NONSYMMETRIC ELLIPTIC OPERATORS WITH WENTZELL BOUNDARY CONDITIONS IN GENERAL DOMAINS

. We study nonsymmetric second order elliptic operators with Wentzell bound- ary conditions in general domains with suﬃciently smooth boundary. The ambient space is a space of L p - type, 1 ≤ p ≤ ∞ . We prove the existence of analytic quasicontractive ( C 0 )-semigroups generated by the closures of such operators, for any 1 < p < ∞ . Moreover, we extend a previous result concerning the continuous dependence of these semigroups on the coeﬃcients of the boundary condition. We also specify precisely the domains of the generators explicitly in the case of bounded domains and 1 < p < ∞ , when all the ingredi-ents of the problem, including the boundary of the domain, the coeﬃcients, and the initial condition, are of class C ∞ .

1. Introduction. In the recent years, after the paper [10], much attention was devoted to the study of symmetric elliptic operators of the type with Wentzell boundary condition as generators of analytic semigroups on spaces of L p type, or on C(Ω), defined in a bounded domain Ω of R N with sufficiently smooth boundary ∂Ω, as in e.g. [2,3,8,2476 FAVINI, GOLDSTEIN, GOLDSTEIN, OBRECHT AND ROMANELLI 12,16]. See also [5,9,18]. Here A(·) is the N × N matrix (a ij (·)), ∂ A ν u = (A∇u) · ν is the conormal derivative of u with respect to A, 0 < β and γ are real-valued and all these functions are sufficiently regular. In [11] we discovered that generation results can be also obtained for some classes of nonsymmetric operators of the type having the coefficients c j , r sufficiently regular in Ω, with boundary condition M u + β∂ A ν u + γu = 0 on ∂Ω.
On the other hand, in [1] we dealt with the more general Wentzell boundary condition (W BC) M u + β∂ A ν u + γu − qβL ∂ u = 0 on ∂Ω, where Ω is a general domain, 0 < β, 1 β and γ are bounded, q ∈ [0, ∞) and L ∂ is a suitable generalization of the Laplace-Beltrami operator. Thus, a natural question arises: Can existence and analyticity results be stated in a more general setting concerning the domain (possibly unbounded domain) and, at the same time, the operator (nonsymmetric operator)? Here we give a positive answer under suitable assumptions on the operator and the boundary condition. Indeed, we prove existence and analyticity of the (C 0 ) semigroup generated by the closure of the nonsymmetric elliptic operator M equipped with Wentzell boundary condition (W BC) on the space X p (Ω), 1 < p < ∞. For the definition of the spaces X p (Ω), as well as the general notation and assumptions, we refer to Section 2. The main results are proved in Section 3. In this framework, the continuous dependence on the coefficients of the boundary condition holds, as an extension of [4, Theorem 3.1] and [1, Section 2]. See Section 4. Finally, in the case of bounded domains, under suitable additional assumptions, an explicit representation of the domain of the generator is given, for 1 < p < ∞. See Section 5.
2. Notation and main assumptions. In the following Ω will be a domain of R N having its nonempty boundary ∂Ω consisting of sufficiently smooth (N − 1) dimensional manifolds. Sufficiently smooth means that the divergence theorem can be used in Ω, Stokes' theorem can be used on ∂Ω, and the usual trace theorems for Sobolev classes hold. The assumption that ∂Ω is of class C 2+δ for some δ > 0 is more than enough.
In addition, let us assume that We associate with A, c, r the formal differential operator M defined by and we denote by M 0 the 'homogeneous symmetric version' of M corresponding to replacing each of c i , r by zero, i.e., We associate with B the operator L ∂ , given by Here ∇ τ is the tangential gradient on ∂Ω. The operator L ∂ becomes the Laplace-Beltrami operator ∆ LB when B = I, the identity matrix. We consider the boundary conditions of Wentzell-type as follows, where ν is the unit outer normal on ∂Ω and ∂ A ν u = (A∇u) · ν is the conormal derivative of u with respect to A,r ∈ L ∞ (∂Ω, R), and a ∈ (W 1,∞ (∂Ω, R)) N . (2.5) is governed by a generalized translation semigroup S = (S(t)) t≥0 , w(t, x) = (S(t)h)(x) = h(δ(x, t)) for a suitable δ : ∂Ω × R + → ∂Ω which can be obtained using the theory of characteristics. We assume that S is a (C 0 ) semigroup on Y p := L p (∂Ω, dS β ) for each 1 < p < ∞. It follows that for all t ≥ 0 and some constants K p , ω p .
Ifã has compact support, then S is of class (C 0 ) on each Y p . If, say, Ω = Ω 1 \ Ω 2 , where Ω 1 is a half space, and Ω 2 is a ball whose closure is in the interior of Ω 1 , then (which can be viewed as R N −1 ), and ∂Ω 2 is bounded. Then, for anyã satisfying (2.5), S is of class (C 0 ) on Y p for all 1 < p < ∞. But when the part of ∂Ω near infinity is curved, it is not clear that S will be of class (C 0 ) on Y p , thus we assume this is as in (A 5 ).
The purpose of (A 5 ) is to allow the boundary conditions (W BC) 0 and (W BC) to be nonsymmetric, due to the presence of the first order terms involvingã. Condition (A 5 ) allows for these terms to determine a Kato perturbation of the basic symmetric operator. But the full semigroup governed by M and its boundary conditions need not be quasicontractive. Therefore we delay the use of (A 5 ) until Theorem 3.5.
Following [10] and [8], let us introduce some notation and spaces. We identify every u ∈ C(Ω) with U = (u| Ω , u| ∂Ω ) and define X p (Ω), or simply X p , to be the completion of {u ∈ C(Ω) : u Xp < ∞} where the norm || · || Xp is given by if Ω is bounded, or if Ω is unbounded, where C 0 (Ω) is the space of all continuous functions on Ω vanishing at infinity. In any case, X ∞ (Ω), or briefly X ∞ , is equipped with the sup norm Here dS β denotes the natural surface measure dS on ∂Ω with weight 1 β . Note that, for p < ∞, f may not have a trace on ∂Ω, and even if f does, this trace needs not equal g. For p = 2, X 2 is a Hilbert space equipped with the inner product Observe that for ε ≥ 0, the subspace C 2+ε (Ω) need not vanish anywhere on ∂Ω.
3. Nonsymmetric operators in general domains. The following theorem is proved in detail in [10], [8,Section 3] in the case of bounded domains, provided that in [8] we replace the Laplace-Beltrami operator by L ∂ . The extension to general domains in the X 2 case is done in [1]. Combining the arguments of these papers proves the more general and more comprehensive statement of it. We point out that the analyticity holds in X p , for 1 < p < ∞, by the Stein interpolation theorem, as discussed in [8].
Moreover, the semigroup generated by G p is analytic for any 1 < p < ∞.
if Ω is bounded and q = 0, the analyticity of the semigroup holds on X 1 and X ∞ provided ∂Ω and all the coefficients are of class C ∞ . (ii) If Ω is bounded and q > 0, the analyticity of the semigroup holds on X 1 and X ∞ by the results in [13].
In the following, we shall consider an unbounded domain Ω which satisfies the assumptions stated at the beginning of Section 2. For this kind of Ω, by [1, Section 2] and similar arguments as in [8], analogous results as in Theorem 3.1 hold, provided that the assumptions (A 1 )-(A 4 ) remain true. Now, let us focus on the main results of the paper concerning the nonsymmetric operator M in general domains as above. We will need some preliminary results. Lemma 3.3. According to the notation introduced in Section 2, we have Proof. The statement (i) follows from the definition of the spaces X p , 1 ≤ p ≤ ∞.
Let f satisfy (3.1). Let us define f n := f χ Λn , where χ Λn is the characteristic function of Λ n , n ≥ 1. Since |f n | ≤ f ∞ the sequence |f n | 1+ 1 m → |f n | a.e. as m → ∞ and |f n | 1+ 1 m ≤ max{ f 2 ∞ , f ∞ } on Λ n for any m ≥ 1. By the dominated convergence theorem, Thus it follows that f n → f a.e., f ∈ L 1 (Λ) and f 1 ≤ k. Then, for 1 ≤ p ≤ ∞, the operator M p − ωI is dissipative on X p for some ω ∈ R, and the closure N p of M p is quasi-m-dissipative on X p .
Proof. Let us consider the case p = 2. Let U, V ∈ D(M 2 ), U = (u| Ω , u| ∂Ω ), V = (v| Ω , v| ∂Ω ) and compute by the divergence theorem, the boundary condition and Stokes' theorem on ∂Ω. Moreover, we use the fact that each u in the domain is in C 2 c (Ω). In particular, for all U = (u| Ω , u| ∂Ω ) ∈ D(M 2 ) we deduce that Now, for any ε > 0, Re where γ − (resp. r + ) denotes the negative part of γ (resp. the positive part of r). Assume ε > 0 is such that ε c ≤ α 0 2 , and define It follows that Hence M 2 is quasi-dissipative on X 2 and thus its closure N 2 is also quasidissipative. In order to prove the range condition for N 2 , we consider Re λ > c + r + ∞ + γ − ∞ . We must solve the equation for all H in a dense subspace of X 2 . Let us take h : Ω → C, h sufficiently smooth on Ω such that H = (h| Ω , h| ∂Ω ) ∈ X 1 ∩ X ∞ . For each U ∈ D(M 2 ) let us evaluate the inner product of both hand sides of (3.3) by V ∈ D(M 2 ). By using similar arguments as above we find that

(3.4)
Define the sesquilinear form B λ (U, V ) as the left hand side of (3.4) and define as C(V ) the right hand side. For q ≥ 0, let us introduce V q as The norm defined by for V = (v| Ω , v| ∂Ω ), makes V q a Hilbert space continuously embedded into X 2 . Then for q ≥ 0, according to our previous calculations, B λ (·, ·) is a sesquilinear form on V q which is bounded and coercive and C(·) is a bounded linear functional on V q .
Let H be such that h ∈ C 2+ε c (Ω). Hence the Lax-Milgram Lemma (see e.g. [15, Theorem 6, p.57]) implies that there exists a unique weak solution U of (3.3) for all H such that h ∈ V q . If h ∈ C 2+ε (Ω), then a standard elliptic regularity argument shows that u ∈ C 2+ε (Ω) and satisfies the elliptic equation a.e. Furthermore, since u| ∂Ω ∈ H 1 (∂Ω, dS β ) (when q > 0), it is possible to apply the divergence theorem obtaining that N 2 U ∈ X 2 . Thus the assertion follows for p = 2. Now let us show that M p is quasidissipative on X p , for all p ∈ [1, ∞). We start with the case p > 2. Let JU := |U | p−2 U χ {U =0} be the duality map of X p (modulo a positive constant multiple which depends on ||U || Xp , for U = 0). Take 0 = U = (u| Ω , u| ∂Ω ) ∈ D(M p ). We have by the divergence theorem, the boundary condition and Stokes' theorem. Observe that, if we call and the positive definiteness of A(x), it follows that Re Z ≥ 0. Arguing in a similar way, one can also deduce that Thus, taking into account that all c i and r + are essentially bounded, we deduce that Hence M p is quasidissipative for p > 2. The same result holds for p ∈ [1, 2) according, for instance, to the proof of [7,Lemma 4.4.3]. Hence, for any p ∈ [1, ∞), M p is quasi-dissipative. Now, M p quasi-dissipative implies for Re λ > c + r + ∞ + γ − ∞ . Hence the range of λ − N p is closed, where N p = M p . Let us show that it is dense, too. Let H ∈ X p for p ∈ (1, ∞). We can argue for any H = (h| Ω , h| ∂Ω ), where h ∈ C 2+ε c (Ω), ε > 0, sufficiently small. Then, by the previous case, there exists U ∈ D(N 2 ), U = (u| Ω , u| ∂Ω ) such that for Re λ sufficiently large and ∇ · A∇u + c · ∇u + ru = h.
By elliptic regularity for the Wentzell problem in bounded domains (see [8]), we have that there exists u ∈ D(N p ) such that (Ω) is dense on each X p , then N p is quasi-m-dissipative on X p , 1 < p < ∞. In the case p = 1, let h ∈ C 2+ε c (Ω) and λ sufficiently large, say λ > ω. Then, for any p ∈ (1, ∞) there exists a unique U ∈ D(N p ) such that λU − N p U = h for any p ∈ (1, ∞). Moreover, By Lemma 3.1, we deduce that U ∈ X 1 and U X1 ≤ k. Hence N 1 , the closure of M 1 , is quasi-m-dissipative on X 1 . Finally, let M ∞ be the realization of M in X ∞ with domain D(M ∞ ), obtained by replacing X p by X ∞ in the definition of D(M p ) and let N ∞ be the closure of M ∞ on X ∞ . Then, as a consequence of (3.5), as p → ∞ we deduce that M ∞ , and hence, N ∞ , is quasidissipative on X ∞ . Moreover, since the range condition for each N p is essentially p-independent, N ∞ is quasi-m-dissipative on X ∞ . We shall show that N p is a Kato perturbation of the operator N p in Theorem 3.4 in X p , for 1 < p < ∞. The proof will require an extension of the Kallman-Rota inequality, which we present now.
Lemma 3.6. Let G generate a uniformly bounded (by k > 0) (C 0 ) semigroup W = (W (t)) t≥0 on a Banach space E. Then, for all f ∈ D(G 2 ) we have Proof. The classical Kallman-Rota inequality is for the case of k = 1, in which case the constant in (3.6) is 4. Let f ∈ D(G 2 ), t > 0. By Taylor's formula, It follows that If G 2 f = 0, letting t → ∞ shows Gf = 0 and there is nothing to prove. If G 2 f = 0, let us consider the polynomial in t, Due to (3.7) we have that P (t) ≥ 0 for all t ∈ R. Hence and (3.6) follows.
Remark 3.7. As a consequence of the previous Lemma, for any ε > 0, Thus if B is any closable operator with D(G) ⊂ D(B), then (by the closed graph theorem) B is a Kato perturbation of G 2 (see [14,Corollary 6.9]). Now, let us return to the proof of Theorem 3.5.
Proof of Theorem 3.5. We can interpret M p as the operator matrix where P 1 (resp. P 2 ) represents the lower order terms of the operator M p acting on Ω (resp. ∂Ω). Now M 0,p includes the terms withã andr described in (A 5 ).
Observe that, by [1, Section 2], the closure G 2 of M 0,2 generates an analytic semigroup which is analytic in the sector Σ(θ) = {z ∈ C : Re z > 0, |arg z| < θ} with θ = π 2 . Similar arguments as in [10,8] allow us to obtain that G p generates a semigroup analytic in the sector Σ(θ p ) = Σ(θ p ) where for 2 ≤ p < ∞, θ p = π p and 1 p + 1 p = 1. Alternatively, Neuberger's theorem could be used here (cf. [14]). When q = 0, P 2 is a bounded operator and then P = P 1 0 0 0 + 0 0 0 P 2 is a Kato perturbation of M 0,p and it follows that N p generates a (C 0 ) semigroup analytic in Σ(θ p ), 1 < p < ∞. When q > 0 andã is nonzero, the argument following the proof of Lemma 3.6 shows that P is a Kato perturbation of M 0,p , but the quasidissipativity of N p need not be valid in this case. Still, N p generates a (C 0 ) semigroup analytic in Σ(θ p ), 1 < p < ∞. The theorem is now proved. 4. Continuous dependence. As a consequence of the Trotter-Neveu-Kato approximation theorem (see e.g. [14,Theorem 7.3]) and according to the results in [4] and [1], one can easily deduce the following theorem. in X p in the following sense. Let β k , γ k ∈ C 1 (∂Ω) be real for k ∈ N 0 = {0, 1, 2, ..} and suppose that uniformly as k → ∞ with β k (x) > 0 for all k and x. Let q k ∈ (0, ∞) for k ∈ N 0 or q k = 0 for k ∈ N 0 with q k → q 0 and Let N kp be the corresponding N p for each k ∈ N 0 , p ∈ [1, ∞]. Note that X kp and X 0p are equal as sets and have uniformly equivalent norms. Let T k = (T k (t)) t≥0 , k ∈ N 0 , be the semigroup generated by N kp on X kp . Then, for all 1 ≤ p ≤ ∞, for any f ∈ X 0p , T k (t)f → T 0 (t)f for all t ≥ 0, uniformly for t in bounded intervals.

Domain characterization.
Let Ω be a bounded domain and assume that a,r vanish and ∂Ω, all the coefficients (a ij ), (b ij ), c i , r, β, γ and the initial function u(0, x) = f (x) are all of class C ∞ , either on Ω and on ∂Ω. In the symmetric case, we showed under these hypotheses that We proved this in [4] for p = 2 and the symmetric case because we were focussing on hyperbolic problems such as the wave and telegraph equations. The proof was based on the theory of (uniformly) elliptic boundary value problems, developed by Agmon, Douglis, Nirenberg, Lions and others, and extended by Triebel and others for 1 < p < ∞. This is explained in detail in H. Triebel's book [17]. The context was that of a bounded domain, with the boundary and all functions appearing in the problem being of class C ∞ on their maximal domains. The proof was based on pseudodifferential operator theory, which requires everything to be C ∞ .
But now we are considering parabolic problems in the L p context. The theory in Triebel's book works for Ω bounded, 1 < p < ∞ and M nonsymmetric (but still uniformly elliptic) and everything being C ∞ . Thus with much work on the considerable technical details, we mimicked the proof in [4] and obtained the following result.