Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 3, 539–555

Mixed Boundary Value Problems for Nonlinear Elliptic Systems in n-Dimensional Lipschitzian Domains

C. Ebmeyer

Abstract. Let u : Ω → ℝ^{N} be the solution of the nonlinear elliptic system

− ∑_{i=1}^{n} ∂_{i}F_{i}(x, ∇u) = f(x) + ∑_{i=1}^{n} ∂_{i}f_{i}(x),

where Ω ⊂ ℝ^{n} is a bounded domain with a piecewise smooth boundary (e.g., Ω is a polyhedron). It is assumed that a mixed boundary value condition is given. Global regularity results in Sobolev and in Nikolskii spaces are proven, in particular [W^{s,2}(Ω)]^{N}-regularity (s < \frac{3}{2} ) of u.

Keywords: Mixed boundary value problems, piecewise smooth boundaries, Nikolskii spaces AMS subject classification: Primary 35 J 55, 35 J 65, secondary 35 J 25

0. Introduction

We treat the nonlinear elliptic system

\begin{matrix}− ∑_{i=1}^{n} ∂_{i}F_{i}(x, ∇u) & = f(x) + ∑_{i=1}^{n} ∂_{i}f_{i}(x)  in Ω \\ u(x) & = 0  on Γ_{D} \\ − ∑_{i=1}^{n} F_{i}(x, ∇u)ν_{i} & = ∑_{i=1}^{n} f_{i}ν_{i}  on Γ_{N} \end{matrix} \}  (0.1)

where Ω ⊂ ℝ^{n} (n ≥ 3) is bounded, u : Ω → ℝ^{N} is a vector-valued function, ∂_{i} = \frac{∂}{∂x_{i}}, ∂Ω = Γ_{D} ∪ Γ_{N} where Γ_{D} is the Dirichlet boundary and Γ_{N} is the Neumann boundary, and ν is the outward normal of ∂Ω. We suppose that ∂Ω is piecewise smooth (e.g., Ω is a polyhedron or has a Lipschitz boundary).

In this paper we investigate the regularity of the solution u of (0.1). Refining the method of [8] we obtain regularity results in Nikolskii spaces and in Sobolev spaces [W^{s,2}(Ω)]^{N}, especially [W^{s,2}(Ω)]^{N}-regularity (s < \frac{3}{2}) of u up to the boundary.

C. Ebmeyer: Universität Bonn, Mathematisches Seminar, Nußallee 15, D - 53115 Bonn ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

Solutions of mixed boundary value problems in non-smooth domains may have singularities on the boundary at such points where the boundary condition is changing or where ∂Ω is not smooth.

In the case of a linear elliptic equation various authors have investigated the regularity of the solution. They have given a decomposition of the solution u into a regular and a singular part.In particular, for Ω ⊂ ℝ^{2} this provides an explicit description of the behaviour of u near the boundary (cf. [4, 7, 9, 11]). In the case when Ω ⊂ ℝ^{n} (n ≥ 3) there are difficulties by finding such a decomposition which describes all the singularities of u (see [2, 3, 10, 14, 17]).

In the case of nonlinear equations there are only few results. Semilinear Dirichlet problems on corner domains are treated in [12, 15] and in [5, 6], where results in weighted Sobolev spaces are given. Further, nonlinear mixed boundary value problems are investigated in [8]. Regularity results in Sobolev spaces are proven.

In this paper we generalize some results given in [8]. Let the boundary of Ω consist of smooth (n − 1)-dimensional manifolds with piecewise smooth boundaries such that each boundary manifold is either a Dirichlet or a Neumann boundary manifold. Let us fix some point P ∈ ∂Ω. Then we suppose that there is a ball B(P) and a smooth mapping which maps Ω onto a domain Ω̂ such that B(P)∩Ω̂ is the intersection of B(P) and a polyhedron. In contrast to [8] we consider the case that B(P) ∩ ∂Ω̂ contains more than one Dirichlet boundary manifold. Further, we admit that B(P) ∩ Ω̂ is probably not convex. But we assume that each inner angle between a Dirichlet and a Neumann boundary manifold is not greater than π.

We suppose that there is a function F (x, p) such that F_{i}^{r}(x, p) is the partial derivative of F (x, p) with respect to the component corresponding to p_{i}^{r} (here F_{i}^{r}(x, p) denotes the r-th component of the vector F_{i}(x, p)). Hence, we deal with the variational case.

The aim of this paper is to show that u ∈ [W^{s,2}(Ω)]^{N} for s < \frac{3}{2}. This result is the best possible, for we admit that Ω̂ can be a polyhedron where the inner angle between a Dirichlet and a Neumann boundary manifold is equal to π. Otherwise, if all such angles are less than π, we prove that u ∈ H^{\frac{3}{2} ,2}(Ω), where H^{s,p}(Ω) denotes a Nikolskii space. Moreover, in the case when N = 1 the solution u of equation (0.1) is Hölder continuous. Then we show that u ∈ L^{p}(Ω) for some p > 3.

This paper is organized as follows. In Section 1 we state the assumptions on the data and the main results. Section 2 contains notations. In Section 3 the proofs of the main results are given. Finally, in Section 4 we explain the proofs with examples of tree-dimensional domains.

1. Assumptions on the data and main results

We need the following assumptions on the data.

(A1) Ω ⊂ ℝ^{n} (n ≥ 3) is a connected open domain with Lipschitz boundary.

(A2) ∂Ω = ⋃_{1≤i≤M} \overline{Γ_{i}}, where Γ_{i} are open (n − 1)-dimensional manifolds, and Γ_{i} ∩ Γ_{j} = ∅ holds for i ≠ j.

Mixed Boundary Value Problems 541

(A3) ∂Γ_{i} (1 ≤ i ≤ M) are (n − 2)-dimensional Lipschitz continuous manifolds. (A4) Γ_{1}, …, Γ_{σ} ⊂ Γ_{D} and Γ_{σ+1}, …, Γ_{M} ⊂ Γ_{N}. (A5) P ∈ ⋂_{i∈Λ} ∂Γ_{i} implies that |Λ| ≤ n. (A6) To each point P ∈ ∂Ω there exists a mapping ϕ and a ball B_{R}(ϕ(P)) such that:

(i) B_{R}(ϕ(P)) ∩ ϕ(∂Ω) is the intersection of B_{R}(ϕ(P)) and a polyhedron.

(ii) B_{R}(ϕ(P)) ∩ ϕ(∂Ω) is simply connected.

(iii) ϕ, ϕ^{−1} ∈ W_{loc}^{2,∞}(ℝ^{n}) and the Jacobian of ϕ is positive definite.

(iv) If Γ_{i} ∈ Γ_{D}, Γ_{j} ∈ Γ_{N}, and ∂Γ_{i} ∩ ∂Γ_{j} ≠ ∅, then angle(ϕ(Γ_{i}), ϕ(Γ_{j})) ≤ π.

(v) At most one pair of boundary manifolds Γ_{i}, Γ_{j} (i ≠ j, ∂Γ_{i} ∩ ∂Γ_{j} ≠ ∅) satisfies angle(ϕ(Γ_{i}), ϕ(Γ_{j})) = π.

Remark.

(i) By angle(ϕ(Γ_{i}), ϕ(Γ_{j})) we denote the inner angle between ϕ(Γ_{i})∩B_{R}(ϕ(P)) and ϕ(Γ_{j})∩B_{R}(ϕ(P)) where it is assumed that ϕ(Γ_{i})∩B_{R}(ϕ(P)) ≠ ∅ and ϕ(Γ_{j})∩B_{R}(ϕ(P)) ≠ ∅.

(ii) We assume that the inner angle between a boundary manifold of ϕ(Γ_{D}) and another one of ϕ(Γ_{N}) is not greater than π (cf. assumption (A6)/(ii)). But it is admitted that the inner angle between two boundary manifolds is greater than π if there is no change of the boundary value condition.

(iii) It is also possible to treat domains with a slit. Then instead of assumption (A6)/(v) we need the assumption that at most one pair of boundary manifolds Γ_{i}, Γ_{j} (i ≠ j, ∂Γ_{i} ∩ ∂Γ_{j} ≠ ∅) satisfies angle(ϕ(Γ_{i}), ϕ(Γ_{j})) = μπ, μ ∈ {1, 2}.

Let x ∈ \overline{Ω} and p ∈ ℝ^{nN} with components x_{i} (1 ≤ i ≤ n) and p_{i}^{r} (1 ≤ r ≤ N), respectively. We suppose that there is a C^{2}-function F (x, p) : Ω × ℝ^{nN} → ℝ such that \frac{∂}{∂p_{i}^{r}} F (x, p) = F_{i}^{r}(x, p) for all 1 ≤ i ≤ n and 1 ≤ r ≤ N, where F_{i}^{r}(x, p) denotes the r-th component of F_{i}(x, p) ∈ ℝ^{N}. We set

F_{x_{i}}(x, p) = \frac{∂}{∂x_{i}} F (x, p), F_{i,x_{k}}(x, p) = \frac{∂}{∂x_{k}} F_{i}(x, p), F_{i,k}^{rs}(x, p) = \frac{∂}{∂p_{k}^{s}} F_{i}^{r}(x, p)

for 1 ≤ i, k ≤ n and 1 ≤ r, s ≤ N. Furthermore, we suppose that there are functions g_{0}, g_{x_{i}}, g_{i} and g_{i,x_{k}} (1 ≤ i, k ≤ n) such that: (H1) c_{0} + c′_{0}|p|^{2} ≤ F (x, p) ≤ g_{0}(x) + c|p|^{2} for g_{0} ∈ L^{∞}(Ω) and c′_{0} > 0. (H2) |F_{x_{i}}(x, p)| ≤ g_{x_{i}}(x) + c|p|^{2} for g_{x_{i}} ∈ L^{1}(Ω). (H3) |F_{i}(x, p)| ≤ g_{i}(x) + c|p| for g_{i} ∈ L^{2}(Ω). (H4) |F_{i,x_{k}}(x, p)| ≤ g_{i,x_{k}}(x) + c|p| for g_{i,x_{k}} ∈ L^{2}(Ω). (H5) |F_{i,k}^{rs}(x, p)| ≤ c. (H6) There is a constant k_{0} > 0 independent of x and p such that for all ξ ∈ ℝ^{nN}

k_{0}|ξ|^{2} ≤ ∑_{r,s=1}^{N} ∑_{i,k=1}^{n} F_{i,k}^{rs}(x, p)ξ_{i}^{r}ξ_{k}^{s}.

(H7) f^{r}(x) ∈ L^{2}(Ω) and f_{i}^{r}(x) ∈ W^{1,2}(Ω) ∩ L^{∞}(Ω) for 1 ≤ i ≤ n and 1 ≤ r ≤ N.

Remark. Hypothesis (H6) can be replaced by the weaker condition

(H6′) There are constants k_{0} > 0 and k_{1} independent of x and p such that for all ξ ∈ [H^{1}(Ω)]^{N}

k_{0} ∫_{Ω} |∇ξ|^{2}dx − k_{1} ∫_{Ω} |ξ|^{2}dx ≤ ∫_{Ω} ∑_{r,s=1}^{N} ∑_{i,k=1}^{n} F_{i,k}^{rs}(x, ∇u)∂_{i}ξ^{r}∂_{k}ξ^{s}dx.

Let us note that the changes to be made in the proofs are obvious.

Under the above hypotheses there exists a unique weak solution u ∈ [W^{1,2}(Ω)]^{N} of problem (0.1) (see [16]).

We use the usual Sobolev spaces W^{s,p}(Ω) and the Nikolskii spaces H^{s,p}(Ω) (cf. [1]). In detail, let s be no integer, let z ∈ ℝ^{n}, s = m + σ where 0 < σ < 1 and m is an integer, Ω_{η} = {x ∈ Ω : dist(x, ∂Ω) ≥ η}, and 1 ≤ p < ∞. The spaces W^{s,p}(Ω) and H^{s,p}(Ω) consist of all functions u for which the norms

∥u∥_{W^{s,p}(Ω)} = (∥u∥_{W^{m,p}(Ω)}^{p} + ∑_{|α|=m} ∫_{Ω} ∫_{Ω} \frac{|∂^{α}u(x) − ∂^{α}u(y)|^{p}}{|x − y|^{n+pσ}} dxdy)^{\frac{1}{p}}

and

∥u∥_{H^{s,p}(Ω)} = (∥u∥_{L^{p}(Ω)}^{p} + ∑_{|α|=m} sup_{{η>0 \atop 0<|z|<η}} ∫_{Ω_{η}} \frac{|∂^{α}u(x + z) − ∂^{α}u(x)|^{p}}{|z|^{σp}} dx)^{\frac{1}{p}}

are finite.

We will prove the following results:

Theorem 1.1.

a) The solution u of equation (0.1) satisfies

u ∈ [W^{s,2}(Ω)]^{N}  for all s < \frac{3}{2}.  (1.1)

b) If angle(Γ_{i}, Γ_{j}) ≠ π for each pair of boundary manifolds Γ_{i}, Γ_{j} (i ≠ j, ∂Γ_{i}∩∂Γ_{j} ≠ ∅), then

u ∈ [H^{\frac{3}{2} ,2}(Ω)]^{N}  (1.2)

holds.

Remark.

(i) By assumption we consider the case when n ≥ 3. But our proofs of (1.1) and (1.2) also hold when n = 2.

(ii) angle(Γ_{i}, Γ_{j}) ≠ π implies that angle(ϕ(Γ_{i}), ϕ(Γ_{j})) ≠ π, for ϕ is smooth.

Using the Sobolev imbedding theorem and (1.1) we get u ∈ [W^{1,s}(Ω)]^{N} for s < \frac{2n}{n−1}. Let us note that s < 3 for n ≥ 3. The next theorem improves this result in the case when N = 1.

Theorem 1.2. Let N = 1 and let the functions g_{x_{i}}, g_{i}, g_{i,x_{k}}, f and f_{k} given in hypotheses (H1) - (H7) satisfy

g_{i} ∈ L^{\frac{n}{1−δ}} (Ω),  g_{x_{i}}, g_{i,x_{k}}, f, ∂_{i}f_{k} ∈ L^{\frac{2n}{3−δ}} (Ω)  (1.3)

for 1 ≤ i, k ≤ n and some δ > 0. Then there exists a constant ε_{0} > 0 independent of n such that the solution u of equation (0.1) satisfies

∇u ∈ L^{s}(Ω)  for s = 3 + ε_{0}.  (1.4)

Remark. The results of Theorem 1.1 and Theorem 1.2 also hold for solutions u(x, t) of parabolic systems. Let u(x, 0) ∈ [W^{1,2}(Ω)]^{N}. Then we get the results (1.1), (1.2), and (1.4) in the spaces [L^{2}(0, T; W^{s,p}(Ω))]^{N} and [L^{2}(0, T; H^{s,p}(Ω))]^{N}.

2. Notations

Let B_{R}(x) = {y ∈ ℝ^{n} : |x − y| < R}. The boundary of Ω is piecewise smooth. By assumption to each point P ∈ ∂Ω there is a constant R_{0} > 0 and a W^{2,∞}-mapping

ϕ^{∗} : x → x̂

such that B_{R_{0}}(P̂) ∩ Ω̂ is the intersection of B_{R_{0}}(P̂) and a polyhedron. (We use the denotations P̂ = ϕ^{∗}(P), Ω̂ = ϕ^{∗}(Ω) etc. and we will write B_{R} instead of B_{R}(P̂).)

In the sequel we suppose that P̂ and R_{0} ∈ (0, 1] are fixed such that P̂ is the only vertex of B_{R_{0}}(P̂) ∩ ∂Ω̂ or that there is no vertex of ∂Ω̂ in B_{R_{0}}(P̂). Further, let P̂ ∈ ∂Γ̂_{k} for some k ∈ {1, …, M}.

We need appropriate basis vectors {ζ^{1}, …, ζ^{n}} in B_{R_{0}}(P̂). Let Λ_{1}, Λ_{2}, and Λ_{3} be disjoint index sets (some of them possibly empty) such that ∪_{i=1}^{3}Λ_{i} = {1, …, n}. Let α^{∗} > 0, |ζ^{i}| = 1 for 1 ≤ i ≤ n, and angle(ζ^{i}, ζ^{j}) ≥ α^{∗} for 1 ≤ i < j ≤ n. We assume the following:

1) y + sζ^{i} ∈ (Ω̂ ∪ ∂Ω̂) for y ∈ (∂Ω̂ ∩ B_{R_{0}}), 0 < s < R_{0}, and 1 ≤ i ≤ n.

2) If Γ̂_{D} ∩ B_{R_{0}} ≠ ∅, then ζ^{i} (i ∈ Λ_{1}) is parallel to Γ̂_{D} ∩ B_{R_{0}}.

3) If Γ̂_{D} ∩ B_{R_{0}} = ∅, then Λ_{1} = {1, …, n}.

4) If i ∈ Λ_{1}, y ∈ (Γ̂_{D} ∩ B_{R_{0}}), s > 0, and y + sζ^{i} ∈ B_{R_{0}}, then y + sζ^{i} ∈ Γ̂_{D}.

5) If Γ̂_{N} ∩ B_{R_{0}} ≠ ∅, then ζ^{i} (i ∈ Λ_{2}) is parallel to Γ̂_{N} ∩ B_{R_{0}}.

6) If Γ̂_{N} ∩ B_{R_{0}} = ∅, then Λ_{2} = {1, …, n}.

7) ζ^{i} (i ∈ Λ_{2}) satisfies i) angle(ζ^{i}, Γ̂_{D} ∩ B_{R_{0}}) ≥ α^{∗} ii) y − sζ^{i} ∉ (Ω̂ ∪ ∂Ω̂) for y ∈ (Γ̂_{D} ∩ B_{R_{0}}), and 0 < s < R_{0}.

8) If angle(Γ̂_{i}, Γ̂_{j}) = π (i ≠ j, Γ̂_{i} ∩ Γ̂_{j} ∩ B_{R_{0}} ≠ ∅), then Λ_{3} = {n}, otherwise Λ_{3} = ∅.

9) ζ^{n} (n ∈ Λ_{3}) satisfies angle(ζ^{n}, (Γ̂_{i} ∪ Γ̂_{j}) ∩ B_{R_{0}}) ≥ α^{∗} where i, j are given in Assumption 8).

Remark.

i) Let us note that there is such a basis. Some examples how to choose the basis vectors are given in Section 4.

ii) We can find a constant α^{∗} depending only on n and on the geometry of ∂Ω.

In the sequel let h > 0. We define E_{i}^{σ}y = y + σζ^{i}, E_{i}^{σ}f(y) = f(y + σζ^{i}),

D_{i}^{h}f(y) = \frac{E_{i}^{h}f(y) − f(y)}{h}  and  D_{i}^{−h}f(y) = \frac{f(y) − E_{i}^{−h}f(y)}{h}

and we will write E_{i}^{σ}f(y)g(y) instead of (E_{i}^{σ}f(y))g(y).

We set R = \frac{R_{0}}{8} , B = B_{R} ∩ Ω̂, B′ = B_{4R} ∩ Ω̂, and
Ω̂_{i}^{h} = {y ∈ B_{R_{0}} : y ≠ x + hζ^{i}, x ∈ B_{R_{0}}}

Ω̂_{i}^{−h} = {y ∈ B_{R_{0}} \ Ω̂ : y = x − hζ^{i}, x ∈ B_{R_{0}} ∩ Ω̂}.

Let τ_{0} be a cut-off function with τ_{0} ≡ 1 in B, supp τ_{0} = B_{4R}, and |∇τ_{0}| ≤ c, where c depends only on R_{0}. By τ we denote the restriction of τ_{0} onto Ω̂ ∪ ∂Ω̂.

Moreover, we need appropriate extensions of functions into Ω̂_{i}^{−h} for i ∈ Λ_{2}. Let the function g(y) be defined on Ω̂. Let z_{0} ∈ ∂Ω̂ ∩ B_{R_{0}} and z_{0} − λζ^{i} ∈ Ω̂_{i}^{−h} for 0 < λ ≤ h. Then we set

g(z_{0} − λζ^{i}) = g(z_{0} + λζ^{i}).  (2.1)

This is an W^{1,2}-extension if g ∈ W^{1,2}(Ω̂). In particular, it holds that ∥g∥_{W^{1,2}(Ω̂_{i}^{−h})} ≤ c∥g∥_{W^{1,2}(Ω̂)}, where the constant c depends only on the data, for α^{∗} depends only on n and on the geometry of ∂Ω.

Next, we define an appropriate extension of v = u ∘ (ϕ^{∗})^{−1} into Ω̂_{i}^{−h} for i ∈ Λ_{2}. Let y ∈ ∂Ω̂ ∩ ∂Ω̂_{i}^{−h}, 0 < λ ≤ h, and y − λζ^{i} ∈ Ω̂_{i}^{−h}. We set

v(y − λζ^{i}) = 0.  (2.2)

This provides an W^{1,2}-extension of v, for i ∈ Λ_{2} implies that (∂Ω̂ ∩ ∂Ω̂_{i}^{−h}) ⊂ Γ̂_{D}. In particular, it holds for 1 ≤ r ≤ N that

∥v^{r}∥_{H^{\frac{3}{2} ,2}(Ω̂_{i}^{−h})} ≤ c∥v^{r}∥_{H^{\frac{3}{2} ,2}(Ω̂)}

where c and c′ depend only on the data and v^{r} is the r-th component of v. Thus, extension (2.2) is an H^{\frac{3}{2} ,2}-extension (cf. [8]).

In what follows we will write ∑_{i,k,l} and ∑_{r,s} instead of ∑_{i,k,l=1}^{n} and ∑_{r,s=1}^{N}, respectively. Further, ∇v is an ℝ^{nN}-vector and |∇v|^{2} = ∑_{r} ∑_{i} |∂_{i}v^{r}|^{2}. The point · denotes the Euclidean scalar product and c denotes a constant which will be allowed to vary from equation to equation.

3. The regularity of the solution

In this section we prove Theorem 1.1 and Theorem 1.2.

Let A be the matrix whose elements are defined by a_{ik} = \frac{∂}{∂x_{i}} (ϕ^{∗k}), where ϕ^{∗k} denotes the k-th component of ϕ^{∗}(x). Let y = x̂. In the sequel we only deal with functions defined onto Ω̂. For simplicity we will write f(y) instead of f((ϕ^{∗})^{−1}(y)) etc. The function v = u ∘ (ϕ^{∗})^{−1} is the weak solution of

− ∑_{i} ˜∂_{i}F_{i}(y, ˜∇v) = f(y) + ∑_{i} ˜∂_{i}f_{i}(y)  (3.1)

where ˜∂_{i}v(y) = ∑_{k} a_{ik}(y)∂_{k}v(y).

In detail, A is positive definite, the smallest eigenvalue λ_{0} > 0 depends only on the geometry of ∂Ω, and

a_{ik}(y) ∈ W^{1,∞}(Ω̂)  (3.2)

holds. Further, let us note that v(y) ∈ [W^{1,2}(Ω̂)]^{N}.

We need several propositions.

Proposition 3.1. It holds that

sup_{0<h<4R} ∫_{B′} τh|D_{i}^{h}∇v|^{2}dy ≤ c  for i ∈ Λ_{1}  (3.3)

where the constant c depends only on R_{0} and the data.

Proof. Let 0 < h < 4R. First, we suppose that 1 ∈ Λ_{1} and we prove (3.3) for i = 1. The Taylor expansion of F (y, p) (p ∈ ℝ^{nN}) entails

\begin{matrix}∑_{r} ∑_{i}(p′ − p)_{i}^{r}F_{i}^{r}(y, p) = F (y, p′) − F (y, p) \\ − ∑_{r,s} ∑_{i,k}(p′ − p)_{i}^{r}(p′ − p)_{k}^{s} ∫_{0}^{1}(1 − t)F_{i,k}^{rs}(y, tp′ + (1 − t)p) dt. \end{matrix}  (3.4)

Let

m_{ik}^{rs}(h) = ∫_{0}^{1}(1 − t)F_{i,k}^{rs}(y, tE_{1}^{h} ˜∇v + (1 − t) ˜∇v) dt

for 1 ≤ i, k ≤ n and 1 ≤ r, s ≤ N. We set p = ˜∇v and p′ = E_{1}^{h} ˜∇v. Thus, (p′ − p)_{i}^{r} = hD_{1}^{h} ˜∂_{i}v^{r} ≡ ∑_{l} hD_{1}^{h}(a_{il}∂_{l}v^{r}) and

\begin{matrix}∑_{r} ∑_{i,l} F_{i}^{r}(y, ˜∇v)D_{1}^{h}(a_{il}∂_{l}v^{r}) = \frac{F (y, E_{1}^{h} ˜∇v) − F (y, ˜∇v)}{h} \\ − ∑_{r,s} ∑_{i,k} h (∑_{l} D_{1}^{h}(a_{il}∂_{l}v^{r})) (∑_{l} D_{1}^{h}(a_{kl}∂_{l}v^{s})) m_{ik}^{rs}(h). \end{matrix}  (3.5)

546 C. Ebmeyer

The function φ = τD_{1}^{h}v is an admissible test function. Multiplying (3.1) by φ yields

∑_{i,l} ∫_{B′} F_{i}(y, ˜∇v) · ∂_{l}(a_{il}τ)D_{1}^{h}v + ∑_{i,l} ∫_{B′} F_{i}(y, ˜∇v) · (a_{il}τ)∂_{l}D_{1}^{h}v

= ∫_{B′} τf · D_{1}^{h}v − ∑_{i,l} ∫_{B′} f_{i} · ∂_{l} (a_{il}τD_{1}^{h}v)

where the point · denotes the Euclidean scalar product in ℝ^{N}. Applying (3.5) we obtain

(I) = ∫_{B′} τ ∑_{r,s} ∑_{i,k} h (∑_{l} D_{1}^{h}(a_{il}∂_{l}v^{r})) (∑_{l} D_{1}^{h}(a_{kl}∂_{l}v^{s})) m_{ik}^{rs}(h)

= ∫_{B′} τ \frac{F (y, E_{1}^{h} ˜∇v) − F (y, ˜∇v)}{h} − ∑_{i,l} ∫_{B′} τF_{i}(y, ˜∇v) · D_{1}^{h}a_{il}∂_{l}E_{1}^{h}v

+ ∑_{i,l} ∫_{B′} F_{i}(y, ˜∇v) · ∂_{l}(a_{il}τ)D_{1}^{h}v − ∫_{B′} τf · D_{1}^{h}v + ∑_{i,l} ∫_{B′} f_{i} · ∂_{l} (a_{il}τD_{1}^{h}v)

= (II) + … + (V I). The identity D_{1}^{h}(gg̃) = D_{1}^{h}gE_{1}^{h}g̃ + gD_{1}^{h}g̃ yields

\begin{aligned}(I) & = ∫_{B′} τ ∑_{r,s} ∑_{i,k} h (∑_{l} (D_{1}^{h}a_{il}∂_{l}E_{1}^{h}v^{r} + a_{il}D_{1}^{h}∂_{l}v^{r})) \\ & × (∑_{l} (D_{1}^{h}a_{kl}∂_{l}E_{1}^{h}v^{s} + a_{kl}D_{1}^{h}∂_{l}v^{s})) m_{ik}^{rs}(h). \end{aligned}

By (3.2) and hypothesis (H5) it follows that

|∫_{B′} τ ∑_{r,s} ∑_{i,k} h (∑_{l} D_{1}^{h}a_{il}∂_{l}E_{1}^{h}v^{r}) (∑_{l} D_{1}^{h}a_{kl}∂_{l}E_{1}^{h}v^{s}) m_{ik}^{rs}(h)|

≤ ch∥∇E_{1}^{h}v∥_{L^{2}(B′)}^{2}

and

|∫_{B′} τ ∑_{r,s} ∑_{i,k} h (∑_{l} D_{1}^{h}a_{il}∂_{l}E_{1}^{h}v^{r}) (∑_{l} a_{kl}D_{1}^{h}∂_{l}v^{s}) m_{ik}^{rs}(h)|

≤ \frac{ch}{η}∥∇E_{1}^{h}v∥_{L^{2}(B′)}^{2} + ηh ∫_{B′} τ |D_{1}^{h}∇v|^{2}

for η > 0. Hypothesis (H6) entails

∫_{B′} τ ∑_{r,s} ∑_{i,k} h (∑_{l} a_{il}D_{1}^{h}∂_{l}v^{r}) (∑_{l} a_{kl}D_{1}^{h}∂_{l}v^{s}) m_{ik}^{rs}(h)

≥ \frac{k_{0}}{2} ∫_{B′} τ ∑_{r} ∑_{i} h (∑_{l} a_{il}D_{1}^{h}∂_{l}v^{r})^{2}

= \frac{k_{0}}{2} ∫_{B′} τ ∑_{r} hD_{1}^{h}∇v^{r} · (A^{T} A) D_{1}^{h}∇v^{r}

≥ \frac{k_{0}λ_{0}^{2}}{2} ∫_{B′} τh |D_{1}^{h}∇v|^{2} .

Altogether we obtain

(I) ≥ c ∫_{B′} τh |D_{1}^{h}∇v|^{2} − ch

for a sufficiently small η > 0. Further, using Taylor expansion and summation by parts we get

(II) = ∫_{B′} τ \frac{F (y, E_{1}^{h} ˜∇v) − F (E_{1}^{h}y, E_{1}^{h} ˜∇v)}{h} + ∫_{B′} τD_{1}^{h}F (y, ˜∇v)

= ∫_{B′} τ ∑_{k} ζ^{1k} ∫_{0}^{1} F_{x_{k}}(ty + (1 − t)E_{1}^{h}y, E_{1}^{h} ˜∇v)dtdy

+ ∫_{B′} D_{1}^{h}(τF (y, ˜∇v)) − ∫_{B′} D_{1}^{h}τF (E_{1}^{h}y, E_{1}^{h} ˜∇v)

= (II)_{1} + (II)_{2} + (II)_{3}

where ζ^{1k} denotes the k-th component of the basis vector ζ^{1}. Hypotheses (H2) and (H1) entail

\begin{aligned}|(II)_{1}| & ≤ c (∑_{k} sup_{0≤t≤1} ∥g_{x_{k}}(y + thζ^{1})∥_{L^{1}(B′)} + ∥E_{1}^{h} ˜∇v∥_{L^{2}(B′)}^{2}) ≤ c \\ (II)_{2} & = −h^{−1} ∫Ω̂_{1}^{h} τF (y, ˜∇v) \\ |(II)_{3}| & ≤ c ∫_{B′} (|E_{1}^{h}g_{0}| + |E_{1}^{h} ˜∇v|^{2}) ≤ c. \end{aligned}  (3.6)

By (3.2) and hypotheses (H3) and (H7) we get

\begin{aligned}|(III)| & ≤ c (∑_{i} ∥g_{i}∥_{L^{2}(B′)}^{2} + ∥ ˜∇v∥_{L^{2}(B′)}^{2} + ∥∇E_{1}^{h}v∥_{L^{2}(B′)}^{2}) ≤ c \\ |(IV)| & ≤ c (∑_{i} ∥g_{i}∥_{L^{2}(B′)}^{2} + ∥ ˜∇v∥_{L^{2}(B′)}^{2} + ∥D_{1}^{h}v∥_{L^{2}(B′)}^{2}) ≤ c \\ |(V)| & ≤ c (∥f∥_{L^{2}(B′)}^{2} + ∥D_{1}^{h}v∥_{L^{2}(B′)}^{2}) ≤ c. \end{aligned}

Next, summation by parts yields

\begin{aligned}(V I) & = ∑_{i,l} ∫_{B′} f_{i} · ∂_{l}(τa_{il})D_{1}^{h}v − ∑_{i,l} ∫_{B′} D_{1}^{h}(τa_{il}f_{i}) · ∂_{l}E_{1}^{h}v + ∑_{i,l} ∫_{B′} D_{1}^{h}(τa_{il}f_{i} · ∂_{l}v) \\ & = (V I)_{1} + (V I)_{2} + (V I)_{3}. \end{aligned}

Due to (3.2) and hypothesis (H7) we obtain

\begin{aligned}|(V I)_{1}| & ≤ c (∑_{i} ∥f_{i}∥_{L^{2}(B′)}^{2} + ∥D_{1}^{h}v∥_{L^{2}(B′)}^{2}) ≤ c \\ |(V I)_{2}| & ≤ c (∑_{i} ∥f_{i}∥_{L^{2}(B′)}^{2} + ∑_{i} ∥D_{1}^{h}f_{i}∥_{L^{2}(B′)}^{2} + ∥∇E_{1}^{h}v∥_{L^{2}(B′)}^{2}) ≤ c. \end{aligned}

Applying hypothesis (H1) we get for η > 0

\begin{aligned}|(V I)_{3}| & = |\frac{1}{h} ∑_{i,l} ∫Ω̂_{1}^{h} τa_{il}f_{i} · ∂_{l}v| \\ & ≤ \frac{c}{ηh} |Ω̂_{1}^{h}| ∑_{i} ∥f_{i}∥_{L^{∞}(Ω̂_{1}^{h})}^{2} + \frac{η}{h} ∫Ω̂_{1}^{h} τ| ˜∇v|^{2} \\ & ≤ c + \frac{η}{c′_{0}h} ∫Ω̂_{1}^{h} τF (y, ˜∇v). \end{aligned}  (3.7)

Let η = \frac{c′_{0}}{2} . Then (3.6), (3.7), and hypothesis (H1) yield

(II)_{2} + |(V I)_{3}| ≤ c − \frac{1}{2h} ∫Ω̂_{1}^{h} τF (y, ˜∇v) ≤ c − \frac{c_{0}}{2h} |Ω̂_{1}^{h}| ≤ c.

Altogether we obtain assertion (3.3) for i = 1. Finally, let us note that the proof of (3.3) for arbitrary i ∈ Λ_{1} follows in the same way

Proposition 3.2. There exists a constant c depending only on R_{0} and the data such that

sup_{0<h<4R} ∫_{B′} τh |D_{i}^{−h}∇v|^{2} dy ≤ c  for i ∈ Λ_{2}.  (3.8)

Proof. Let 0 < h < 4R. We give the proof of (3.8) for some fixed number i ∈ Λ_{2}, say i = 1.

First, we extend v into Ω̂_{1}^{−h} by using (2.2), and the functions F (·, p), g_{0}, τ, a_{ik} (1 ≤ i, k ≤ n) by using (2.1). Now, let us verify that φ = −τD_{1}^{−h}v is an admissible test function. The conditions on ζ^{i} (i ∈ Λ_{2}) imply that y − hζ^{1} ∉ Ω̂ ∪ ∂Ω̂ for y ∈ Γ̂_{D} ∩ B′. Hence, the extension (2.2) yield

v(y − hζ^{1}) = 0 for y ∈ Γ̂_{D} ∩ B′,

Mixed Boundary Value Problems 549

thus

φ(y) = τh^{−1} (v(y − hζ^{1}) − v(y)) = 0 for y ∈ Γ̂_{D} ∩ B′.

Multiplying (3.1) by φ and integrating over Ω̂ we get

\begin{matrix}− ∫_{B′} τf · D_{1}^{−h}v + ∑_{i,l} ∫_{B′} f_{i} · ∂_{l}(a_{il}τD_{1}^{−h}v) + ∑_{i,l} ∫_{B′} F_{i}(y, ˜∇v) · ∂_{l}(a_{il}τ)D_{1}^{−h}v \\ = − ∑_{i,l} ∫_{B′} F_{i}(y, ˜∇v) · (τa_{il})∂_{l}D_{1}^{−h}v \\ = ∑_{i,l} ∫_{B′} τF_{i}(y, ˜∇v) · [ − D_{1}^{−h}(a_{il}∂_{l}v) + D_{1}^{−h}a_{il}E_{1}^{−h}∂_{l}v] \end{matrix} (3.9)

where we have used the identity D_{1}^{−h}(gg̃) = D_{1}^{−h}gE_{1}^{−h}g̃+gD_{1}^{−h}g̃. The Taylor expansion of F (y, ·) yields

∑_{r} ∑_{i}(p′ − p)_{i}^{r}F_{i}^{r}(y, p)

= F (y, p′) − F (y, p)

− ∑_{r,s} ∑_{i,k}(p′ − p)_{i}^{r}(p′ − p)_{k}^{s} ∫_{0}^{1}(1 − t)F_{i,k}^{rs}(y, tp′ + (1 − t)p) dt.

We set

m_{ik}^{rs}(−h) = ∫_{0}^{1}(1 − t)F_{i,k}^{rs}(y, tE_{1}^{−h} ˜∇v + (1 − t) ˜∇v) dt

for 1 ≤ i, k ≤ n and 1 ≤ r, s ≤ N. Let us put p = ˜∇v and p′ = E_{1}^{−h} ˜∇v. Then we obtain

− ∑_{r} ∑_{i,l} F_{i}^{r}(y, ˜∇v)D_{1}^{−h} (a_{il}∂_{l}v^{r})

= \frac{1}{h}(F (y, E_{1}^{−h} ˜∇v) − F (y, ˜∇v))

− ∑_{r,s} ∑_{i,k} h( ∑_{l} D_{1}^{−h}(a_{il}∂_{l}v^{r}))( ∑_{l} D_{1}^{−h}(a_{kl}∂_{l}v^{s}))m_{ik}^{rs}(−h).

Thus, (3.9) yields

(I) = ∫_{B′} τh ∑_{r,s} ∑_{i,k} ( ∑_{l} D_{1}^{−h}(a_{il}∂_{l}v^{r}))( ∑_{l} D_{1}^{−h}(a_{kl}∂_{l}v^{s}))m_{ik}^{rs}(−h)

550 C. Ebmeyer

= ∫_{B′} τh^{−1}(F (y, E_{1}^{−h} ˜∇v) − F (y, ˜∇v))

+ ∑_{i,l} ∫_{B′} τF_{i}(y, ˜∇v) · D_{1}^{−h}a_{il}∂_{l}E_{1}^{−h}v

− ∑_{i,l} ∫_{B′} F_{i}(y, ˜∇v) · ∂_{l}(a_{il}τ)D_{1}^{−h}v

+ ∫_{B′} τf · D_{1}^{−h}v

− ∑_{i,l} ∫_{B′} f_{i} · ∂_{l}(a_{il}τD_{1}^{−h}v)

= (III) + … + (V I).

Hypothesis (H6) entails

(I) ≥ \frac{k_{0}}{2} ∫_{B′} τhD_{1}^{−h} ˜∇v · D_{1}^{−h} ˜∇v = \frac{k_{0}}{2} ∫_{B′} τh ∑_{r} |D_{1}^{−h}(A∇v^{r})|^{2} .

We use

∫_{B′} τhAD_{1}^{−h}∇v^{r} · AD_{1}^{−h}∇v^{r} ≥ λ_{0}^{2} ∫_{B′} τh |D_{1}^{−h}∇v^{r}|^{2}

∫_{B′} τh(D_{1}^{−h}A)∇E_{1}^{−h}v^{r} · (D_{1}^{−h}A)∇E_{1}^{−h}v^{r} ≤ c ∫_{B′} τh |∇E_{1}^{−h}v^{r}|^{2} ≤ c

2 ∫_{B′} τh(D_{1}^{−h}A)∇E_{1}^{−h}v^{r} · AD_{1}^{−h}∇v^{r} ≤ \frac{c}{η} ∫_{B′} τh |∇E_{1}^{−h}v^{r}|^{2} + η ∫_{B′} τh |D_{1}^{−h}∇v^{r}|^{2}

for η > 0. Putting η = \frac{k_{0}λ_{0}^{2}}{4} it follows that

(I) ≥ \frac{k_{0}λ_{0}^{2}}{4} ∫_{B′} τh |D_{1}^{−h}∇v|^{2} − c.

Next,

\begin{aligned}(II) & = − ∫_{B′} τD_{1}^{−h}F (y, ˜∇v) + ∫_{B′} τh^{−1}(F (y, E_{1}^{−h} ˜∇v) − F (E_{1}^{−h}y, E_{1}^{−h} ˜∇v)) \\ & = (II)_{1} + (II)_{2}. \end{aligned}

Summation by parts entails

(II)_{1} = − ∫_{B′∪B′′} τD_{1}^{−h}F (y, ˜∇v)

= − ∫_{B′∪B′′} D_{1}^{−h}(τF (y, ˜∇v)) + ∫_{B′∪B′′} D_{1}^{−h}τF (E_{1}^{−h}y, E_{1}^{−h} ˜∇v)

= (II)_{11} + (II)_{12}

Mixed Boundary Value Problems 551

where

B′′ = {y ∈ B_{R_{0}} \ B′ : y = x + hζ^{1}, x ∈ B′}.

The extensions (2.1) and (2.2) entail

|(II)_{11}| = \frac{1}{h} | ∫Ω̂_{1}^{−h} τF (y, ˜∇v)| ≤ \frac{1}{h} ∫Ω̂_{1}^{−h} |g_{0}| ≤ ∥g_{0}∥_{L^{∞}(Ω̂_{1}^{h})} \frac{1}{h}|Ω̂_{1}^{−h}| ≤ c. Further, using hypothesis (H1) we obtain

|(II)_{12}| ≤ c ∫_{B′} |F (E_{1}^{−h}y, E_{1}^{−h} ˜∇v)| ≤ c ∫_{B′} (|E_{1}^{−h}g_{0}| + |E_{1}^{−h} ˜∇v|^{2}) ≤ c. Let ζ^{1k} be the k-th component of the basis vector ζ^{1}. Hypothesis (H2) and the Taylor expansion entail

|(II)_{2}| ≤ ∫_{B′} τ ∑_{k} |ζ^{1k}| ∫_{0}^{1} |F_{x_{k}}(ty + (1 − t)E_{1}^{−h}y, E_{1}^{−h} ˜∇v)| dtdy

≤ c( ∑_{k} sup_{0≤t≤1} ∥g_{x_{k}}(y − thζ^{1})∥_{L^{1}(B′)} + ∥E_{1}^{−h} ˜∇v∥_{L^{2}(B′)}^{2})

≤ c.

By (3.2) and Hypotheses (H3) and (H7) we get

\begin{aligned}|(III)| & ≤ c( ∑_{i} ∥g_{i}∥_{L^{2}(B′)}^{2} + ∥ ˜∇v∥_{L^{2}(B′)}^{2} + ∥∇E_{1}^{−h}v∥_{L^{2}(B′)}^{2}) ≤ c \\ |(IV)| & ≤ c( ∑_{i} ∥g_{i}∥_{L^{2}(B′)}^{2} + ∥ ˜∇v∥_{L^{2}(B′)}^{2} + ∥D_{1}^{−h}v∥_{L^{2}(B′)}^{2}) ≤ c \\ |(V)| & ≤ c (∥f∥_{L^{2}(B′)}^{2} + ∥D_{1}^{−h}v∥_{L^{2}(B′)}^{2}) ≤ c. \end{aligned}

Next,

\begin{aligned}(V I) & = − ∑_{i,l} ∫_{B′} f_{i} · ∂_{l}(a_{il}τ)D_{1}^{−h}v − ∑_{i,l} ∫_{B′} τa_{il}f_{i} · D_{1}^{−h}∂_{l}v \\ & = (V I)_{1} + (V I)_{2}. \end{aligned}

Due to (3.2) and Hypothesis (H1)

|(V I)_{1}| ≤ c( ∑_{i} ∥f_{i}∥_{L^{2}(B′)}^{2} + ∥D_{1}^{−h}v∥_{L^{2}(B′)}^{2}) ≤ c

follows. Using summation by parts we obtain

(V I)_{2} = − ∑_{i,l} ∫_{B′∪B′′} τa_{il}f_{i} · D_{1}^{−h}∂_{l}v

= ∑_{i,l} ∫_{B′∪B′′} D_{1}^{−h}(τa_{il}f_{i})∂_{l}E_{1}^{−h}v − ∑_{i,l} ∫_{B′∪B′′} D_{1}^{−h}(τa_{il}f_{i}∂_{l}v)

= (V I)_{3} + (V I)_{4}.

In view of hypothesis (H7) we get

|(V I)_{3}| = ∑_{i,l} ∫_{B′} (D_{1}^{−h}(τa_{il})f_{i} + E_{1}^{−h}(τa_{il})D_{1}^{−h}f_{i})∂_{l}E_{1}^{−h}v

≤ c( ∑_{i} ∥f_{i}∥_{L^{2}(B′)}^{2} + ∑_{i} ∥D_{1}^{−h}f_{i}∥_{L^{2}(B′)}^{2} + ∥∇E_{1}^{−h}v∥_{L^{2}(B′)}^{2})

≤ c.

The extension (2.2) yields ∂_{l}v = 0 in Ω̂_{1}^{−h}. This implies that

(V I)_{4} = \frac{1}{h} ∑_{i,l} ∫Ω̂_{1}^{−h} τa_{il}f_{i}∂_{l}v = 0.

Thus, the assertion follows

Proposition 3.3. Let Λ_{3} = \{n\} and 0 < δ < \frac{1}{2}. Then there exists a constant c depending only on R_{0}, δ, and the data such that

sup_{0<h<4R} ∫_{B} h^{1+δ}|D_{n}^{h}∇v|^{2}dy ≤ c.  (3.10)

The proof of this proposition follows as in [8] using (3.1), (3.3), (3.8), and Fourier series.

Now, we are able to prove the main results.

Proof of Theorem 1.1. a) Recall that Ω_{η} = {x ∈ Ω : dist(x, ∂Ω) ≥ η} and note that the basis vectors ζ^{i} fulfil angle(ζ^{i}, ζ^{j}) ≥ α^{∗} for 1 ≤ i < j ≤ n, where the constant α^{∗} depends only on the geometry of ∂Ω. It holds that τ ≡ 1 in B. Thus, (3.3), (3.8), and (3.10) yield for all δ ∈ (0, \frac{1}{2})

sup_{{η>0 \atop 0<|z|<η}} ∫_{((ϕ^{∗})^{−1}(B))_{η}} \frac{|∇u(x + z) − ∇u(x)|^{2}}{|z|^{1−δ}} dx ≤ c  (3.11)

where the constant c depends only on the data, δ, and on R_{0}. Further, let us note that R_{0} depends only on the shape of ∂Ω.

Next, there are a finite set of points {P̂_{1}, …, P̂_{k}} and a set of balls B_{R_{i}}(P̂_{i}) such that

∂Ω ⊂ ⋃_{i=1}^{k}(B^{i} ∩ ∂Ω), where B^{i} = (ϕ^{∗})^{−1}(B_{R_{i}}(P̂_{i})),

and P̂_{i} is the only vertex of ∂Ω̂ in B_{R_{i}}(P̂_{i}) or B_{R_{i}}(P̂_{i}) ∩ ∂Ω̂ contains no vertex of ∂Ω̂. Further, the radii R_{i} (1 ≤ i ≤ k) depend only on the data, for they are determined by the geometry of Ω. Thus,

u ∈ [H^{\frac{3}{2} − \frac{δ}{2} ,2}(Ω)]^{N} for δ ∈ (0, \frac{1}{2})

follows. The imbedding theorem of Nikolskii spaces into Sobolev spaces (cf. [1])

H^{s,p}(Ω) → W^{s−ε,p}(Ω) for ε > 0

entails u ∈ [W^{s,2}(Ω)]^{N} for all s < \frac{3}{2}. This yields assertion (1.1).

b) Using (3.3) and (3.8) we get (3.11) for δ = 0. Proceeding as above we obtain u ∈ [H^{\frac{3}{2} ,2}(Ω)]^{N}

Proof of Theorem 1.2. We only sketch the proof. Assumption (1.3) yields f ∈ L^{q}(Ω) and f_{i} ∈ L^{2q}(Ω) for some q > \frac{n}{2}. Now, N = 1 holds. Following [13] we see that u ∈ C^{0,α}(\overline{Ω}) for some α > 0. Thus, we can proceed as in [8]. The Hölder continuity and the equation yield

∫_{B_{r}(y_{0})∩Ω̂} \frac{|∇v(y)|^{2}}{|y − y_{0}|^{n−2+2ε}} dy ≤ c

for some ε > 0. Replacing the test functions φ by r^{−ε}φ in Propositions 3.1 and 3.2 and recalling the proof of Proposition 3.3 we get

∫_{B_{r}(P̂)∩Ω̂} r^{3−ε−n}|h^{\frac{1+δ}{2}} D_{i}^{h}∇v|^{2} ≤ c

for 1 ≤ i ≤ n, 0 < r ≤ \frac{R_{0}}{8} and 0 < δ < \frac{1}{2}. Applying an imbedding theorem of Morrey-Nikolskii type we obtain the assertion

4. Examples

In this section we give some explicit examples of the index sets Λ_{1}, Λ_{2}, Λ_{3}, and the basis vectors ζ^{1}, …, ζ^{n}.

Let Ω ⊂ ℝ^{3} be a polyhedron. We consider three typical situations: an edge of ∂Ω (Example 1), the case when angle(Γ_{D}, Γ_{N}) = π (Example 2), and a corner point (Example 3).

Let P = (0, 0, 0)^{T}, B_{R_{0}} = \{y : |y| < \frac{1}{2}\}, and let e_{k} (1 ≤ k ≤ 3) be the k-th unit vector in ℝ^{3}.

Example 1. Let

\begin{aligned}Γ_{∗}^{1} & = \{y ∈ B_{R_{0}} : y_{1} = 0, y_{3} > 0\} \\ Γ_{∗}^{2} & = \{y ∈ B_{R_{0}} : y_{3} = 0, y_{1} > 0\} \end{aligned}

and

Ω ∩ B_{1} = {y ∈ B_{1} : y_{1} > 0, y_{3} > 0}.

Case 1: Γ_{D} ∩ B_{R_{0}} = \overline{Γ_{∗}^{1}} and Γ_{N} ∩ B_{R_{0}} = Γ_{∗}^{2}. Let us put ζ^{1} = e_{2} and ζ^{2} = e_{3}. Then ζ^{1} and ζ^{2} are parallel to Γ_{D} ∩ B_{R_{0}}, thus, Λ_{1} = {1, 2}. Next, we put Λ_{2} = {3}. We must choose ζ^{3} such that ζ^{3} is parallel to Γ_{N} ∩ B_{R_{0}} and angle(ζ^{3}, Γ_{D} ∩ B_{R_{0}}) ≥ α^{∗} for some suitable large constant α^{∗} > 0 (i.e., α^{∗} ∼ angle(Γ_{∗}^{1}, Γ_{∗}^{2})). Thus, let ζ^{3} = e_{3}.

Case 2: Γ_{D} ∩ B_{R_{0}} = ∅ and Γ_{N} ∩ B_{R_{0}} = \overline{Γ_{∗}^{1}} ∪ \overline{Γ_{∗}^{2}}. It holds that Λ_{1} = \{1, 2, 3\}. We must choose ζ^{i} (1 ≤ i ≤ 3) such that

i) y + sζ^{i} ∈ \overline{Ω} for y ∈ ∂Ω ∩ B_{R_{0}} and 0 < s < R_{0}

ii) angle(ζ^{i}, ζ^{j}) ≥ α^{∗} for 1 ≤ i < j ≤ 3 and some suitable constant α^{∗} > 0.

Thus, let ζ^{i} = e_{i} for 1 ≤ i ≤ 3.

Case 3: Γ_{D} ∩ B_{R_{0}} = \overline{Γ_{∗}^{1}} ∪ \overline{Γ_{∗}^{2}} and Γ_{N} ∩ B_{R_{0}} = ∅. Now, it holds that Λ_{2} = \{1, 2, 3\}. The basis vectors ζ^{i} (1 ≤ i ≤ 3) must fulfil

554 C. Ebmeyer

i) y + sζ^{i} ∈ \overline{Ω} for y ∈ ∂Ω ∩ B_{R_{0}} and 0 < s < R_{0}
ii) angle(ζ^{i}, ζ^{j}) ≥ α^{∗} for 1 ≤ i < j ≤ 3 and α^{∗} > 0
iii) angle(ζ^{i}, Γ_{D} ∩ B_{R_{0}}) ≥ α^{∗}

where α^{∗} > 0 is suitable. Thus, let ζ^{1} = \frac{\sqrt{3}}{2}e_{1} + \frac{1}{2}e_{3}, ζ^{2} = \frac{1}{2}e_{1} + \frac{\sqrt{3}}{2}e_{3}, and ζ^{3} = \frac{1}{3}e_{2} + \frac{2}{3}(e_{1} + e_{3}).

Example 2. Let

Ω ∩ B_{R_{0}} = {y ∈ B_{R_{0}} : y_{3} > 0}

and

\begin{aligned}Γ_{D} ∩ B_{R_{0}} & = \{y ∈ B_{R_{0}} : y_{3} = 0, y_{1} ≥ 0\} \\ Γ_{N} ∩ B_{R_{0}} & = \{y ∈ B_{R_{0}} : y_{3} = 0, y_{1} < 0\}. \end{aligned}

We choose ζ^{1} = e_{1} and ζ^{2} = e_{2}. Then y + sζ^{i} ∈ Γ_{D} ∩ B_{R_{0}} holds for y ∈ Γ_{D} ∩ B_{R_{0}}, s > 0, and y + sζ^{i} ∈ B_{R_{0}}. Thus, Λ_{1} = {1, 2}. Further, Λ_{2} = ∅ and Λ_{3} = {3}. Let us put ζ^{3} = e_{3}.

Example 3. Let Ω = [0, 1]^{3}.

Case 1: Γ_{D} = {y ∈ ∂Ω : y_{3} = 0} and Γ_{N} = ∂Ω \ Γ_{D}. The two vectors e_{1} and e_{2} are parallel to Γ_{D} ∩ B_{R_{0}} and e_{3} is parallel to Γ_{N} ∩ B_{R_{0}}. Thus, let Λ_{1} = {1, 2}, ζ^{1} = e_{1}, ζ^{2} = e_{2}, Λ_{2} = {3}, and ζ^{3} = e_{3}.

Case 2: Γ_{D} = {y ∈ ∂Ω : y_{2} = 0 ∨ y_{3} = 0} and Γ_{N} = ∂Ω \ Γ_{D}. Now, e_{1} is parallel to Γ_{D} ∩ B_{R_{0}}, thus, Λ_{1} = {1} and ζ^{1} = e_{1}. Further, the two vectors e_{2} and e_{3} are parallel to Γ_{N} ∩ B_{R_{0}}, thus, Λ_{2} = {2, 3}. We must choose ζ^{i} (i = 2, 3) such that

i) angle(ζ^{i}, Γ_{D} ∩ B_{R_{0}}) ≥ α^{∗}

ii) angle(ζ^{2}, ζ^{3}) ≥ α^{∗}

for some suitable constant α^{∗} > 0. Thus, let ζ^{2} = \frac{\sqrt{3}}{2}e_{2} + \frac{1}{2}e_{3} and ζ^{3} = \frac{1}{2}e_{2} + \frac{\sqrt{3}}{2}e_{3}.

Case 3: Γ_{D} = ∅ and Γ_{N} = ∂Ω. It holds that Λ_{1} = {1, 2, 3}. Let ζ^{i} = e_{i} for 1 ≤ i ≤ 3.

Case 4: Γ_{D} = ∂Ω and Γ_{N} = ∅. Now, it holds that Λ_{2} = {1, 2, 3}. We choose ζ^{i} (1 ≤ i ≤ 3) such that

i) angle(ζ^{i}, Γ_{D} ∩ B_{R_{0}}) ≥ α^{∗}

ii) angle(ζ^{i}, ζ^{j}) ≥ α^{∗} for 1 ≤ i < j ≤ 3 and α^{∗} > 0
iii) y + sζ^{i} ∈ Ω for y ∈ ∂Ω ∩ B_{R_{0}} and 0 < s < R_{0}

where α^{∗} > 0 is suitable.

Mixed Boundary Value Problems 555

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Received 08.07.1998; in revised form 22.03.1999