ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 8, Pages 47–51 (December 19, 2002) S 1079-6762(02)00104-X

MAXIMAL REGULARITY FOR PARABOLIC EQUATIONS WITH INHOMOGENEOUS BOUNDARY CONDITIONS IN SOBOLEV SPACES WITH MIXED L_{p}-NORM

PETER WEIDEMAIER

(Communicated by Michael E. Taylor)

Abstract. We determine the exact regularity of the trace of a function u ∈ L_{q} (0, T; W_{p}^{2}(Ω)) ∩ W_{q}^{1} (0, T; L_{p} (Ω)) and of the trace of its spatial gradient on ∂Ω × ( 0, T ) in the regime p ≤ q. While for p = q both the spatial and temporal regularity of the traces can be completely characterized by fractional order Sobolev-Slobodetskii spaces, for p ≠ q the Lizorkin-Triebel spaces turn out to be necessary for characterizing the sharp temporal regularity.

Introduction

The space W_{p,q}^{2,1}(Ω_{T}) := L_{q} (0, T; W_{p}^{2}(Ω)) ∩ W_{q}^{1} (0, T; L_{p} (Ω)), Ω_{T} := Ω × ( 0, T ), Ω ⊂ ℝ^{n}, is often employed in the theory of evolution equations which are of first order in time and second order in space; see von Wahl [15] for parabolic equations, Sohr [12], Iwashita [8] for the Navier-Stokes equation, and Clément and Prüss [3] for parabolic Volterra equations. For the heat equation (as model problem) this space corresponds to maximal regularity if the inhomogeneous part in the equation belongs to L_{q} (0, T; L_{p} (Ω)). Results of maximal regularity type have been established under various conditions ([2], [4], [6], [9]), but always for homogeneous boundary conditions or the Cauchy problem. Combining these results with Theorems 2.3 and 2.4 stated below, maximal regularity follows also for problems with inhomogeneous boundary conditions.

1. Trace theory in the classical case p = q ∈ ( 1, ∞)

The trace of u ∈ W_{p,p}^{2,1}(Ω_{T}) (space denoted W_{p}^{2,1}(Ω_{T}) in [11]) belongs to the space W_{p}^{2−1/p,(2−1/p)/2}(Γ_{T}), where W_{p}^{α,β}(Γ_{T}) := L_{p} (0, T; W_{p}^{α}(Γ)) ∩ W_{p}^{β} (0, T; L_{p} (Γ)), W_{p}^{s} denoting the Sobolev-Slobodetskii spaces and Γ_{T} := Γ × ( 0, T ), Γ := ∂Ω. This result is sharp. The analogous result holds for the trace of a spatial derivative ∂_{j}u, j ∈ {1, …, n}, which is an element of W_{p}^{1−1/p,(1−1/p)/2}(Γ_{T}). For these results see Ladyzhenskaya, Solonnikov, and Ural’tseva [11, Chapter II, Lemma 3.4] and Grisvard [5, Théorème 4.2]. While the Russian authors used the “method of integral representation”, Grisvard applied interpolation theory.

\begin{matrix}c \\ ◯ \end{matrix}2002 American Mathematical Society

Received by the editors October 16, 2002.

2000 Mathematics Subject Classification. Primary 35K20, 46E35; Secondary 26D99.

Key words and phrases. Maximal regularity, inhomogeneous boundary conditions, trace theory, mixed norm, Lizorkin-Triebel spaces.

2. Trace theory in the case 1 < p ≤ q < ∞

The present author [16] has shown continuity of the map

W_{p,q}^{2,1}(Ω_{T}) ∋ u ↦ u Γ_{T} ∈ L_{q} (0, T; W_{p}^{2−1/p}(Γ)) ∩ W_{q}^{(2−1/p)/2} (0, T; L_{p} (Γ)),

but had to leave open whether this map is onto. In a retrospective view, this target space was not the optimal one. It turned out that the sharp result is obtained if the regularity of the trace in the time variable is described by the Lizorkin-Triebel space F_{q,p}^{(2−1/p)/2} (0, T; L_{p} (Γ)), whose definition follows.

Definition 2.1. A function g ∈ L_{q} (0, T; L_{p} (Γ)) belongs to the class

F_{q,p}^{β} (0, T; L_{p} (Γ)) , β ∈ ( 0, 1 )

iff

(∫_{0}^{T} (∫_{0}^{T −t}h^{−1−pβ}∥g(·, t + h) − g(·, t)∥_{L_{p} (Γ)}^{p} dh)^{q/p} dt)^{1/q} < ∞ . (2.1)

Remark 2.2. The Lizorkin-Triebel spaces F_{q,p}^{β}(ℝ, ℝ) defined by Fourier analysis have a finite difference characterization of the type occurring in the previous definition (see Triebel [14, 2.5.10 Theorem]). This fact motivated us to introduce the Lizorkin-Triebel spaces on the bounded domain ( 0, T ) as above. Since β < 1, one can use first order discrete differences in the definition instead of second order ones, which the reader might have expected.

Let us introduce

W F_{p,q}^{α,β}(Γ_{T}) := L_{q} (0, T; W_{p}^{α}(Γ)) ∩ F_{q,p}^{β} (0, T; L_{p} (Γ))

and endow this space with the norm

∥g∥_{W F_{p,q}^{α,β}(Γ_{T})} = ∥g∥_{L_{q} (0,T; W_{p}^{α}(Γ))} + |g|_{F_{q,p}^{β} (0,T; L_{p} (Γ))},

where |g|_{F_{q,p}^{β} (0,T; L_{p} (Γ))} is given by the integral in (2.1).

We are going to formulate our two main theorems, in which we assume Ω ⊂ ℝ^{n} is an open subset (not necessarily bounded) with a compact boundary of the class C^{1,1} (see [10, 6.2.2 Definition] for details ). Moreover, ν denotes the vectorfield of outer unit normals on Γ.

Theorem 2.3. i) For 3/2 < p ≤ q < ∞ the map u ⟼ u Γ_{T}, well-defined for

those functions u ∈ W_{p,q}^{2,1}(Ω_{T}) which are continuous w.r.t. x ∈ \overline{Ω}, has a continuous extension

γ_{D} : W_{p,q}^{2,1}(Ω_{T}) ⟶ W F_{p,q}^{2−1/p,(2−1/p)/2}(Γ_{T}).

ii) For 3 < p ≤ q < ∞ the map u ⟼ ∇u · ν Γ_{T}, well-defined for those functions

u ∈ W_{p,q}^{2,1}(Ω_{T}) which are continuously differentiable w.r.t. x ∈ \overline{Ω}, has a continuous extension

γ_{N} : W_{p,q}^{2,1}(Ω_{T}) ⟶ W F_{p,q}^{1−1/p,(1−1/p)/2}(Γ_{T}).

Theorem 2.4. Let 1 ≤ p ≤ q < ∞. For k = 0, 1 there exist continuous linear operators

Z_{k} : W F_{p,q}^{2−k−1/p,(2−k−1/p)/2}(Γ_{T}) ⟶ W_{p,q}^{2,1}(Ω_{T})

such that γ_{D}Z_{0} = I_{0} for the p, q as detailed in Theorem 2.3 i) and γ_{N}Z_{1} = I_{1} for the p, q as detailed in Theorem 2.3 ii), where I_{k} denotes the identity map in W F_{p,q}^{2−k−1/p,(2−k−1/p)/2}(Γ_{T}).

This latter theorem shows in particular that the trace operators γ_{D} and γ_{N} from Theorem 2.3 are onto, thus implying that the trace spaces found are sharp. The first theorem is proved (in a half-space situation) by the “method of integral representation” introduced by the Russian school [7]. The full details are given in [21]. The Dirichlet case in Theorem 2.4 is proved in [18]. Crucial in the proofs of both theorems are certain weighted Hardy inequalities (see [1] and [20]).

3. Existence results of maximal regularity in the space W_{p,q}^{2,1}(Ω_{T}) for

linear parabolic boundary value problems of second order with inhomogeneous boundary conditions

We consider the following problems:

(3.1) u_{t} − a_{ij}(x, t)∂_{i}∂_{j}u + a_{i}(x, t)∂_{i}u + a_{0}(x, t)u = f(x, t)  in Ω_{T} ,
(3.2)  u(x, 0) = u_{0}(x)  in Ω,

with Dirichlet boundary condition

(3.3)  u(ξ, t) = g(ξ, t)  on Γ_{T}

or conormal boundary condition

(3.4)  a_{ij}(ξ, t)ν_{i}(ξ)∂_{j}u(ξ, t) + b_{0}(ξ, t)u(ξ, t) = g(ξ, t)  on Γ_{T} .

For convenience we write u ∈ P_{D} ( u_{0}, f, g) if u solves (3.1), (3.2) with the Dirichlet boundary condition (3.3), and we write u ∈ P_{N} ( u_{0}, f, g) if u solves the same problem with the boundary condition (3.4). We are interested in solutions u ∈ W_{p,q}^{2,1}(Ω_{T}). The theory for q = p is classical; see Ladyzhenskaya, Solonnikov, and Ural’tseva [11, Chapter IV, § 9] for the Dirichlet problem and Solonnikov [13] for the conormal boundary condition. More recently the case q ≠ p was treated, e.g., within the theory of analytic semigroups by interpolation methods (see Cannarsa and Vespri [2]) or by vector-valued Calderón-Zygmund theory (see Hieber and Prüss [6] and Krylov [9]). In these papers only homogeneous boundary conditions were considered. Combining these results with our sharp trace results from Section 2, we can incorporate inhomogeneous boundary conditions and prove the following existence theorems:

Theorem 3.1. Consider (3.1) → (3.3). Assume that Ω is a bounded domain in ℝ^{n} of class C^{2+ε} for some ε > 0 and that

(AD) (a_{ij}(x, t))_{1≤i,j≤n} is symmetric and positive definite uniformly in \overline{Ω} × [ 0, T ],
a_{ij}(x, t) ∈ C^{0}([ 0, T ], C^{0}(\overline{Ω})), a_{i}(x, t) ∈ C^{0}([ 0, T ], L_{r} (Ω)) with r > n,
a_{0}(x, t) ∈ C^{0}([ 0, T ], L_{s} (Ω)) with s > \frac{n}{2};
(ED) 3/2 < p ≤ q < ∞;
(F) f ∈ L_{q} (0, T; L_{p} (Ω));
(gD) g ∈ W F_{p,q}^{2−1/p,(2−1/p)/2}(Γ_{T});
(Iv) u_{0} ∈ B_{p,q}^{2(1−1/q)}(Ω);
(CD) u_{0}(·) = g(·, 0) on Γ.

Then P_{D} ( u_{0}, f, g) has a unique solution u ∈ W_{p,q}^{2,1}(Ω_{T}) and there is a constant c_{D}^{∗}(p, q, T) with

∥u∥_{W_{p,q}^{2,1}(Ω_{T})} ≤ c_{D}^{∗} · (∥u_{0}∥_{B_{p,q}^{2(1−1/q)}(Ω)} + ∥f∥_{L_{q} (0,T; L_{p} (Ω))} + ∥g∥_{W F_{p,q}^{2−1/p,(2−1/p)/2}(Γ_{T})}).

In the next theorem B([ 0, T ], X) denotes the bounded X-valued functions.

Theorem 3.2. Consider (3.1), (3.2), (3.4). Let Ω be as in the previous theorem. Let (F), (Iv) from the previous theorem hold unaltered and assume further that

(AN) (a_{ij}(x, t))_{1≤i,j≤n} is symmetric and positive definite uniformly in \overline{Ω} × [ 0, T ],

a_{ij}(x, t) ∈ C^{0}([ 0, T ], C^{μ}(\overline{Ω})) ∩ B([ 0, T ], C^{1+ε}(Γ)) ∩ C^{α}([ 0, T ], C^{0}(Γ)), b_{0} ∈ B([ 0, T ], C^{μ}(Γ)) ∩ C^{α}([ 0, T ], C^{0}(Γ)) for some ε > 0, α > \frac{1}{2}(1 − 1/p), μ > 1 − 1/p, and a_{i}(x, t), a_{0}(x, t) are as specified in the previous theorem;

(EN) 3 < p ≤ q < ∞;
(gN) g ∈ W F_{p,q}^{1−1/p,(1−1/p)/2}(Γ_{T});
(CN) a_{ij}(ξ, 0)ν_{i}(ξ)∂_{j}u_{0}(ξ) + b_{0}(ξ, 0)u_{0}(ξ) = g(ξ, 0) on Γ .

Then P_{N} ( u_{0}, f, g) has a unique solution u ∈ W_{p,q}^{2,1}(Ω_{T}) and there exists a constant c_{N}^{∗}(p, q, T) with

∥u∥_{W_{p,q}^{2,1}(Ω_{T})} ≤ c_{N}^{∗} · (∥u_{0}∥_{B_{p,q}^{2(1−1/q)}(Ω)} + ∥f∥_{L_{q} (0,T; L_{p} (Ω))} + ∥g∥_{W F_{p,q}^{1−1/p,(1−1/p)/2}(Γ_{T})}).

Remark 3.3. a) The conditions on the coefficients as formulated in the last two theorems are those obtained in [19]. We do not claim that they are sharp.

b) The Besov space specified in Theorem 3.1 (Iv) is sharp; see [17] and the literature cited therein.

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Fraunhofer-Institut Kurzzeitdynamik, Eckerstr. 4, D-79104 Freiburg, Germany

E-mail address: weide@emi.fhg.de