General Wentzell boundary conditions , differential operators and analytic semigroups in C [ 0 , 1 ]

We are concerned with the study of the analyticity of the (C0) semigroup generated by the realizations of the operators Au = αu′′ + βu′ or Au = b(au′)′ + βu′ in C[0, 1] with general Wentzell boundary conditions of the type lim x→j Au(x)+ b̃(x)u′(x) = 0 for j = 0, 1 in C[0, 1]. Here the functions a, α, β, b, b̃ are assumed to be in C[0, 1], with a, α ∈ C1(0, 1), a(x) > 0, α(x) > 0, in (0, 1), b(x) > 0 in [0, 1] and a, or α, possibly degenerate at the endpoints, i.e. a, or α, allowed to vanish at 0 and 1.


Introduction
Inspired by their previous results concerning generation and analyticity for (C 0 ) semigroups generated by degenerate elliptic second order differential operators with Wentzell boundary conditions, the authors started a systematic investigation of analogous problems for some classes of linear or nonlinear, possibly degenerate at the boundary, second order differential operators with general Wentzell boundary conditions in different function spaces.General Wentzell boundary condition for such an operator A reads as follows where Ω is a bounded subset of R N with sufficiently smooth boundary ∂Ω, β, γ are nonnegative continuous functions on ∂Ω with β > 0 and n is the unit outer normal.In [9] we proved generation results in C[0, 1] which extended substantially earlier results dealing with Dirichlet, Neumann, Robin and Wentzell boundary conditions.Subsequently generation and regularity in C(Ω), in suitable L p (Ω, µ) spaces and in H1 (Ω) were obtained in the papers [10]1 , [12] and [13] , while in [11] the wave equation with Wentzell boundary conditions (i.e.β = γ = 0 on ∂Ω) for Ω = (0, 1) was considered.After these results, in a couple of years, an increasing and wide interest for the above topics led to significant and widespread progress, by using different approaches and techniques: see e.g.Warma [20] , Arendt, Metafune, Pallara and Romanelli [1] , Xiao and Liang [21] , Engel [5] , [6] , Vogt and Voigt [19] , and Batkai and Engel [2] .In particular, it is worthwhile to point out that the recent results by Xiao and Liang [21] and Batkai and Engel [2] revealed that, thanks to a suitable abstract framework, analyticity with general Wentzell boundary condition for second order (in time) equations in C[0, 1] can be obtained as a byproduct of the study of cosine families with general Wentzell boundary conditions.For wave equations, see also Gal, G. R. Goldstein and J. A. Goldstein [15] .Despite so many different efforts, the problem of analyticity in the degenerate case remains not completely solved even in , with a > 0, or α > 0, in (0, 1), b > 0 in [0, 1] and a, or α, possibly degenerate (i.e.a, or α, allowed to vanish at the endpoints 0 and 1).Our method relies in the idea of replacing the considered operator by a related new operator with pure Wentzell boundary conditions (i.e.Au(j) = 0 at j = 0, 1) which generates an analytic semigroup, and therefore into coming back to the original operator by a suitable perturbation which saves analyticity.To this aim in Section 1 we present analyticity results stated in previous papers, while Section 2 is devoted to the proof of the main results.

Preliminary results
In order to make more clear the tools involved in the proof of the main theorem, we recall two theorems, one due to Campiti and Metafune [ [3] , Theorem 4.2] and the other to Favini and Romanelli [ [14] , Theorems 2.2 and 2.5] (see also Favini, G. R. Goldstein, J. A. Goldstein and Romanelli [8] ).
, and β is Hölder continuous at x = 0, 1.Then the operator (L, D(L)) given by Then the operator (G, D(G)) given by generates an analytic semigroup on C[0, 1].

Main Results
Let us start by considering an operator of the type Au := αu where the coefficient α may degenerate at the boundary, but with degeneracy of low order.
First we show that every for suitable positive constants c o , c 1 .Since and, consequently, dt .
Therefore our assertion holds.Moreover every u ∈ D(L) verifies the inequality and fix a (large) positive integer n.For sake of brevity, denote by u Ji,∞ , j = 0, 1, ..., n − 1 the supremum norm in the space C(J i ), where Therefore, in view of (5), there exists a suitable constant C > 0 such that for any x ∈ J 0 , we have and this yields On the other hand, we can repeat the argument on J 1 , if x ∈ J 1 and obtain that Thus, in general, for i = 0, 1, ..., n − 1 and x ∈ J i , we have Taking n sufficiently large so that 1 Therefore Since we can choose n ∈ N arbitrarily large, our claim is proved.
The above result can be extended to operators of more general type.Indeed the following results hold.
Corollary 3A Let us consider α, β, b in C[0, 1] such that α satisfies assumptions (4) and ( 5) and, in addition, suppose that Then the operator (C, D(C)) given by Proof According to Theorem 1, the operator , L 1 u(j) = 0 for j = 0, 1} generates an analytic semigroup on C[0, 1].Now, in analogy with the previous proof, we will show that the operator B 1 u := −bu with domain D(B 1 ) := D(L 1 ) is L 1 -bounded with L 1 -bound equal to zero.Indeed, arguing as in Theorem 3, it is still true that every u ∈ C[0, 1] ∩ C 2 (0, 1) with L 1 u ∈ C[0, 1] has its first derivative u in C[0, 1] and its second derivative u in L 1 (0, 1).Let β0 := max By similar arguments as in Theorem 3) we can find a suitable positive constant C such that On the other hand, for n sufficiently large, we rewrite the inequality (10) where β is replaced by β + b, Bu by (β + b)u and L by L 1 .We obtain that Then, plugging ( 13) into (12) gives Since we can choose n ∈ N sufficiently large, the above calculations imply our assertion.
Proof It suffices to observe that the operator with domain D(C 1 ) satisfies the assumptions of the previous Corollary.Now, let us consider the case of the coefficient α with degeneracy of higher order at 0 and 1.
Then the operator (A, D(A)) given by and observe that where Therefore there exist two positive constants C 0 , C 1 such that W 1 (t) dt is strictly increasing and differentiable, and as in [14] (see also [8] ), this allows to deduce that the operator (B 0 , D(B 0 )) is similar to the operator (B ∞ , D(B ∞ )) given by Consequently, according to [ [7] , Chapter II Corollary 4.9], the operator (B 2 0 , D(B 2 0 )) generates an analytic semigroup on C[0, 1].Here we have that and, arguing as in [14] , or in [8] , it follows that D(B 2 0 ) = D(L).In order to conclude our proof we have only to show that the operator Bu := −bu with domain D(A) is L-bounded with L-bound equal to zero.According to (17) and above considerations, for any > 0 there exists a suitable k > 0 such that bu Hence the assertion holds true.