Parabolic problems with general Wentzell boundary conditions and diffusion on the boundary

We show a result of maximal regularity in spaces of H\"older continuous function, concerning linear parabolic systems, with dynamic or Wentzell boundary conditions, with an elliptic diffusion term on the boundary.


Introduction
In this paper we want to study second order parabolic systems in the forms (1.1) and (1.2) Here for every t ∈ [0, T ], A(t, x, D x ) is a second order linear strongly elliptic operator in the open, bounded subset Ω of R n , L(t) is a second order linear strongly elliptic tangential operator in ∂Ω, B(t, x ′ , D x ) is a first order (not necessarily tangential) operator in ∂Ω. It is clear that, at least formally, (1.1) and (1.2) are strictly related. A large amount of papers has been devoted to parabolic problems with dynamic and Wentzell boundary conditions in the form (1.1)- (1.2) in the case that the summand L(t)(u(t, ·) |Γ ) does not appear. We refer to the bibliographies in [6] and [8]. In our knowledge, a problem in the form (1.1) was introduced for the first time in [10], in the particular case that A(t, x, D x ) = α(x)∆ x , with α positively valued, B(t, x ′ , D x ) = b(x ′ )D ν , with ν unit normal vector to ∂Ω, pointing outside Ω, L(t) = a(x)∆ LB u, where we indicate with ∆ LB the Laplace-Beltrami operator. [10] contains a physical interpretation of the problem: briefly, a heat equation with a heat source on the boundary, that depends on the heat flow along the boundary, the heat flux across the boundary and the temperature at the boundary.
The first paper where a problem in the form (1.2) is really studied seems to be [3]. In it it was considered the system with , A strongly elliptic, β(x ′ ) > 0 in ∂Ω, D νA conormal derivative, q ∈ [0, ∞). It is proved that, if 1 ≤ p ≤ ∞ the closure a suitable realisation of the problem in the space L p (Ω × ∂Ω) (1 ≤ p ≤ ∞), gives rise to an analytic semigroup (not strongly continuous if p = ∞). The continuous dependence on the coefficients had already been considered in [2].
In [1] the authors generalised some of the results in [3], considered also the case that the first equation in (1.3) is the telegraph equation (with two initial conditions) and studied the asymptotic behaviour of solutions.
In [12] the author considered the case of a domain Ω with merely Lipschitz boundary, with a strongly elliptic operator A (independent of t). It was shown that a realisation of A with the general boundary condition (Au) |∂Ω − γ∆ LB u + D νA u + βu = g in ∂Ω generates a strongly continuous compact semigroup in C(Ω). Semilinear problems were studied in [13] and [14].
Finally, in the paper [11] the authors treated (1.1) in the particular case with k which may be negative (in contrast with the previously quoted literature). They showed that, if the initial datum u 0 is in H 1 (Ω) and u 0|∂Ω ∈ H 1 (Γ), The main aim of this paper is to show that, in a suitable functional setting, the role of the operator B(t, x ′ , D x ) in (1.1) and (1.2) is minor, in the sense that these equations can be treated as perturbations of the corresponding problems with B(t, x ′ , D x ) ≡ 0. In fact, we shall see that B(t, x ′ , D x ) may be, apart some limitations on the regularity of its coefficients, an arbitrary first order linear differential operator. Moreover we shall consider problems with coefficients depending on t and we shall obtain results of maximal regularity, that is, results establishing the existence of linear and topological isomorphisms between classes of data and classes of solutions. Following the lines of [7] and [8], we shall work in spaces of Hölder continuous functions. Now we are going to state our main results. We begin by introducing the following assumptions: (A1) Ω is an open, bounded subset of R n (n ∈ N, n ≥ 3), lying on one side of its topological boundary Γ, which is a compact submanifold of R n of class C 2+β , for some β ∈ (0, 1).
is a second order, partial differential operator in Γ. More precisely: for every we suppose, moreover, that, if |α| = 2, l α,Φ is real valued, for every open subset V of U , with V ⊂⊂ U , l α,Φ|V ∈ C β/2,β ((0, T ) × V ), and there exists ν(V ) positive such that, We want to prove the following Theorem 1.1. Suppose that (A1)-(A4) are fulfilled. Then the following conditions are necessary and sufficient in order that (1.1) have a unique solution u belonging to C 1+β/2,2+β ((0, T ) × Ω): Theorem 1.2. Suppose that (A1)-(A4) are fulfilled. Then the following conditions are necessary and sufficient in order that (1.2) have a unique solution in C 1+β/2,2+β ((0, T ) × Ω): Now we are going to describe the organisation of the paper. We begin by considering in Section 2 the parabolic problem with L strongly elliptic in Γ. We do not impose the variational form of L. We show that the operator L, defined as is the infinitesimal generator of an analytic semigroup in C(Γ). This can be easily obtained, by local charts methods, employing well known analogous results in R n (see [9], Chapter 3). Employing maximal regularity techniques in spaces of continuous and Hölder continuous functions (see again [9]), we determine in Proposition 2.3 necessary and sufficient conditions (analogous to well known conditions in R n−1 ), in order that (1.4) have a unique solution in C 1+β/2,2+β ((0, T ) × Γ) (we shall recall the definition of these classes in the following). This first step is admittedly simply, but we were not able to find it in literature.
In Section 3 we study systems (1.1) and (1.2). Employing the results of Section 2 we begin by determining in Theorem 3.1 necessary and sufficient conditions such that system (1.1) have a unique solution in C 1+β/2,2+β ((0, T )×Ω) in the particular case A(t, x, D x ) = A(x, D x ), B(t, x ′ , D x ) = 0, L(t) = L (independent of t). Finally, we obtain Theorem 1.1 from this particular case, by perturbation arguments. Theorem 1.2 is a simple consequence of Theorem 1.1.
We conclude this preliminary section by specifying some notations and by recalling some facts that we shall use.
C, C 0 , C 1 , . . . will indicate positive constants that we are not interested to precise and may be different from time to time. We shall write C(α, β, . . . ) to indicate that the constant depends on α, β, . . . .
If L is a tangential differential operator in the boundary Γ and u is defined in [0, T ] × Ω, we shall write Lu(t, x ′ ) (x ′ ∈ Γ), instead of L(u(t, ·) |Γ )(x ′ ). If X and Y are Banach spaces, we shall indicate with L(X, Y ) the Banach space of linear, bounded operator from X to Y . In case X = Y , we shall write L(X).
Let Ω be an open subset of R n . We shall indicate with B(Ω) and C(Ω) the spaces of (respectively) complex valued, bounded and complex valued, uniformly continuous and bounded functions with domain Ω. If f ∈ C(Ω), it is continuously extensible to its topological closure Ω. We shall identify f with this extension. If m ∈ N, we indicate with C m (Ω) the class of functions f in C(Ω), whose derivatives D α f , with order |α| ≤ m, belong to C(Ω). We shall equip these spaces with their natural norms: The previous definition can be extended in an obvious way to functions f : ). This is a consequence of the embedding (se Theorem 4 in [7] and the indicated references).
If I is an open interval in R, Ω is an open subset of R n and α, β are nonnegative, we set This is a Banach space, with the norm The following facts hold (see [7]), Lemma 1: (II) Suppose α, β ≥ 0 with β ∈ Z and Ω such that there exists a common linear bounded extension operator, mapping C(Ω) into C(R n ) and C β (Ω) into C β (R n ). Then, Let β ∈ (0, 1) and suppose that there exists a common linear bounded extension operator mapping The previous definitions and results can be extended (by local charts) to functions f : Γ → X, with Γ suitably smooth differentiable manifold. In Section refse2 we shall also deal with the Besov space B 2 ∞,∞ (Γ). This space can be defined by local charts, employing the following definition of the space It is known (see COMPLETARE ) that B 2 ∞,∞ (R n−1 ) properly contains the space W 2,∞ (R n ) of elements of C 1 (R n ) with Lipschitz continuous first order derivatives. On the other hand, B 2 ∞,∞ (R n−1 ) ⊆ C 2−ǫ (R n ) ∀ǫ ∈ (0, 2]. We shall consider also spaces W 2,p (Γ), just in the case of Γ compact and of class C 2+β (β > 0), which can be again defined by local charts. In Section 2 we shall employ Besov spaces B α ∞,∞ (Γ), with α ∈ [0, 2]. We shall need the following facts, which can be easily deduced from analogous statements in R n−1 (see [5]): (a) (b) in any case, We shall employ the following version of the continuation principle: is a linear and topological isomorphism between X and Y .

Parabolic problems in Γ
In this section, we study the parabolic system (1.4). We introduce the following as (A4'). L is a second order, partial differential operator in Γ. More precisely: for every local chart We begin by considering the elliptic system depending on the parameter λ We shall prove the following Theorem 2.1. Suppose that (A1) and (A4) hold. Then: (II) g belongs also to the Besov space B 2 ∞,∞ (Γ); moreover, for every γ ∈ [0, 2) there exists C > 0, depending on φ 0 and γ, such that Proof. We take an arbitrary x 0 ∈ Γ and consider a local chart (U, Φ) around x 0 , with U open subset of Γ and Φ diffeomorphism between U and Φ(U ), open subset in R n−1 . We introduce in Φ(U ) the strongly elliptic operator B ♯ , By shrinking U (if necessary), we may assume that the coefficients of L ♯ are in C β (Φ(U )) and are extensible to elements l β in C β (R n ), in such a way that the operator which we continue to call L ♯ = |α|≤2 l β (y)D β y is strongly elliptic in R n . Now we consider the problem Then, (see Chapter 3 in [9]), for every ; v belongs also to the Besov space B 2 ∞,∞ (R n−1 ) (see [4], Proposition 2.5, on account of the embedding C(R n−1 ) ֒→ B 0 ∞,∞ (R n−1 )) ; moreover, for every γ ∈ [0, 2) there exists C > 0, depending on φ 0 and γ, such that with v solving (2.2). We observe that (α 1 ) : , h ∈ C(Γ) and g vanishes outside U 1 , then g = S(x 0 , λ)h.
As a simple consequence of Theorem 2.1, we deduce the following Proof. (I) and (II) immediately follow from Theorem 2.1. We observe also that D(L) contains C 2 (Γ) and so it is dense in C(Γ).
We take v(t, x) := u(t, x) − u 0 as new unknown. v should solve the system to which the conclusion in (β) is applicable.