Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 21, 1996, 179–187

AN INHOMOGENEOUS DIRICHLET PROBLEM FOR A NON-HYPOELLIPTIC LINEAR PARTIAL DIFFERENTIAL OPERATOR

Reinhard Hochmuth

Freie Universität Berlin, Department of Mathematics Arnimallee 3, D-14195 Berlin, Germany; hochmuth@math.fu-berlin.de

Abstract. In this paper we state an inhomogeneous Dirichlet problem for a class of linear partial differential operators which are non-hypoelliptic. We prove uniqueness, existence and regularity results for its solutions.

1. Introduction

In [1] K. Doppel and the present author stated a homogeneous Dirichlet problem for non-hypoelliptic linear partial differential operators, especially for a product of uniformly elliptic differential operators with smooth coefficient functions. Here we state an inhomogeneous Dirichlet problem for the same class of partial differential operators. The related homogeneous problem turns out to be an equivalent reformulation of the homogeneous Dirichlet problem in [1]. But this new formulation yields more general results than the former one. For further details of this problem and for some literature relevant in this context we refer to the introduction in [1].

2. The inhomogeneous problem

Most of the definitions used here are taken from [1]. We recapitulate only some notation necessary for understanding the subsequent theorems.

Let Ω_{1} and Ω_{2} be bounded domains in R^{n_{1}} or R^{n_{2}} (n_{1}, n_{2} ≥ 2, respectively) with boundaries ∂Ω_{μ} (μ = 1, 2) of class C^{∞} (cf. e.g. Grisvard [2, Definition 1.2.1.1]). Thus the domains Ω_{μ} satisfy the uniform cone condition and the product domain Ω := Ω_{1} × Ω_{2} ⊂ R^{n} (n = n_{1} + n_{2} ) has the same property (cf. Hochmuth [3, Satz 3.1]). Furthermore, Ω is a Lipschitz domain (cf. Grisvard [2, Theorem 1.2.2.2]).

On each of the domains Ω_{μ} we consider a uniformly elliptic differential operator P_{μ}( · , D_{x_{μ}}) of second order

P_{μ}( · , D_{x_{μ}}) := − ∑_{i,j=1}^{n_{μ}} D_{j}(a_{ij}^{(μ)}(·)D_{i}) + ∑_{i=1}^{n_{μ}} b_{i}^{(μ)}(·)D_{i} + c^{(μ)}(·)

1991 Mathematics Subject Classification: Primary 35B30; Secondary 35A05.

with D_{x_{μ}} = (D_{1}, …, D_{n_{μ}}) and D_{j} = ∂/∂x_{j}, where a_{ij}^{(μ)}, b_{i}^{(μ)}, c^{(μ)} ∈ C^{∞}( \overline{Ω}_{μ}) are given real-valued functions with a_{ij}^{(μ)} = a_{ji}^{(μ)} . Note that then there are constants ϱ_{μ} ∈ R^{+} with

∑_{i,j=1}^{n_{μ}} a_{ij}^{(μ)}(x_{μ})ξ_{i}ξ_{j} ≥ ϱ_{μ} ∑_{i=1}^{n_{μ}} ξ_{i}^{2} for all x_{μ} ∈ Ω_{μ} and (ξ_{1}, …, ξ_{n_{μ}}) ∈ R^{n_{μ}}.

By P̃_{μ}( · , D_{x_{μ}}) we denote the formal adjoint operators

P̃_{μ}( · , D_{x_{μ}}) := − ∑_{i,j=1}^{n_{μ}} D_{j}(a_{ji}^{(μ)}(·)D_{i}) − ∑_{i=1}^{n_{μ}} D_{i}(b_{i}^{(μ)}(·)) + c^{(μ)}(·).

We consider the classical homogeneous elliptic Dirichlet problems:

Problem (Ω_{μ}) (μ = 1, 2). For f_{μ} ∈ C^{0;λ}(Ω_{μ}) ∩ C( \overline{Ω}_{μ}) (λ ∈ (0, 1]) find a function u_{μ} ∈ C^{2}(Ω_{μ}) ∩ C( \overline{Ω}_{μ}) such that

P_{μ}(x_{μ}, D_{x_{μ}})u_{μ}(x_{μ}) = f_{μ}(x_{μ})  for x_{μ} ∈ Ω_{μ},
u_{μ}(x_{μ}) = 0  for x_{μ} ∈ ∂Ω_{μ}.

Now we formulate the inhomogeneous Dirichlet problem of classical type for the non-hypoelliptic product operator P ( · , D_{x}) defined by

P (x, D_{x}) := P_{1}(x_{1}, D_{x_{1}})P_{2}(x_{2}, D_{x_{2}}) for x = (x_{1}, x_{2}) ∈ Ω_{1} × Ω_{2}.

For P ( · , D_{x}) the formal adjoint operator is denoted by P̃( · , D_{x}).

Problem (Ω). For f ∈ C(Ω) and g ∈ C(∂Ω) find a function u ∈ C^{2,2}(Ω) ∩ C(\overline{Ω}) such that

P (x, D_{x})u(x) = f(x)  for x ∈ Ω,
u(x) = g(x)  for x ∈ ∂Ω

is valid.

To give a Hilbert space formulation of problem (Ω) we define

(2.1)  Γ := {σ_{1} ∈ N_{0}^{n_{1}} | |σ_{1}| ≤ 1} × {σ_{2} ∈ N_{0}^{n_{2}} | |σ_{2}| ≤ 1}.

A simple calculation shows that there are functions a_{στ} ∈ C^{∞}( \overline{Ω} ) (σ, τ ∈ Γ) with which we can write the operator P ( · , D_{x}) in the form

(2.2)  P ( · , D_{x}) = ∑_{σ,τ∈Γ}(−1)^{|σ|}D^{σ}(a_{στ} (·)D^{τ} ).

Inhomogeneous Dirichlet problem for non-hypoelliptic differential operator 181

Thus, partial integration on C^{∞}( \overline{Ω} ) × C_{0}^{∞}(Ω) gives the bilinear form b( · , ·):

(2.3)  b(u, ϕ) := (P ( · , D_{x})u, ϕ)_{0;Ω} = ∑_{σ,τ∈Γ}(a_{στ} D^{τ} u, D^{σ}ϕ)_{0;Ω}.

We denote by α^{μ} = (α_{1}^{μ}, …, α_{n}^{μ}) ∈ N_{0}^{n} (μ = 1, 2) multi-indices for which α_{j}^{1} = 0 for all j ∈ {n_{1} + 1, …, n} and α_{k}^{2} = 0 for all k ∈ {1, …, n_{1}}, respectively, and define the (anisotropic) Sobolev space H^{s,t}(Ω) (s, t ∈ N_{0}) as the completion of C_{∗}^{s+t}(Ω) with respect to the norm

∥u∥_{s,t;Ω} := ( ∑_{{α=α^{1}+α^{2} \atop |α^{1}|≤s, |α^{2}|≤t}} ∥D^{α}u∥_{0;Ω}^{2})^{1/2},

and H_{0}^{1,1}(Ω) as the smallest closed subspace of H^{1,1}(Ω) including C_{0}^{∞}(Ω) (cf. [1]).

Because of the boundedness of the functions a_{στ}, there exists a constant c ∈ R^{+} with which

|b(u, φ)| ≤ c∥u∥_{1,1;Ω}∥φ∥_{1,1;Ω} for all (u, φ) ∈ C^{∞}( \overline{Ω} ) × C_{0}^{∞}(Ω),

and the bilinear form b( · , ·) can be continuously extended to H^{1,1}(Ω) × H_{0}^{1,1}(Ω). This extension will also be denoted by b( · , ·). Because the product domain Ω has a Lipschitz boundary ∂Ω and because of the continuous embedding H^{1,1}(Ω) ↪ H^{1}(Ω) we can introduce the trace operator γ: H^{1,1}(Ω) → H^{1/2}(∂Ω) (cf. Grisvard [2, Theorem 1.5.1.3]).

For the weak formulation of the inhomogeneous Dirichlet problem (Ω) we call an admissible boundary function every g ∈ L^{2}(Ω) for which the through

ℓ_{g}(φ) := (g, P̃( · , D_{x})φ)_{0;Ω} for all φ ∈ C_{0}^{∞}(Ω)

dense in H_{0}^{1,1}(Ω) defined linear functional is bounded with respect to ∥ · ∥_{1,1;Ω} , i.e. with some constant c ∈ R^{+} one has

|(g, P̃( · , D_{x})φ)_{0;Ω}| < c∥φ∥_{1,1;Ω} for all φ ∈ C_{0}^{∞}(Ω).

Problem (W). For f ∈ L_{2}(Ω) and for an admissible boundary function g ∈ L_{2}(Ω) find a function u ∈ L_{2}(Ω) such that w := u − g ∈ H^{1,1}(Ω) fulfills the equations

b(w, φ) = (f, φ)_{0;Ω} + (g, P̃( · , D_{x})φ)_{0;Ω}  for all φ ∈ C_{0}^{∞}(Ω),
γw = 0.

For the present it seems that the related homogeneous problem with g = 0 differs from the weak problem (B) in [1]. However, the two formulations are equivalent (cf. Theorem 3.2). But, by the one given here, one can easily discern that the star-shapedness of the domain Ω assumed in [1] is not necessary for the fact that the solutions of problem (Ω) are also solutions of problem (W). Furthermore, it is easier to handle the inhomogeneous problem by this formulation.

3. Uniqueness and existence of solutions

First we state a uniqueness result.

Theorem 3.1. Assume that problems (Ω_{μ}) (μ = 1, 2) are uniquely solvable. Then a function w ∈ H^{1,1}(Ω) with γw = 0 and

b(w, φ) = 0 for all φ ∈ C_{0}^{∞}(Ω)

vanishes, i.e. w = 0.

Proof. Let φ_{μ} ∈ C_{0}^{∞}(Ω_{μ}) (μ = 1, 2) and φ := φ_{1} ⊗ φ_{2} ∈ C_{0}^{∞}(Ω). Then partial integration and the theorem of Fubini give

(3.1)  \begin{aligned}0 & = b(w, φ) = (w, P̃_{2}P̃_{1}φ)_{0;Ω} \\ & = ∫_{Ω_{2}} P̃_{2}φ_{2}(x_{2})(∫_{Ω_{1}} w(x_{1}, x_{2})P̃_{1}φ_{1}(x_{1}) dx_{1}) dx_{2} = (w̃, P̃_{2}φ_{2})_{0;Ω_{2}} \end{aligned}

with w̃ ∈ H^{1}(Ω_{2}) defined by w̃(x_{2}) := (w( · , x_{2}), P̃_{1}φ_{1})_{0;Ω_{1}} for x_{2} ∈ Ω_{2} . Because of γw = 0, i.e. especially

∫_{∂Ω_{2}}(∫_{Ω_{1}} |w(x_{1}, η)|^{2} dx_{1}) dη = 0,

we get for almost all η ∈ ∂Ω_{2}

(3.2)  ∫_{Ω_{1}} |w(x_{1}, η)|^{2} dx_{1} = 0.

Bearing in mind (3.2) and observing

|w̃(x_{2})| = |(w( · , x_{2}), P̃_{1}φ_{1})_{0;Ω_{1}} | ≤ ∥w( · , x_{2})∥_{0;Ω_{1}}∥P̃_{1}φ_{1}∥_{0;Ω_{2}}

we get w̃ ∈ H_{0}^{1}(Ω_{2}). Therefore relation (3.1), the unique solvability of problem (Ω_{2} ) and elliptic regularity yield w̃ = 0, and we have for x_{2} ∈ Ω_{2}

(w( · , x_{2}), P̃_{1}φ_{1})_{0;Ω_{1}} = 0 for φ_{1} ∈ C_{0}^{∞}(Ω_{1}).

Furthermore, we have w( · , x_{2}) ∈ H_{0}^{1}(Ω_{1}) for almost all x_{2} ∈ Ω_{2} . Finally, the unique solvability of problem (Ω_{1} ) and the elliptic regularity give w( · , x_{2}) = 0 for almost all x_{2} ∈ Ω_{2} , i.e. w = 0.

The next theorem shows that H_{0}^{1,1}(Ω) can be characterized by the trace operator γ. A consequence of this theorem is that problem (B) in [1] and problem (W) with g = 0 are equivalent.

Theorem 3.2. A function w ∈ H^{1,1}(Ω) lies in H_{0}^{1,1}(Ω) if and only if γw = 0.

Proof. Because C_{0}^{∞}(Ω) is dense in H_{0}^{1,1}(Ω), the functions w ∈ H_{0}^{1,1}(Ω) have the property γw = 0. Thus we only have to prove that functions w ∈ H^{1,1}(Ω) with γw = 0 lie in H_{0}^{1,1}(Ω). First we remark that the bilinear form b( · , ·) := ( · , ·)_{1,1;Ω} is induced by the Laplace operators P_{μ}( · , D_{x_{μ}}) = −Δ. The boundary value problems (Ω_{μ}) with respect to this partial differential operators are uniquely solvable. On the other hand, for w ∈ H^{1,1}(Ω) a unique w̃ ∈ H_{0}^{1,1}(Ω) exists with

(w − w̃, φ)_{1,1;Ω} = 0 for all φ ∈ H_{0}^{1,1}(Ω).

By γ(w − w̃) = 0 and Theorem 3.1 we have w = w̃, i.e. w ∈ H_{0}^{1,1}(Ω).

To state an existence result we have to introduce the coerciveness: we say that the bilinear form b( · , ·) is H_{0}^{1,1}(Ω)-coercive if there are constants ϱ ∈ R^{+} and q ∈ R with which

b(u, u) ≥ ϱ∥u∥_{1,1;Ω}^{2} − q∥u∥_{0;Ω}^{2} for all u ∈ H_{0}^{1,1}(Ω).

Theorem 3.3. If problems (Ω_{μ}) are uniquely solvable and if the bilinear form b( · , ·) is H_{0}^{1,1}(Ω)-coercive, then for every f ∈ L_{2}(Ω) and for every admissible boundary function g ∈ L_{2}(Ω) there exists a unique solution u = w + g ∈ L_{2}(Ω) of problem (W) with w ∈ H_{0}^{1,1}(Ω).

Proof. The inhomogeneous problem is equivalent to finding a function w ∈ H_{0}^{1,1}(Ω) (cf. Theorem 3.2) with

b(w, φ) = (f, φ)_{0;Ω} + (g, P̃( · , D_{x})φ)_{0;Ω} for all φ ∈ H_{0}^{1,1}(Ω).

The H_{0}^{1,1}(Ω)-coerciveness of the bilinear form b( · , ·) implies the Fredholm property for the latter problem. Therefore, because of Theorem 3.1, there is a unique solution w ∈ H_{0}^{1,1}(Ω). Thus the assertion of the theorem follows with u = w+g.

Remark. Especially every function g ∈ H^{1,1}(Ω) is an admissible boundary function. For g ∈ H^{1,1}(Ω) the proof of Theorem 3.3 implies the existence of a constant c ∈ R^{+} independent of f and g with which

(3.3)  ∥w∥_{1,1;Ω} ≤ c(∥f∥_{0;Ω} + ∥g∥_{1,1;Ω}).

Furthermore, one gets u ∈ H^{1,1}(Ω) and the estimate (3.3) for solutions u.

4. Regularity of solutions

Obviously, the solutions u ∈ C^{∞}( \overline{Ω} ) of the homogeneous boundary value problem considered in [1, Theorem 3.5], are also solutions of problem (W) for g = 0. Analogously to [1] one can show by (3.3) and the density of C^{∞}( \overline{Ω} ) in H^{s,t}(Ω) (cf. Grisvard [2, Theorem 1.4.2.1]):

Theorem 4.1. Let us suppose that problems (Ω_{μ}) are uniquely solvable and that the bilinear form b( · , ·) is H_{0}^{1,1}(Ω)-coercive. Let further f ∈ L_{2}(Ω) and an admissible boundary function g ∈ L_{2}(Ω) be such that for some s, t ∈ N_{0} there is a constant c ∈ R^{+} with

(4.1)  |(f, φ)_{0;Ω} + (g, P̃( · , D_{x})φ)_{0;Ω}| ≤ c∥φ∥_{s,t;Ω}  for all φ ∈ C^{∞}( \overline{Ω} ).

Then there exists a unique solution u = w + g ∈ L_{2}(Ω) of problem (W) with

w ∈ H^{s+2,t+2}(Ω).

Remark. Condition (4.1) is obviously satisfied for functions f ∈ H^{s,t}(Ω) and g ∈ H^{s+2,t+2}(Ω). In this case one gets u ∈ H^{s+2,t+2}(Ω) and a constant c_{s,t} ∈ R^{+} independent of f and g with

∥u∥_{s+2,t+2;Ω} ≤ c_{s,t}(∥f∥_{s,t;Ω} + ∥g∥_{s+2,t+2;Ω}).

By using Theorem 4.1, the remark above, the continuous embedding H^{s,t}(Ω) ↪ H^{min(s,t)}(Ω) and the usual Sobolev embedding one gets regularity results similar to the classical ones. But the product property of the domain Ω allows us to prove more appropriate embedding and regularity theorems.

Theorem 4.2. For the domains Ω_{μ} ⊂ R^{n_{μ}} (μ = 1, 2) satisfying the uniform cone condition and for Ω = Ω_{1} × Ω_{2} we have a continuous embedding

H^{s_{1},s_{2}}(Ω) ↪ C( \overline{Ω} )

for numbers s_{μ} ∈ N_{0} with s_{μ} > n_{μ}/2.

Proof. By assumption also the domain Ω satisfies the uniform cone condition and therefore C^{∞}( \overline{Ω} ) lies dense in H^{s_{1},s_{2}}(Ω). Thus it is enough to prove for the functions u ∈ C^{∞}( \overline{Ω} ) the estimate

(4.2)  sup_{x∈ \overline{Ω}} |u(x)| ≤ c∥u∥_{s_{1},s_{2};Ω}

for some constant c ∈ R^{+} . Then continuation yields (4.2) on H^{s_{1},s_{2}}(Ω), and thus the statement of the theorem is valid.

To give a detailed proof we have to introduce some notation concerning the assumed uniform cone condition. Here cones C(r_{μ}, Σ_{μ}) with vertex at x_{μ} = 0 are sets

C(r_{μ}, Σ_{μ}) := B(r_{μ}) ∩ {λy_{μ} | y_{μ} ∈ Σ_{μ}, λ > 0},

where Σ_{μ} is an open set on the surface of the ball B(r_{μ}) = {y_{μ} ∈ R^{n_{μ}} | |y_{μ}| < r_{μ}}.

Let x = (x_{1}, x_{2}) ∈ \overline{Ω}_{1} × \overline{Ω}_{2} ⊂ \overline{Ω} and let C_{x_{μ}} = C(r_{μ}, Σ_{μ}) be cones related to the uniform cone condition with respect to Ω_{μ}, especially x_{μ} − C_{x_{μ}} ⊂ Ω_{μ}.

Furthermore, let f_{μ} be fixed functions in C_{0}^{∞}(R^{n_{μ}}) with the properties

1. f_{μ}(y_{μ}) = 1 for |y_{μ}| < 1,

2. f_{μ}(y_{μ}) = 0 for |y_{μ}| ≥ 2,

3. |D^{α}f_{μ}(y_{μ})| ≤ M for y_{μ} ∈ R^{n_{μ}}, |α| ≤ s_{μ} + 1 and with some M ∈ R^{+} . We define the functions e_{μ}(y_{μ}) := f_{μ}(2(y_{μ} − x_{μ})/r_{μ}) on R^{n_{μ}}. Thus we have

e_{μ}(y_{μ}) = \{ \begin{matrix}1 & for |y_{μ} − x_{μ}| < r_{μ}/2, \\ 0 & for |y_{μ} − x_{μ}| ≥ r_{μ} \end{matrix}

and

(4.3)  |D^{α}e_{μ}(y_{μ})| ≤ \frac{M}{r_{μ}^{|α|}}  for |α| ≤ s_{μ}.

For the unit vectors η_{μ} with x_{μ} + η_{μ}ϱ_{μ} ∈ Ω_{μ} for 0 < ϱ_{μ} < r_{μ} we have

\begin{aligned}u(x_{1}, x_{2}) & = −e_{1}(x_{1} + η_{1}ϱ_{1})u(x_{1} + η_{1}ϱ_{1}, x_{2})|_{ϱ_{1}=0}^{ϱ_{1}=r_{1}} \\ & = − ∫_{0}^{r_{1}} \frac{∂(e_{1}(x_{1} + η_{1}ϱ_{1})u(x_{1} + η_{1}ϱ_{1}, x_{2}))}{∂ϱ_{1}} dϱ_{1}, \end{aligned}

and by partial integration we get

u(x_{1}, x_{2}) = \frac{(−1)^{s_{1}}}{(s_{1} − 1)!} ∫_{0}^{r_{1}} ϱ_{1}^{s_{1}−1} \frac{∂^{s_{1}}(e_{1}(x_{1} + η_{1}ϱ_{1})u(x_{1} + η_{1}ϱ_{1}, x_{2}))}{∂ϱ_{1}^{s_{1}}} dϱ_{1}.

By using e_{2} and the Fubini theorem one analogously gets

\begin{aligned}u(x_{1}, x_{2}) & = \frac{(−1)^{s_{1}+s_{2}}}{(s_{1} − 1)!(s_{2} − 1)!} ∫_{0}^{r_{1}} ∫_{0}^{r_{2}} ϱ_{1}^{s_{1}−1}ϱ_{2}^{s_{2}−1} \frac{∂^{s_{1}+s_{2}}}{∂ϱ_{1}^{s_{1}}∂ϱ_{2}^{s_{2}}} [e_{1}(x_{1} + η_{1}ϱ_{1}) \\ & × e_{2}(x_{2} + η_{2}ϱ_{2})u(x_{1} + η_{1}ϱ_{1}, x_{2} + η_{2}ϱ_{2})] dϱ_{1}dϱ_{2}. \end{aligned}

Integration over Σ_{1} × Σ_{2} , again the Fubini theorem and the Schwarz inequality yield with C^{x_{μ}} := x_{μ} − C_{x_{μ}} (4.4)

|Σ_{1} × Σ_{2}| |u(x_{1}, x_{2})|

= \frac{1}{(s_{1} − 1)!(s_{2} − 1)!}|∫_{C^{x_{1}}} ∫_{C^{x_{2}}} ϱ_{1}^{s_{1}−n_{1}}ϱ_{2}^{s_{2}−n_{2}} \frac{∂^{s_{1}+s_{2}}}{∂ϱ_{1}^{s_{1}}∂ϱ_{2}^{s_{2}}} [e_{1}e_{2}u] dy_{1} dy_{2}|

≤ c ∫_{C^{x_{1}}} ϱ_{1}^{s_{1}−n_{1}} |∫_{C^{x_{2}}} ϱ_{2}^{2(s_{2}−n_{2})} dy_{2}|^{1/2}|∫_{C^{x_{2}}} |\frac{∂^{s_{1}+s_{2}}}{∂ϱ_{1}^{s_{1}}∂ϱ_{2}^{s_{2}}} [e_{1}e_{2}u]|^{2} dy_{2}|^{1/2} dy_{1}

≤ c|∫_{C^{x_{1}}} ϱ_{1}^{2(s_{1}−n_{1})} dy_{1}|^{1/2}|∫_{C^{x_{2}}} ϱ_{2}^{2(s_{2}−n_{2})} dy_{2}|^{1/2}

× |∫_{C^{x_{1}}} ∫_{C^{x_{2}}} |\frac{∂^{s_{1}+s_{2}}}{∂ϱ_{1}^{s_{1}}∂ϱ_{2}^{s_{2}}} [e_{1}e_{2}u]|^{2} dy_{1} dy_{2}|^{1/2}

≤ c|∫_{C^{x_{1}}} ϱ_{1}^{2(s_{1}−n_{1})} dy_{1}|^{1/2}|∫_{C^{x_{2}}} ϱ_{2}^{2(s_{2}−n_{2})} dy_{2}|^{1/2}∥e_{1}e_{2}u∥_{s_{1},s_{2};Ω}.

We have (cf. Wloka [4, Section 6])

(4.5)  ∫_{C^{xμ}} ϱ_{μ}^{2(s_{μ}−n_{μ})} dy_{μ} = \frac{2π^{n_{μ}/2}}{Γ(n_{μ}/2)(2s_{μ} − n_{μ})}r_{μ}^{2s_{μ}−n_{μ}}.

If we apply the Leibniz product rule to e_{1}e_{2}u and refer back to (4.3) we obtain

(4.6)  ∥e_{1}e_{2}u∥_{s_{1},s_{2};Ω} ≤ c · max_{|α_{1}|≤s_{1}} r_{1}^{−|α_{1}|} · max_{|α_{2}|≤s_{2}} r_{2}^{−|α_{2}|} · ∥u∥_{s_{1},s_{2};Ω}.

Taking account of (4.5) and (4.6) in (4.4), we get

|Σ_{1} ×Σ_{2}| |u(x_{1}, x_{2})| ≤ c·r_{1}^{s_{1}−n_{1}/2} · max_{|α_{1}|≤s_{1}} r_{1}^{−|α_{1}|} ·r_{2}^{s_{2}−n_{2}/2} · max_{|α_{2}|≤s_{2}} r_{2}^{−|α_{2}|} ·∥u∥_{s_{1},s_{2};Ω},

i.e. (4.2).

Corollary 4.3. For domains Ω_{μ} ⊂ R^{n_{μ}} (μ = 1, 2) satisfying the uniform cone condition and for Ω = Ω_{1} × Ω_{2} we have the continuous embedding

H^{s_{1},s_{2}}(Ω) ↪ C^{t_{1},t_{2}}( \overline{Ω} )

for s_{μ}, t_{μ} ∈ N_{0} with s_{μ} − t_{μ} > n_{μ}/2.

Acknowledgement. I thank Professor K. Doppel and Professor I.S. Louhivaara for their stimulating and valuable comments during the preparation of this paper.

References

[1] Doppel, K., and R. Hochmuth: On the regularity of solutions of a homogeneous Dirichlet problem for a non-hypoelliptic linear partial differential operator. - Ann. Acad. Sci. Fenn. Ser. A I Math. 16, 1991, 183–198.

[2] Grisvard, P.: Elliptic problems in nonsmooth domains. - Monographs and Studies in Mathematics 24, Pitman, Boston, 1985.

[3] Hochmuth, R.: Regularitätsresultate für ein Randwertproblem einer nicht-hypoellip-tischen linearen partiellen Differentialgleichung. - Dissertation, Freie Universität Berlin, 1989.

[4] Wloka, J.: Partial differential equations. - Cambridge University Press, Cambridge, 1987.

Received 24 October 1994