ | + τ − τ ′ t . Hence1 + | x − y | t ≤ − τ ′ | x ′ − y ′ | t + τ − τ ′ ≤ τ − τ ′ (cid:16) | x ′ − y ′ | t (cid:17) and | K t ( x ′ , y ′ ) | ≤ c t − ( d − (cid:16) | x ′ − y ′ | t (cid:17) d e ωt ≤ c (1 + τ ) d (1 − τ ′ ) d t − ( d − (cid:16) | x − y | t (cid:17) d e ωt . Therefore | K t ( x, y ) − K t ( x ′ , y ′ ) | ≤ c (1 + τ ) d (1 − τ ′ ) d t − ( d − (cid:16) | x − y | t (cid:17) d e ωt . Next, it follows from (11) and (12) that | K t ( x, y ) − K t ( x ′ , y ′ ) | ≤ c t − ( d − (cid:16) | x − x ′ | + | y − y ′ | t (cid:17) ν ′ e ωt . Then | K t ( x, y ) − K t ( x ′ , y ′ ) | ≤ c ′ t − ( d − (cid:16) | x − x ′ | + | y − y ′ | t (cid:17) ν ′ ε (cid:16) | x − y | t (cid:17) d (1 − ε ) e ωt by interpolation, where c ′ = 2 c (1+ τ ) d (1 − ε ) (1 − τ ′ ) d (1 − ε ) . Note that1 t = 1 t + | x − y | (cid:16) | x − y | t (cid:17) . Therefore | K t ( x, y ) − K t ( x ′ , y ′ ) | ≤ c ′ t − ( d − (cid:16) | x − x ′ | + | y − y ′ | t + | x − y | (cid:17) ν ′ ε (cid:16) | x − y | t (cid:17) d (1 − ε ) − ν ′ ε e ωt t ∈ (0 , c > ν ∈ (0 ,
1) such that k T t k L (Γ) → C ν (Γ) ≤ k T k L (Γ) → C ν (Γ) k T t − k L (Γ) → L (Γ) k T k L (Γ) → L (Γ) ≤ c e − λ t for all t ≥
3. Hence | K t ( x, y ) − K t ( x ′ , y ) | ≤ c e − λ t | x − x ′ | ν for all x, x ′ , y ∈ Γ and t ≥ | x − x ′ | ≤
1. By duality there exists a c ′ > | K t ( x, y ) − K t ( x ′ , y ′ ) | ≤ c ′ e − λ t ( | x − x ′ | + | y − y ′ | ) ν for all x, x ′ , y, y ′ ∈ Γ and t ≥ | x − x ′ | ≤ | y − y ′ | ≤
1. Since Γ is bounded, therequired H¨older Poisson bounds follow for t ≥ A The chain condition
Let Ω ⊂ R d be open and connected. We say that Ω satisfies the chain condition if thereexists a c > x, y ∈ Ω and n ∈ N there are x , . . . , x n ∈ Ω such that x = x , x n = y and | x k +1 − x k | ≤ cn | x − y | for all k ∈ { , . . . , n − } . Obviously in generalΩ does not satisfy the chain condition. Proposition A.1.
Let Ω ⊂ R d be open bounded connected with Lipschitz boundary. Then Ω satisfies the chain condition. The proof requires some preparation. Let Ω ⊂ R d be open bounded connected withLipschitz boundary. If T > γ : [0 , T ] → Ω is a Lipschitz curve, then γ is differentiablealmost everywhere. We define the length of γ by ℓ ( γ ) = R T | γ ′ ( t ) | dt . Define the geo-metric distance d : Ω × Ω → [0 , ∞ ) by d ( x, y ) is the infimum of ℓ ( γ ), where T > γ : [0 , T ] → Ω is a Lipschitz curve with γ (0) = x and γ ( T ) = y . Obviously | x − y | ≤ ℓ ( γ )and hence | x − y | ≤ d ( x, y ).We first consider a special Lipschitz chart. Lemma A.2.
Let U ⊂ R d be an open set and Φ be a bi-Lipschitz map from an openneighbourhood of U onto an open subset of R d such that Φ( U ) = E and Φ(Ω ∩ U ) = E − .Then there are c , c > such that d ( x, y ) ≤ c | x − y | and | x − y | ≤ c for all x, y ∈ Ω ∩ U . Proof . Let L ∈ R be larger than both the Lipschitz constant for Φ and Φ − . Further, let x, y ∈ Ω ∩ U . Define γ : [0 , → Ω by γ ( t ) = Φ − ((1 − t ) Φ( x ) + t Φ( y )). Then γ (0) = x and γ (1) = y . Moreover, γ is Lipschitz continuous and | γ ′ ( t ) | ≤ L | Φ( y ) − Φ( x ) | ≤ L | y − x | for almost every t ∈ [0 , d ( x, y ) ≤ ℓ ( γ ) ≤ L | x − y | . Also | x − y | ≤ L | Φ( x ) − Φ( y ) | ≤ L .We next show that the geometric distance is equivalent with the induced Euclideandistance on Ω. Lemma A.3.
There exists a c > such that | x − y | ≤ d ( x, y ) ≤ c | x − y | for all x, y ∈ Ω . roof . By a compactness argument there are N ∈ N and for all k ∈ { , . . . , N } there areopen U k ⊂ R d and a bi-Lipschitz map Φ k from an open neighbourhood of U k onto an opensubset of R d such that Φ k ( U k ) = E and Φ k (Ω ∩ U k ) = E − ; and moreover, Γ ⊂ S Nk =1 U k . Forall k ∈ { , . . . , N } fix w k ∈ Ω ∩ U k . Again by compactness there are N ′ ∈ { N +1 , N +2 , . . . } and for all k ∈ { N + 1 , . . . , N ′ } there are w k ∈ Ω and r k > B ( w k , r k ) ⊂ Ω andΩ ⊂ N [ k =1 U k ∪ N ′ [ k = N +1 B ( w k , r k ) . By Lemma A.2 there are c , c ≥ d ( x, y ) ≤ c | x − y | and | x − y | ≤ c for all k ∈ { , . . . , N } and x, y ∈ Ω ∩ U k . Without loss of generality we may assume that 2 r k ≤ c for all k ∈ { N + 1 , . . . , N ′ } . For simplicity write U k = B ( w k , r k ) for all k ∈ { N + 1 , . . . , N ′ } .Then d ( x, y ) ≤ c | x − y | and | x − y | ≤ c for all k ∈ { N + 1 , . . . , N ′ } and x, y ∈ U k .We next prove that the geometric distance d is bounded on Ω. Define M = 2 c +max { d ( w k , w l ) : k, l ∈ { , . . . , N ′ }} . Let x, y ∈ Ω. Then there are k, l ∈ { , . . . , N ′ } suchthat x ∈ U k and y ∈ U l . Hence d ( x, y ) ≤ d ( x, w k ) + d ( w k , w l ) + d ( w l , y ) ≤ M . Therefore d is bounded by M .Finally suppose that there is no c > d ( x, y ) ≤ c | x − y | for all x, y ∈ Ω.Then for all n ∈ N there are x n , y n ∈ Ω such that d ( x n , y n ) > n | x n − y n | . It follows that | x n − y n | ≤ Mn for all n ∈ N . The sequence ( x n ) n ∈ N is bounded since Ω is bounded. Passingto a subsequence if necessary, we may assume that the sequence ( x n ) n ∈ N is convergent. Let x = lim n →∞ x n . Then lim n →∞ y n = x and x ∈ Ω. Since Ω ⊂ S Nk =1 U k ∪ S N ′ k = N +1 B ( w k , r k ),there exists a k ∈ { , . . . , N ′ } such that x ∈ U k . Because U k is open there exists an N ∈ N such that x n ∈ U k and y n ∈ U k for all n ∈ N with n ≥ N . Finally choose n ∈ N such that n ≥ max { N , c } . Then n | x n − y n | < d ( x n , y n ) ≤ c | x n − y n | ≤ n | x n − y n | . This is a contradiction.Now we are able to prove the proposition.
Proof of Proposition A.1.
Let c > x, y ∈ Ω and n ∈ N .Since the case x = y is trivial, we may assume that x = y . There exist T > γ : [0 , T ] → Ω such that γ (0) = x , γ (1) = y and ℓ ( γ ) ≤ d ( x, y ). For all k ∈ { , . . . , n − } let t k = min { t ∈ [0 , T ] : ℓ ( γ | [0 ,t ] ) = k ℓ ( γ ) n } , which exists by continuity. Set x k = γ ( t k ). Further define x = x and x n = y . Then | x k +1 − x k | ≤ d ( x k +1 , x k ) ≤ ℓ ( γ ) n ≤ d ( x, y ) n ≤ cn | x − y | for all k ∈ { , . . . , n − } , as required. 25 cknowledgements The authors wish to thank Wolfgang Arendt for many discussions at various stages of thisproject. The authors wish to thank Mourad Choulli for helpful comments regarding thechain condition.The first-named author is most grateful for the hospitality extended to him during afruitful stay at the University of Ulm. He wishes to thank the University of Ulm for financialsupport. Part of this work is supported by an NZ-EU IRSES counterpart fund and theMarsden Fund Council from Government funding, administered by the Royal Society ofNew Zealand. Part of this work is supported by the EU Marie Curie IRSES program,project ‘AOS’, No. 318910.
References [AE1]
Arendt, W. and
Elst, A.F.M. ter , Gaussian estimates for second orderelliptic operators with boundary conditions.
J. Operator Theory (1997),87–130.[AE2] , Sectorial forms and degenerate differential operators. J. Operator Theory (2012), 33–72.[AE3] , The Dirichlet problem without the maximum principle. Annales del’Institut Fourier (2019), 763–782.[AEKS] Arendt, W., Elst, A.F.M. ter, Kennedy, J. B. and
Sauter, M. , TheDirichlet-to-Neumann operator via hidden compactness.
J. Funct. Anal. (2014), 1757–1786.[Aro]
Aronson, D. G. , Bounds for the fundamental solution of a parabolic equation.
Bull. Amer. Math. Soc. (1967), 890–896.[Aus] Auscher, P. , Regularity theorems and heat kernels for elliptic operators.
J.London Math. Soc. (1996), 284–296.[AT] Auscher, P. and
Tchamitchian, P. , Gaussian estimates for second orderelliptic divergence operators on Lipschitz and C domains. In Evolution equa-tions and their applications in physical and life sciences (Bad Herrenalb, 1998) ,vol. 215 of Lecture Notes in Pure and Appl. Math., 15–32. Marcel Dekker, NewYork, 2001.[BE]
Behrndt, J. and
Elst, A. F. M. ter , Jordan chains of elliptic partialdifferential operators and Dirichlet-to-Neumann maps.
J. Spectr. Theory (2019).In press.[CK]
Choulli, M. and
Kayser, L. , Gaussian lower bound for the Neumann Greenfunction of a general parabolic operator.
Positivity (2015), 625–646.[Dan1] Daners, D. , Heat kernel estimates for operators with boundary conditions.
Math. Nachr. (2000), 13–41.26Dan2] , Inverse positivity for general Robin problems on Lipschitz domains.
Arch.Math (2009), 57–69.[Dav] Davies, E. B. , Heat kernels and spectral theory . Cambridge Tracts in Mathe-matics 92. Cambridge University Press, Cambridge etc., 1989.[EO1]
Elst, A.F.M. ter and
Ouhabaz, E.-M. , Partial Gaussian bounds for degen-erate differential operators II.
Ann. Sc. Norm. Super. Pisa Cl. Sci. (2015),37–81.[EO2] , Poisson bounds for the Dirichlet-to-Neumann operator on a C κ -domain. J. Differential Equations (2019), 4224–4273.[ERe]
Elst, A.F.M. ter and
Rehberg, J. , H¨older estimates for second-order op-erators on domains with rough boundary.
Adv. Diff. Equ. (2015), 299–360.[ERo] Elst, A.F.M. ter and
Robinson, D.W. , Local lower bounds on heat kernels.
Positivity (1998), 123–151.[Gia] Giaquinta, M. , Multiple integrals in the calculus of variations and nonlinearelliptic systems . Annals of Mathematics Studies 105. Princeton University Press,Princeton, 1983.[Neˇc]
Neˇcas, J. , Direct methods in the theory of elliptic equations . Corrected 2ndprinting edition, Springer Monographs in Mathematics. Springer-Verlag, Berlin,2012.[Nit]
Nittka, R. , Regularity of solutions of linear second order elliptic and parabolicboundary value problems on Lipschitz domains.
J. Differential Equations (2011), 860–880.[Ouh]