Generation of semigroups for the thermoelastic plate equation with free boundary conditions

We consider the linear thermoelastic plate equations with free boundary conditions in uniform $C^4$-domains, which includes the half-space, bounded and exterior domains. We show that the corresponding operator generates an analytic semigroup in $L^p$-spaces for all $p\in(1,\infty)$ and has maximal $L^q$-$L^p$-regularity on finite time intervals. On bounded $C^4$-domains, we obtain exponential stability.


Introduction
Let Ω ⊂ R N be a domain with boundary Γ. We consider the linear thermoelastic plate equations u tt + ∆ 2 u + ∆θ = f 1 in (0, ∞) × Ω, with initial conditions u| t=0 = u 0 in Ω, u t | t=0 = u 1 in Ω, θ| t=0 = θ 0 in Ω. (1-2) System (1-1) serves as a standard simplified model for thin elastic plates with thermoelastic effects, see [10], Chapter 2, or [1], for a discussion of this and similar models. In (1-1), u(t, x) stands for the vertical displacement at time t ≥ 0 and at position x = (x 1 , . . . , x N ) ∈ Ω, while θ(t, x) denotes the temperature (relative to some reference temperature) at time t and position x. Note that we omitted all physical constants for simplicity. Among the physically relevant boundary conditions, the maybe most complicated are the so-called free boundary conditions ∆u − (1 − β)∆ ′ u + θ = g 1 on (0, ∞) × Γ, ( [1][2][3] which will be considered in the present paper. In (1)(2)(3), ∆ and ∆ ′ stand for the Laplace operator in Ω and the Laplace-Beltrami operator on the boundary Γ, respectively, and ∂ ν denotes the derivative in outer normal direction. For a survey on other types of boundary conditions and generation of semigroups for them, we refer, e.g., to [11]. The physically relevant situation is the two-dimensional case N = 2, but we can consider (1-1)- (1)(2)(3) in any dimension.
One of the standard approaches to (1-1)-(1-3) is to set v := ∂ t u and obtain the first-order system acting on U := (u, u t , θ) ⊤ and being of the form The natural space for the L p -realization of the mixed-order boundary value problem (A(D), B(D)) is given by E (0) p (Ω) and its solution space by E More precisely, we define A p,Ω as an unbounded operator in E We consider uniform C 4 -domains, see Definition 3.1 below. The main result of the present paper shows that for all p ∈ (1, ∞), the operator A p,Ω generates an analytic C 0 -semigroup. This is a consequence of the stronger result that A p,Ω has maximal L q -L p -regularity (Theorem 3.3). On bounded C 4 -domains, we obtain exponential stability (Theorem 3.7).
The thermoelastic plate equations has been studied by many authors, mostly in an L 2 -setting. Many results deal with exponential stability of the associated semigroup, e.g., [8], [18], [16], [11], [22]. For the analyticity of the semigroup, we refer to [17], [14], and [15] in the L 2 -setting. For the treatment of nonlinear problems, corresponding results in L p are of relevance. In the whole-space case, analyticity of the generated semigroup in L p was shown in [3]. In the case of the half-space and of bounded domains, equations (1-1) with Dirichlet (clamped) boundary conditions were studied in [20] and [19]. In the paper [13], a rather complete analysis in the L p -setting can be found for hinged boundary conditions u = ∆u = θ = 0.
System (1-1)-(1-3), i.e. the thermoelastic plate equations with free boundary conditions in the L p -setting, has been studied recently by the authors in [5]. It was shown that the second-order (in time) system (1-1)-(1-3) has maximal L q -L p -regularity. However, this does not imply that the first-order system (1-4)- (1)(2)(3)(4)(5) generates an analytic C 0 -semigroup. This was also observed in the case of the structurally damped plate equation with clamped boundary conditions in [4]. In fact, in the situation of [4], we have maximal regularity, but no generation of semigroup unless additional conditions are included in the basic space. Roughly speaking, this is due to the fact that the standard resolvent estimates hold only for right-hand sides with vanishing first component, and the reformulation of (1-1) as a first-order system in fact leads to such a right-hand side.
In the present paper, however, we show that the operator related to the first-order system  generates an analytic C 0 -semigroup without additional conditions on the basic space E (0) p (Ω). The proofs are based on Fourier multiplier methods on one hand and on the results from [5] on the other hand. If the domain Ω is bounded, we obtain exponential stability apart from the kernel of the operator. In particular, we obtain generation of an analytic semigroup and exponential stability for the twodimensional system which was studied in [12], in this way generalizing the results in [12] from the L 2 -case to the L p -case.

The whole space case
In this section, we consider the whole-space case, i.e. system (1-1)-(1-2) with Ω = R N . Our approach is based on the Fourier transform and results on vectorvalued Fourier multipliers. In particular, the proof of maximal regularity in the sense of well-posedness in L q -L p -Sobolev spaces make use of the concept of Rboundedness and variants of Michlin's theorem. As standard references, we mention [2] and [9].
The Fourier transform F in R N is given by for Schwartz functions ϕ and extended by duality to tempered distributions. A symbol m ∈ L ∞ (R N ) is called a Fourier multiplier if F −1 mF defines a bounded linear operator in L p (R N ). One of the key ingredients to show R-sectoriality will be the vector-valued version of Michlin's theorem on Fourier multipliers due to Weis [23] and Girardi and Weis [7].
The following definition is a variant of [5], Definition 3.2.
b) Let s ∈ R \ {0}, and let m(ξ, λ) := (λ + |ξ| 2 ) s/2 . Then m ∈ M s (Σ θ ). This can be seen by homogeneity: As m is quasi-homogeneous of order s in the sense that c) By a similar homogeneity argument, we see that (ξ, λ) → |ξ| The following result is one main tool for the results below and was shown in [6], Theorem 3.3.
We consider the whole space resolvent where diag(. . .) stands for the diagonal matrix with the corresponding elements on the diagonal. For the next result, we use the fact that the induced operator S j (D) defines an isometric isomorphism Lemma 2.4. For every ϑ < ϑ 0 , λ 0 > 0 and j ∈ {0, 1, 2} we have Proof. Let j ∈ {0, 1, 2}. In view of (2-4) and Lemma 2.2, we have to show that every entry of the matrix It was shown in [20], Section 2, that for all λ ∈ λ 0 + Σ ϑ we have we obtain m 21 ∈ M 0 (λ 0 +Σ ϑ ). All other entries of the matrix M (j) can be estimated similarly. Therefore, M (j) ∈ M 0 (λ 0 + Σ ϑ ) which finishes the proof.
b) The operator A p,R N is not sectorial for any angle and therefore does not generate a bounded C 0 -semigroup on F p . c) For any λ 0 > 0, the operator A p,R N −λ 0 is R-sectorial with R-angle ϑ 0 . Therefore, A p,R N − λ 0 has maximal L q -L p -regularity in (0, ∞), and A p,R N has maximal L q -L p -regularity in (0, T ) with T < ∞. In particular, A p,R N generates an analytic C 0 -semigroup.
c) The R-sectoriality follows from Lemma 2.4 with j = 2, and the other statements are consequences of the general theory on R-sectorial operators.

The case of a uniform C 4 -domain
Let Ω ⊂ R N be a uniform C 4 -domain with boundary Γ, and let p ∈ (1, ∞). To show that the operator A p : E Here, G = (g 1 , g 2 , g 3 ) ⊤ is defined in the whole of Ω. Similarly to [5], (1.8), we define the spaces The following result on the existence of R-bounded solution operators was shown in [5], Theorem 1.4.
Let R Ω : f → f | Ω denote the restriction of a function defined on R N to Ω. Obviously, we have r Ω ∈ L(H i p (R N ), H i p (Ω)) with norm 1 for any i ∈ N 0 . In fact, r Ω is a retraction as a corresponding co-retraction (extension operator) exists for uniform C 4 -domains. In the following, we fix an extension operator e Ω : L 1 loc (Ω) → L 1 loc (R N ) with the property that for any p ∈ (1, ∞) and f ∈ H i p (Ω), we have r Ω e Ω f = f , and e Ω L(H i p (Ω),H i p (R N )) ≤ C p for i = 0, . . . , 4. For the existence of such an extension operator, we refer to [21], Appendix A.
The following theorem is the main result of the present paper.
Theorem 3.3. There exist λ 0 > 0 and ϑ > π 2 such that the operator A p,Ω − λ 0 is R-sectorial with R-angle ϑ. Therefore, A p,Ω has maximal L q -L p -regularity in every finite time interval. In particular, A p,Ω generates an analytic C 0 -semigroup in E Proof. We first obtain a description of the resolvent (λ − A p,Ω ) −1 . For this, let F ∈ E (0) p (Ω) be given. We apply the extension operator e Ω from above to every component of F and obtain e Ω F ∈ E (0) p (R N ). We set U 1 := r Ω R(λ)e Ω F for λ ∈ Σ ϑ0 with R(λ) being the whole space resolvent defined in (2)(3).
(ii) In the case A ′ (D) = 0, we consider ( A(D), B(D)) as a perturbation of (A(D), B(D)). Let A B and A B denote the corresponding operators, respectively. Note that we have D( A B ) = D(A B ). By the interpolation inequality, for every ε > 0 there exists C ε > 0 such that ). Due to part (i) of the proof, A B is R-sectorial, and by an abstract perturbation result on R-sectorial operators ( [9], Corollary 6.7), the same holds for A B .
Remark 3.5. Whereas the lower-order perturbation of the operator A(D) could be handled by an abstract perturbation result on R-boundedness, to our knowledge there is no such theorem on boundary perturbation which could be applied to our situation. Therefore, the proof of Lemma 3.4 directly uses the structure of the solution operators.

Proof.
A straight-forward calculation shows that B 1 u = −∆ ′ u and B 2 u = ∂ ν ∆ ′ u holds up to lower-order terms. Therefore, we can apply Lemma 3.4 to both boundary value problems.
Finally, we study exponential stability in the case of a bounded domain.

Proof.
As Ω is bounded, the operator A p,Ω has compact resolvent and discrete spectrum. Moreover, the spectrum is independent of p ∈ (1, ∞). It was shown in [12] that A 2,Ω is dissipative which implies σ(A 2,Ω ) ⊂ {λ ∈ C : Re λ ≤ 0}. Moreover, 0 is the only eigenvalue on the imaginary axis. Now the statements of the theorem follow from general semigroup theory.