A PRIORI BOUNDS FOR SOLUTIONS AND SPECTRA

OF ROBIN BOUNDARY VALUE PROBLEMS

E.N. Dancer and D. Daners

October 1994

Abstract. We prove global L_{p}-bounds for weak solutions of Robin boundary value problems which just depend on an upper bound of the volume of the domain rather than on the particular geometry of the domain. We also show that such uniform estimates can not be true in case of the Neumann problem. We also obtain a generalization of the classical Friedrichs inequality giving a uniform positive lower bound on the spectrum of Robin boundary value problems.

1. Introduction

In these notes we give estimates for solutions of the linear Robin boundary value problem

(1.1)  \{ \begin{matrix}A(x, ∂)u & = f & in Ω \\ \frac{∂}{∂ν_{a}} u + b_{0}u & = 0 & on ∂Ω , \end{matrix}

where A(x, ∂) is uniformly elliptic operator, Ω is a bounded Lipschitz domain, ν_{a} the outer conormal vector on the boundary ∂Ω of Ω, b_{0} ≥ β > 0 and f ∈ L_{p}. We shall derive global a priori bounds for weak solutions of (1.1) depending only on the upper bound of the volume |Ω| of Ω and not on the particular geometry of the domain. Usually, for sufficiently smooth domains, a priori bounds are obtained by means of Sobolev embedding theorems. As the embedding constants are highly dependent on the geometry of ∂Ω even at a single point, this can not give good control on the norms of the solution of (1.1). Key to obtain the uniform bounds is an inequality due to Maz’ja [12] which generalizes Sobolev’s inequality as well as the classical isoperimetric inequality.

We also prove that uniform estimates do not hold in case of the Neumann problem, i.e. the case b_{0} ≡ 0. This is done by approaching a domain where the Neumann Laplacian does not have compact resolvent by smooth domains.

Typeset by AMS-TEX

The second author was supported by a grant of the Australian Research Council

We shall also show that there exists a uniform lower bound for the first eigenvalue of the problem

(1.2)  \{ \begin{matrix}A(x, ∂) & = λu & in Ω \\ \frac{∂}{∂ν_{a}} u + b_{0}u & = 0 & on ∂Ω . \end{matrix}

In case the problem is self adjoint we shall prove a lower bound of the spectrum just depending on the width of a strip containing the domain Ω, the ellipticity constant and a lower bound on b_{0} and not on the geometry of the boundary. Note also that a lower bound for the first eigenvalue can be obtained from Maz’ja’s inequality, but this bound depends on the volume of the domain rather than on the width of a strip. To do this we prove a generalization of Friedrichs inequality using ideas of Friedrichs in case of the Dirichlet problem. As we use Sard’s lemma to obtain the inequality we need to assume that the boundary of Ω is of class C^{1}. To show that the inequality is true for an arbitrary Lipschitz domain we make use of the fact that one can approximate arbitrary bounded Lipschitz domains by smooth domains in such a way that also the surface integrals approach the original one. This is a useful technical lemma because it allows one to prove inequalities for Lipschitz domains provided one knows them for smooth domains with constants independent of the domain.

Although an immediate consequence of Maz’ja’s inequality and the variational characterization of the first eigenvalue λ_{1} of (1.2) uniform lower bounds for the spectrum of (1.2) seem not to be known for general Lipschitz domains. Lower bounds for the first eigenvalue in the self adjoint case have been proved under various conditions on the geometry of the domain, so for instance by Bossel [4], Sperb [14]–[16]. These results always require restrictions on the curvature of the boundary ∂Ω. Bandle [3] does not consider the problem of lower bounds of λ_{1} in the case of Robin boundary conditions saying there exist no lower bounds without imposing further restrictions ([3], Section III.2.5).

One reason for the interest in estimates which hold uniformly on the domain is that they are very useful for studying how solutions vary as the domain is varied. In particular, our ideas could be combined with those in Dancer [5], [6] to study how the number of solutions of nonlinear Robin boundary value problems vary with the domain shape. We will discuss this elsewhere.

2. Global bounds for weak solutions

We prove our estimates for weak solutions of the equation (1.1) for a general second order elliptic operator A(x, ∂) of the form

(2.1)  A(x, ∂)u = − ∑_{i,j=1}^{N} ∂_{i}(a_{ij}(x)∂_{j}u) + ∑_{i=1}^{N} a_{i}(x)∂_{i}u + a_{0}u

where ν_{a} the outer conormal vector. We assume that a_{ij} = a_{ji}, a_{i}, a_{0} ∈ L_{∞}(Ω) for i, j = 1, … N, b_{0} ∈ L_{∞}(∂Ω) with

(2.2)  b_{0}(x) ≥ β

for some β > 0 and that there exists a constant α > 0 such that

(2.3)  ∑_{i,j=1}^{N} a_{ij}(x)ξ^{i}ξ^{j} ≥ α|ξ|^{2}

for all x ∈ Ω and ξ ∈ ℝ^{N}. Let us write

[a] := [a_{ij}]_{1≤i,j≤N}  and  a := (a_{1}, …, a_{N}) .

Define as usual

a_{0}(u, v) := ∫_{Ω}([a]∇u|∇v) dx + ∫_{Ω}(a|∇u)v dx + ∫_{Ω} a_{0}uv dx

and

a(u, v) := a_{0}(u, v) + ∫_{∂Ω} b_{0}uv dσ

for all u, v ∈ W_{2}^{1}(Ω). Recall that a weak (sub, super) solution of (1.1) is a function u ∈ W_{2}^{1}(Ω) such that

(2.4)  a(u, v) (≤, ≥)= ⟨f, v⟩

for all v ∈ W_{2}^{1}(Ω) with v ≥ 0, where

⟨f, v⟩ := ∫_{Ω} fv dx .

As already pointed out in the introduction the main ingredient for the proof of our a priori bounds is an inequality due to Maz’ja ([12], see [13], Corollary 3.6.3). It asserts that

(2.5)  ∥u∥_{\frac{N}{N−1}} ≤ \frac{Γ(1 + N/2)^{1/N}}{N\sqrt{π}} (∥∇u∥_{1} + ∥u∥_{1,∂Ω})

for all for all u ∈ W_{2}^{1}(Ω). From this inequality we get the classical isoperimetric inequality by setting u ≡ 1, and the Sobolev inequality by considering only functions in W̊_{2}^{1}(Ω). Applying this inequality to u^{2} we obtain for any Lipschitz domain Ω with volume |Ω| ≤ M

(2.6)  ∥u∥_{\frac{2N}{N−1}} ≤ c(N, M)(∥∇u∥_{2}^{2} + ∥u∥_{2,∂Ω}^{2})^{1/2}

for all u ∈ W_{2}^{1}(Ω) where c(N, M) is a constant depending only on N and M > 0 ([13], Corollary 4.11.1/2).

We shall prove a priori estimates for (sub, super) solutions of (1.1) for f ∈ L_{p}(Ω) just in terms of

(2.7)  α^{−1}∥f∥_{p} ,  δ := \frac{∥a∥_{∞}^{2}}{α^{2}} + \frac{∥a_{0}^{−}∥_{∞}}{α} ,  γ := αβ^{−1}

as well as |Ω|, N and p. Here, a_{0}^{−} := sup{−a_{0}, 0} denotes the negative part of a_{0}. Let us start with the case p < N.

2.1 Theorem. Let M > 0 and p ∈ (\frac{2N}{N+2}, N) be fixed. Then there exists a constant C > 0 only dependent on M, p and N such that for any (sub, super) solution u of (1.1)

(2.8) ∥(u^{+}, u^{−})u∥_{\frac{Np}{N−p}} ≤ C max\{1, γ\}(α^{−1}∥f∥_{p} + max\{1, γ^{μ}\}δ^{1+μ}∥(u^{+}, u^{−})u∥_{2})

for all f ∈ L_{p}(Ω) whenever |Ω| ≤ M, where μ := 1 if p ∈ (\frac{2N}{N+2}, 2] and μ := \frac{N(p−2)}{p} if p > 2. Here, u^{+} and u^{−} are the positive and the negative part of u.

Proof. We only give the proof for subsolutions. The corresponding result for supersolutions follows by replacing u by −u. For solutions combine the results for sub- and supersolutions. For q ≥ 1 and m > 0 define

G_{m}(s) := \{ \begin{matrix}s^{\frac{q+1}{2}} & for s ≤ m \\ m^{\frac{q+1}{2}} + \frac{q+1}{2}m^{\frac{q−1}{2}} (s − m) & for s ≥ m \end{matrix}

and

H_{m}(s) := \{ \begin{matrix}s^{q} & for s ≤ m \\ m^{q} + \frac{q+1}{2}m^{q−1}(s − m) & for s ≥ m \end{matrix}

Set w := G_{m}(u^{+}) and v := H_{m}(u^{+}). Then, v, w ∈ W_{2}^{1}(Ω) (c.f [10], Theorem 7.8). An easy calculation then shows that

([a]∇w|∇w) ≤ \frac{q + 1}{2}([a]∇u|∇v) , (a|∇w)w = \frac{q + 1}{2}(a|∇u)v and uv ≤ w^{2} .

Using the uniform ellipticity (2.3) we thus obtain

α∥∇w∥_{2}^{2} ≤ ∫_{Ω}([a]∇w|∇w) dx ≤ \frac{q + 1}{2} a_{0}(u, v) − ∫_{Ω}(a|∇w)w dx − \frac{q + 1}{2} ∫_{Ω} a_{0}w^{2} dx

≤ \frac{q + 1}{2} a_{0}(u, v) + ∥a∥_{∞}∥∇w∥_{2}∥w∥_{2} + \frac{q + 1}{2}∥a_{0}^{−}∥_{∞}∥w∥_{2}^{2}

≤ \frac{q + 1}{2} a_{0}(u, v) + ∥a∥_{∞}(ε∥∇w∥_{2}^{2} + \frac{1}{4ε}∥w∥_{2}^{2}) + \frac{q + 1}{2}∥a_{0}^{−}∥_{∞}∥w∥_{2}^{2}

for arbitrary ε > 0. If we set ε = \frac{α}{2∥a∥_{∞}} and take into account the definition of δ we get

(2.9)  ∥∇w∥_{2}^{2} ≤ \frac{q + 1}{2}(\frac{2}{α} a_{0}(u, v) + 2δ∥w∥_{2}^{2})

Observe that from the definition of v and w it follows that

w^{2} ≤ \frac{q + 1}{2} uv .

Together with (2.2), (2.6) and (2.9) and using the fact that u is a subsolution we have that

(2.10)  \begin{aligned}∥w∥_{\frac{2N}{N−1}} & ≤ c max\{1, γ\}q(α^{−1} a(u, v) + δ∥w∥_{2}^{2}) \\ & ≤ c max\{1, γ\}q(α^{−1}⟨f, v⟩ + δ∥w∥_{2}^{2}) \end{aligned}

where c depends only on N and M. From the definition of v and w it is easy to see that v ≤ w^{\frac{2q}{q+1}} and therefore by Hölder’s inequality

(2.11)  ⟨f, v⟩ ≤ ∥f∥_{p}∥w^{\frac{2q}{q+1}} ∥_{p′} .

Using again Hölder’s inequality we get the estimate ∥w∥_{2}^{2} ≤ ∥w^{\frac{2}{q+1}} ∥_{p}∥w^{\frac{2q}{q+1}} ∥_{p′}, and since by definition we have G_{m}(s) ≤ s^{\frac{q+1}{2}} we have that

(2.12)  ∥w∥_{2}^{2} ≤ |Ω|^{\frac{1}{p} − \frac{1}{2}} ∥u^{+}∥_{2}∥w^{\frac{2q}{q+1}} ∥_{p′}

if p ∈ (\frac{2N}{N+2}, 2]. If p > 2 we have p′ < 2 < \frac{2N}{(N−1)} and we can apply a well known interpolation inequality (c.f. [10], Sect. 7.1) to obtain

(2.13)  ∥w∥_{2}^{2} ≤ 2(ε∥w∥2_{\frac{2N}{N−1}} + ε^{−μ}∥w∥_{p′}^{2})

with μ := \frac{N(p−2)}{p} and ε > 0 arbitrary. Moreover, by Cauchy-Schwarz inequality and

the fact that G_{m}(s) ≤ s^{\frac{q+1}{2}} we have that

∥w∥_{p′}^{2} ≤ ∥w^{\frac{2}{q+1}} ∥_{p′}∥w^{\frac{2q}{q+1}} ∥_{p′} ≤ |Ω|^{\frac{1}{p′} − \frac{1}{2}} ∥u^{+}∥_{2}∥w^{\frac{2q}{q+1}} ∥_{p′} .

Choosing ε appropriately we get from (2.10) and (2.11)

(2.14)  \begin{matrix}∥w^{\frac{2}{q+1}} ∥_{(q+1)N′}^{q+1} = ∥w∥_{\frac{2N}{N−1}} \\ ≤ Cq^{2} max\{1, γ\}(α^{−1}∥f∥_{p} + max\{1, γ^{μ}\}δ^{1+μ}∥u^{+}∥_{2})∥w^{\frac{2}{q+1}} ∥_{qp′}^{q}, \end{matrix}

where N′ its the dual exponent to N. The constant C depends only on M, p and N. We obtain the same inequality for p < 2 with μ = 0 using (2.12) instead of (2.13). Set now q := \frac{N(p−1)}{(N−p)}. Dividing the inequality by ∥w^{\frac{2}{q+1}} ∥q_{\frac{Np}{N−p}} we finally have

∥w^{\frac{2}{q+1}} ∥_{\frac{Np}{N−p}} ≤ C max\{1, γ\}(α^{−1}∥f∥_{p} + max\{1, γ^{μ}\}δ^{1+μ}∥u^{+}∥_{2}) ,

where C depends only on M, p and N. As G_{m}(s)↗s^{\frac{q+1}{2}} as m tends to infinity we get from the monotone convergence theorem that lim_{m→∞} ∥w^{\frac{2}{q+1}} ∥_{\frac{Np}{N−p}} = ∥u^{+}∥_{\frac{Np}{N−p}} completing the proof of the theorem. □

Next we prove L_{∞}-estimates for the solutions uniform in |Ω| ≤ M provided p > N. The estimates are obtained by means of Moser iteration techniques in conjunction with Maz’ja’s inequality (2.6).

2.2 Theorem. Let M > 0 and p > N be fixed. Then there exists a constant C > 0 only dependent on M, p and N such that for any (sub, super) solution u of (1.1)

(2.15)  ∥(u^{+}, u^{−})u∥_{∞} ≤ C max{1, γ}(α^{−1}∥f∥_{p} + max{1, γ^{μ}}δ^{1+μ}∥(u^{+}, u^{−})u∥_{2})

for all f ∈ L_{p}(Ω) whenever |Ω| ≤ M, where μ := \frac{N(p−2)}{p}.

Proof. We only give the proof for subsolutions. The corresponding result for supersolutions follows by replacing u by −u and for solutions combine the estimates for suband supersolutions. We proceed in the same way as in the proof of the above theorem to obtain (2.14). As G_{m}(s)↗s^{\frac{q+1}{2}} as m tends to infinity we get

∥u^{+}∥_{(q+1)N′}^{q+1} ≤ Cq^{2} max{1, γ}(α^{−1}∥f∥_{p} + max{1, γ^{μ}}δ^{1+μ}∥u^{+}∥_{2})∥u^{+}∥_{qp′}^{q},

saying that u^{+} ∈ L_{qp′} implies the stronger inclusion u^{+} ∈ L_{(q+1)N′}. Replacing u^{+} by

w̃ := \frac{u^{+}}{C max\{1, γ\}(α^{−1}∥f∥_{p} + max\{1, γ^{μ}\}δ^{1+μ}∥u^{+}∥_{2})}

we see that ∥w̃∥_{(q+1)N′}^{q+1} ≤ q^{2}∥w̃∥_{qp′}^{q} and therefore

(2.16)  ∥w̃∥_{(q+1)N′} ≤ q^{\frac{2}{q}} ∥w̃∥qp′^{\frac{q}{q+1}}

for all q ≥ 1. Set now

q_{0} := 1  and  q_{n+1} := \frac{q_{n} + 1}{p′}N′

for all n ∈ ℕ. As p > N (q_{n})_{n∈ℕ} is increasing with lim_{n→∞} q_{n} = ∞. Rewriting (2.16) we obtain

∥w̃∥_{q_{n+1}p′} ≤ q^{\frac{2}{qn}} ∥w̃∥_{q_{n}p′}^{(1+ \frac{1}{qn})^{−1}}

for all n ∈ ℕ. Iterating this inequality yields

∥w̃∥_{q_{n+1}p′} ≤ qn^{\frac{2}{qn}} ∏_{k=0}^{n−1} q_{k}^{\frac{2}{q_{k}} ∏_{i=k+1}^{n}(1+ \frac{1}{q_{i}})^{−1}}∥w̃∥_{p′}^{∏_{k=0}^{n}(1+ \frac{1}{q_{k}} )^{−1}}

for all n ∈ ℕ. As q_{k} ≥ 1 and (1 + \frac{1}{q_{k}})^{−1} < 1 for all k ∈ ℕ this simplifies to

(2.17)  ∥w̃∥_{q_{n+1}p′} ≤ ∏_{k=0}^{n} qk^{\frac{2}{q_{k}}} ∥w̃∥_{p′}^{∏_{k=0}^{n}(1+ \frac{1}{q_{k}} )^{−1}}

for all n ∈ ℕ. Next we estimate the product on the right hand side of (2.17). To do this we need estimates of q_{n}. By definition of q_{n} it is clear that

\frac{N′}{p′} ≤ \frac{q_{n}}{q_{n−1}} ≤ \frac{2N′}{p′}

for all n ∈ ℕ, and therefore α^{n} ≤ q_{n} ≤ (2α)^{n} for all n ∈ ℕ, where α := \frac{N′}{p′} > 1. Therefore,

c_{0} := ∏_{k=0}^{n} qk^{\frac{2}{q_{k}}} ≤ (2α)^{2 ∑_{k=0}^{n} kα^{−k}} < ∞.

From (2.17) we can now conclude that ∥w̃∥_{∞} ≤ c_{0}∥w̃∥_{p′}^{θ} with θ := ∏_{k=0}^{∞}(1 + \frac{1}{q_{k}})^{−1}.

Obviously, θ < 1 and therefore, using the fact that ∥w̃∥_{p′} ≤ |Ω|^{\frac{1}{p′}} ∥w̃∥_{∞}, we finally arrive at

∥w̃∥_{∞} ≤ (c_{0}|Ω|^{\frac{θ}{p′}} )^{\frac{1}{1−θ}} .

As θ and c_{0} depend only on N and p we easily obtain (2.15) using the definition of w̃. This completes the proof of the theorem. □

The above Theorems take a very simple form if the problem is self adjoint, i.e. a = 0 and a_{0} nonnegative. In this case we obtain the following corollary.

2.3 Corollary. Let M > 0 be fixed, a = 0 and a_{0} ≥ 0. Then there exists a constant C > 0 only dependent on M, p and N such that for any (sub, super) solution u of (1.1)

∥(u^{+}, u^{−})u∥_{m(p)} ≤ C max{α^{−1}, β^{−1}}∥f∥_{p}

for all f ∈ L_{p}(Ω) whenever |Ω| ≤ M, where m(p) = \frac{Np}{N−p} if p ∈ (\frac{2N}{N+2}, N) and m(p) := ∞ if p > N.

2.4 Remark. Note that in order to prove Theorem 2.1 and 2.2 we only need that (2.4) holds for all v ∈ W_{2}^{1}(Ω) such that v ≥ 0 and uv ≥ 0, because we only deal with test functions satisfying this condition.

Next we prove an L_{∞}-estimate for (sub, super) solutions which is independent of ∥(u^{+}, u^{−})u∥_{2}. To do this we use ideas of Trudinger [17], who proved the corresponding result in case of Dirichlet boundary conditions (see [10], Theorem 8.16).

2.5 Theorem. Let M > 0 and p > N be fixed and a_{0} nonnegative. Then there exists a constant C > 0 only dependent on M, p, N, max{1, γ} and δ such that for any (sub, super) solution u of (1.1)

(2.18)  ∥(u^{+}, u^{−})u∥_{∞} ≤ Cα^{−1}∥f∥_{p}

for all f ∈ L_{p}(Ω) whenever |Ω| ≤ M.

In order to prove the above theorem we need the following elementary inequalities.

2.6 Lemma. Let 0 < s < m. Then, we have that

(a) log \frac{m}{m − s} ≤ \frac{s}{m − s} (b) (log \frac{m}{m − s})^{2} ≤ \frac{s^{2}}{m(m − s)} .

Proof. (a) follows from the fact that

log \frac{m}{m − s} = log(1 + \frac{s}{m − s}) ≤ \frac{s}{m − s} .

To prove (b) we set t := \frac{s}{m}. It is then easy to see that (b) is equivalent to the fact that

(log \frac{1}{1 − t})^{2} ≤ \frac{t^{2}}{1 − t}

for all t ∈ (0, 1), which is equivalent to g(t) := t^{−2}(1 − t)( log(1 − t))^{2} ≤ 1 for all t ∈ (0, 1). Using de l’Hôpital’s rule we find that lim_{t→0} g(t) = 1. As g ≥ 0 it is now sufficient to show that g′(t) < 0 for all t ∈ (0, 1). As

g′(t) = −t^{−2} log(1 − t)[2 − t^{−1}(t − 2) log(1 − t)] ,

it remains to show that t^{−1}(t − 2) log(1 − t) > 2 for all t ∈ (0, 1). Using the Taylor expansion of log(1 − t) we obtain

t^{−1}(t − 2) log(1 − t) = 2 + ∑_{k=2}^{∞}(\frac{2}{k + 1} − \frac{1}{k} )t^{k} > 2

for all t ∈ (0, 1) completing the proof of (b). □

Now we are in a position to prove Theorem 2.5.

Proof of Theorem 2.5. As in the previous theorems it is sufficient to give the proof for subsolutions. From Theorem 2.2 we already know that u^{+} ∈ L_{∞}, so that it makes sense to define

v := \frac{u^{+}}{m + k − u^{+}}

where m := ∥u^{+}∥_{∞} and k := α^{−1}∥f∥_{p}. It is easy to see that v ∈ W_{2}^{1}(Ω), so we can use v as a test function in (2.4). By the ellipticity (2.3) of A(x, ∂) we have that

\begin{aligned}∫_{Ω}([a]∇u|∇v) dx & = (m + k) ∫_{Ω} \frac{(m + k)^{2}}{(m + k − u^{+})^{2}} ([a]∇u^{+}|∇u^{+}) dx \\ & ≤ α(m + k) ∫_{Ω} \frac{|∇u^{+}|^{2}}{(m + k − u^{+})^{2}} .dx \end{aligned}

Using the definition of v and that u is a subsolution we obtain

(2.19)

α ∫_{Ω} \frac{|∇u^{+}|^{2}}{(m + k − u^{+})^{2}} dx + \frac{1}{2} \frac{β}{m + k} ∫_{∂Ω} \frac{(u^{+})^{2}}{m + k − u^{+}} dσ

≤ \frac{1}{m + k} a(u, v) + \frac{∥a∥_{∞}}{m + k} ∫_{Ω} \frac{|u^{+}||∇u^{+}|}{m + k − u^{+}} dx

≤ α|Ω|^{\frac{1}{p′}} + ∥a∥_{∞}(ε ∫_{Ω} \frac{|∇u^{+}|^{2}}{(m + k − u^{+})^{2}} dx + \frac{1}{4ε}|Ω|^{2})

= \frac{α}{2} ∫_{Ω} \frac{|∇u^{+}|^{2}}{(m + k − u^{+})^{2}} dx + \frac{α}{2}(δ|Ω|^{2} + 2|Ω|^{\frac{1}{p′}} )

if we set ε := (2\sqrt{δ})^{−1}. We introduce now an auxiliary function w defined by

w := log \frac{m + k}{m + k − u^{+}} .

Then, it is clear that ∇w = \frac{∇u^{+}}{m+k−u^{+}} , and by Lemma 2.6(b)

∫_{∂Ω} w^{2} dσ ≤ \frac{1}{m + k} ∫_{∂Ω} \frac{(u^{+})^{2}}{m + k − u^{+}} dσ .

Therefore, by (2.6) and (2.19) we conclude that

(2.20)  ∥w∥_{2} ≤ c max\{1, γ\}(δM^{2} + 2M^{\frac{1}{p′}} ) .

where c merely depends on M and N. In the next step we show that we can apply Theorem 2.2 to estimate the L_{∞}-norm of w. In order to do this we take

v := \frac{φ}{m + k − u^{+}}

as a test function in (2.4) where φ ∈ W_{2}^{1}(Ω) ∩ L_{∞}(Ω) with φ ≥ 0 and uφ ≥ 0. Then, in the same way as in the proof of [10], Theorem 8.16 we obtain

a(w, φ) ≤ a(u, v) ≤ ⟨f, v⟩ ≤ ⟨fk^{−1}, φ⟩ .

The only difference is that we have to estimate the boundary integral. This is done in an obvious way using Lemma 2.6(a). By a truncation argument we easily see that the above inequality holds for all nonnegative φ ∈ W_{2}^{1}(Ω) such that uφ ≥ 0. Finally note that in this case by definition of w we also have that wφ ≥ 0. Thus, we can apply Theorem 2.2 and Remark 2.4 together with (2.20) to conclude that

(2.21)  ∥w∥_{∞} ≤ C max\{1, γ\}(1 + max\{1, γ^{1+μ}\}δ^{1+μ}(δM^{2} + 2M^{\frac{1}{p′}})) .

The desired result follows now by taking the exponential of (2.21) and the fact that ∥w∥_{∞} = k^{−1}(m + k). □

We shall discuss now some consequences of the above results. In a first remark we discuss briefly further generalizations.

2.7 Remarks. (a) All the above results are still true (with essentially the same proof) if the boundary conditions are Robin on part of the boundary and Dirichlet on the remainder of the boundary. This is because (2.6) is still valid.

(b) One can get better estimates for other classes of domains satisfying an inequality similar to (2.6) with the L_{\frac{2N}{N−1}} -norm replaced by an L_{r}-norm with r > \frac{2N}{N−1}, i.e.

(2.22)  ∥u∥_{r} ≤ c_{0}(∥∇u∥_{2}^{2} + ∥u∥_{2,∂Ω}^{2})^{1/2}

holds for all u ∈ W_{2}^{1}(Ω). Then, an estimate similar to (2.8) holds for p < r^{∗} := \frac{r}{r−2} and (2.15) if p > r^{∗}. More precisely, an inspection of the proofs of Theorem 2.1 and 2.2 shows that for any (sub, super) solution u of (1.1)

∥(u^{+}, u^{−})u∥_{m(r,p)} ≤ C max{1, γ}(α^{−1}∥f∥_{p} + max{1, γ^{μ}}δ^{1+μ}∥(u^{+}, u^{−})u∥_{2}) ,

where m(r, p) = \frac{r^{∗}p}{r^{∗}−p} if p ∈ (\frac{2N}{N−2}, r^{∗}) and m(r, p) = ∞ if p > r^{∗}. The constant C depends only on r, p, c_{0} and an upper bound for |Ω|, and μ has to be replaced by μ := r^{∗} \frac{p−2}{p}. Classes of domains satisfying (2.22) are characterized in [12] and [13].

To discuss further consequences let us introduce some notation. As a: W_{2}^{1} ×W_{2}^{1} → ℝ is a continuous bilinear form it induces an operator A ∈ L(W_{2}^{1}, (W_{2}^{1})′) such that

a(u, v) = ⟨Au, v⟩

for all u, v ∈ W_{2}^{1}. Moreover it is well known that there exists λ_{0} ∈ ℝ such that (u, v) ↦ a(u, v) + λ(u, v)_{L_{2}} is coerzive for all λ ≥ λ_{0}. Therefore by the theorem of Lax-Milgram λ ∈ ϱ(A) whenever λ ≥ λ_{0}. In particular, this implies that A is a closed operator on (W_{2}^{1})′ with domain of definition D(A) = W_{2}^{1}. As Ω is a Lipschitz domain we have from Sobolev embedding theorem that W_{2}^{1} ↪ L_{p} for p ∈ [1, \frac{2N}{N−2}) and hence, by duality, L_{p} ↪ (W_{2}^{1})′ for p > \frac{2N}{N+2}. Consequently, the L_{p}-realization A_{p} of A, defined by

A_{p}u := Au  for  D(A) := {u ∈ D(A) ∩ L_{p}: Au ∈ L_{p}}

is a closed operator on L_{p} for all p > \frac{2N}{N+2}. For the sake of completeness we summarize the spectral properties of A and A_{p} in the following proposition.

2.8 Proposition. The operator A_{p} has compact resolvent and ϱ(A_{p}) = ϱ(A) holds for all p > \frac{2N}{N+2}. Furthermore, (λ − A_{p})^{−1}L_{p} ⊂ L_{m(p)}, where m(p) = \frac{Np}{N−p} if p ∈ (\frac{2N}{N+2}, N) and m(p) = ∞ if p > N and the resolvent is a compact operator from L_{p} into L_{m(p)} if p < N and from L_{p} into L_{q} for all q < ∞ if p > N.

Proof. The smoothing properties of the resolvent follow easily from Theorem 2.1 and 2.2. We prove now the last assertion which in particular implies the compactness of the resolvent. In case p ∈ (\frac{2N}{N+2}, 2] the compactness readily follows from the following diagram whenever λ ∈ ϱ(A) since the embeddings are compact.

(W_{2}^{1})′ ⟵^{⊃} L_{p}
↓^{(λ−A)^{−1}}  ↓^{(λ−A_{p})^{−1}}
W_{2}^{1}  ^{⊂}⟶ L_{\frac{Np}{N−p}}

Compactness in case p ∈ (2, N) follows from a compactness property of the Riesz-Thorin interpolation. Indeed, let 2 < p < q < N be arbitrary. Then, the operator (λ − A_{p})^{−1} ∈ L(L_{p}, L_{\frac{pN}{N−p}}) is obtained by interpolation between the compact operator

(λ − A_{2})^{−1} ∈ L(L_{2}, L_{\frac{2N}{N−2}} ) and the bounded operator (λ − A_{q})^{−1} ∈ L(L_{q}, L_{\frac{qN}{N−q}}), and therefore compact by a result due to Krasnosel’skiĭ [11]. In a similar way we obtain compactness if p > N.

Now we show the p-independence of the spectrum of A_{p}. To do that note first that by Sobolev embedding theorem (λ − A)^{−1}W_{2}^{1} ⊂ L_{p} if p ∈ (\frac{2N}{N+2}, \frac{2N}{N−2}). Then, a bootstrapping argument using the smoothing properties of the resolvent shows that all the eigenfunctions of A are in L_{∞}. By the compactness of the resolvent we know that the spectrum consists only of eigenvalues of finite multiplicity and therefore we see that ϱ(A) = ϱ(A_{p}) for all p > \frac{2N}{N+2}. □

Finally, note that Theorem 2.5 tells us that 0 ∈ ϱ(A) provided a_{0} is nonnegative, and that

(2.23)  ∥A_{p}^{−1}∥_{L(L_{p},L_{∞})} ≤ Cα^{−1}

holds for all p > N with a constant C depending merely on p and an upper bound for |Ω|, δ and γ.

3. Neumann Problems

In this section we show that uniform estimates for weak solutions as in the case of Dirichlet or Robin boundary condition can not be true for Neumann boundary conditions. As the Laplacian under Neumann boundary conditions is not invertible we consider the following problem:

(3.1)  \{ \begin{matrix}−Δu + u & = f & in Ω \\ \frac{∂}{∂ν} u & = 0 & on ∂Ω , \end{matrix}

where Ω is an arbitrary bounded domain in ℝ^{N}. By a weak solution we mean as usual a function u ∈ W_{2}^{1}(Ω) such that

(3.2)  ∫_{Ω}(∇u|∇v) + uv dx = ∫_{Ω} fv dx

for all test functions v ∈ W_{2}^{1}(Ω). Taking the u as a test function we easily see that

(3.3)  ∥u∥_{2} ≤ ∥f∥_{2} ,

which means that we have a uniform L_{2}-bound for the solution. We shall see that there is no uniform L_{p}-bound for all p > 2. To show this we need an approximation theorem for solutions defined in different domains. Let (Ω_{n})_{n∈ℕ} be a sequence of domains such that Ω_{n} ⊂ Ω, Ω_{n} ⊂ Ω_{n+1} for all n ∈ ℕ, and ⋃_{n=1}^{∞} Ω_{n} = Ω. In this case we say that Ω_{n}↗Ω. If Ω_{1}, Ω_{2} are open bounded subsets of ℝ^{N} we write Ω_{1} ⊂⊂ Ω_{2} if \overline{Ω}_{1} ⊂ Ω_{2}.

3.1 Proposition. Suppose that Ω_{n}↗Ω and that u_{n} is the solution of (3.1) with Ω replaced by Ω_{n}. Then,

(3.4)  lim_{n→∞} u_{n}|_{Ω′} = u|_{Ω′}

in W_{2}^{1}(Ω′) for all Ω′ ⊂⊂ Ω, where u is the solution of (3.1).

Proof. Note first that it follows easily from (3.2) using u_{n} as a test function and (3.3) that

(3.5)  ∥u_{n}∥_{W_{2}^{1}(Ω_{n})} ≤ ∥f∥_{2} .

Let (Ω′_{n})_{∈ℕ} be a sequence of domains such that Ω′_{n} ⊂⊂ Ω_{n} and ⋃_{n=1}^{∞} Ω′_{n} = Ω. Then, by interior regularity of the weak solution u_{n} (c.f. [10], Theorem 8.8) for any n ∈ ℕ we find a constant c > 0 such that

∥u_{k}∥_{W_{2}^{2}(Ω′_{n})} ≤ c(∥u_{k}∥_{W_{2}^{1}(Ω′_{n})} + ∥f∥_{2}) ≤ 2c∥f∥_{2}

for all k ≥ n+1, where we used (3.5) for the last inequality. This means that the sequence (u_{k})_{k>n} is bounded in W_{2}^{2}(Ω′_{n}). Thus, we can find a subsequence converging strongly in W_{2}^{1}(Ω′_{n}) to w_{n}. By a diagonalization argument we can make sure that w_{n+1}|_{Ω′_{n}} = w_{n} for all n ∈ ℕ. As ⋃_{n=1}^{∞} Ω′_{n} = Ω we can define a function w ∈ W_{2,loc}^{1}(Ω) in such a way that w|_{Ω′_{n}} = w_{n} for all n ∈ ℕ. From (3.5) we easily deduce that w ∈ W_{2}^{1}(Ω) with ∥w∥_{W_{2}^{1}(Ω)} ≤ ∥f∥_{2}. From the construction of w it is clear that lim_{n→∞} u_{n}|_{Ω′} = w|_{Ω′} in W_{2}^{1}(Ω′) for all Ω′ ⊂⊂ Ω. If we can show that u = w this proves (3.4). By uniqueness of the solution it is therefore sufficient to show that w is a solution of (3.1). To this end let φ ∈ W_{2}^{1}(Ω) be arbitrary. Then, given ε > 0 we find n_{0} ∈ ℕ such that

∫_{Ω\Ω_{n}} |∇φ|^{2} + φ^{2} dx < ε^{2}

for all n ≥ n_{0}. From Cauchy-Schwarz inequality and (3.5) we easily deduce that

∫_{Ω_{n}\Ω_{n_{0}}} (∇u_{n}|∇φ) + u_{n}φ dx < ε∥f∥_{2} .

Hence,

∫_{Ω_{n}}(∇u_{n}|∇φ) + u_{n}φ dx ≤ ∫_{Ω_{n_{0}}} (∇u_{n}|∇φ) + u_{n}φ dx + ε∥f∥_{2} .

A similar argument shows that

∫_{Ω\Ω_{n_{0}}} (∇w|∇φ) + wφ dx ≤ ε∥f∥_{2} .

As ε > 0 was arbitrary we obtain by combining the above results that

lim_{n→∞} ∫_{Ω_{n}}(∇u_{n}|∇φ) + wφ dx = ∫_{Ω}(∇w|∇φ) + wφ dx .

On the other hand, it is clear that

lim_{n→∞} ∫_{Ω_{n}} fφ dx = ∫_{Ω} fφ dx .

which shows that w is the weak solution of (3.1) completing the proof of the proposition. □

3.2 Proposition. If Ω is a bounded domain such that the embedding of W_{2}^{1}(Ω) into L_{2}(Ω) is not compact, then

R := (−Δ_{N} + I )^{−1} ∉ L(L_{2}(Ω), L_{p}(Ω))

for all p > 2. Here, Δ_{N} denotes the Laplacian under Neumann boundary conditions.

For an example of a bounded domain where the embedding of W_{2}^{1}(Ω) into L_{2}(Ω) is not compact see e.g. [7], Theorem V.4.21.

Proof. Suppose that R ∈ L(L_{2}(Ω), L_{p}(Ω)) for some p > 2. Then, using Hölder’s inequality we see that, given ε > 0, there exists a domain Ω′ with \overline{Ω}′ ⊂ Ω such that

(3.6)  ∥Rf∥_{2,Ω\Ω′} ≤ ε

for all f ∈ 𝔹 := {g ∈ L_{2}(Ω): ∥g∥_{2} ≤ 1}. (We choose Ω′ such that Ω \ Ω′ has small measure.) On the other hand, we know that {Rf: f ∈ 𝔹} is a bounded set in W_{2}^{1}(Ω) and therefore, {Rf|_{Ω′}: f ∈ 𝔹} is relatively compact in L_{2}(Ω′). This, together with (3.6) easily implies the compactness of R as an operator on L_{2}(Ω), which means that −Δ_{N} has compact resolvent on L_{2}(Ω). But this is not possible because this is equivalent of W_{2}^{1}(Ω) being compactly embedded in L_{2}(Ω). Therefore our claim follows. □

3.3 Remark. One can use interpolation theory and duality to show that

(−Δ + I )^{−1} ∉ L(L_{p}(Ω), L_{q}(Ω))

for all 1 < p < q ≤ ∞ if Ω satisfies the condition of the above Proposition.

As a corollary we obtain our result that the solution of (3.1) does not satisfy a uniform estimate for all smooth domains with a given bound on the volume.

3.4 Corollary. Let M > 0 be given. Then, the map R := (−Δ_{N} + I )^{−1} can not satisfy a uniform bound with respect to all smooth domains with |Ω| ≤ M as a map from L_{2}(Ω) to L_{p}(Ω) whenever p > 2.

Proof. As the embedding constant L_{q} into L_{p} depends just on the volume of the domain and the exponents p and q it obviously enough to prove the assertion for p arbitrarily close to 2. Choose Ω as in the previous proposition and suppose that (Ω_{n})_{n∈ℕ} is a sequence of smooth domains such that Ω_{n}↗Ω. Suppose that u_{n} is the solution of a (3.1) with Ω replaced by Ω_{n} and u the solution of (3.1) with the same f ∈ L_{2}(Ω). Then, from Proposition 3.1 we conclude that lim_{n→∞} ∥u_{n}∥_{p,Ω′} = ∥u∥_{p,Ω′} for p > 2 sufficiently close to 2 and Ω′ ⊂⊂ Ω (Here we used Sobolev embedding theorem). Therefore, if ∥R∥_{L(L_{2}(Ω_{n}),L_{p}(Ω_{n}))} was uniformly bounded with respect to n ∈ ℕ for some p > 2 this would imply that the same is true for the domain Ω. But this contradicts Proposition 3.2. □

3.5 Remarks. (a) By interpolation and duality it is easily seen that there is no uniform bound for (−Δ_{N} + I )^{−1} as a map from L_{p}(Ω) to L_{q}(Ω) whenever 1 < p < q ≤ ∞.

(b) On the other hand, if we restrict ourselves to consider just domains in a class for which

(3.7)  ∥u∥_{r} ≤ c∥u∥_{W_{2}^{1}}

holds for some r > 2 and a constant c independent of the domain in the given class we can obtain uniform bounds on (−Δ_{N} + I )^{−1} as a map from L_{p}(Ω) to L_{q}(Ω) for suitable q > p. Indeed, proceeding similar as in the proof of Theorem 2.1 we can show that

(3.8)  (−Δ_{N} + I )^{−1} ∈ L(L_{p}(Ω), L_{\frac{r^{∗}p}{r^{∗}−p}} (Ω))

with a norm depending only on c in (3.7), r and p provided p ∈ (\frac{2N}{N+2}, r^{∗}). In case p > r^{∗} we can prove using the same iteration method as in the proof of Theorem 2.2 that

(3.9)  (−Δ_{N} + I )^{−1} ∈ L(L_{p}(Ω), L_{∞}(Ω))

with norm depending on the same quantities as above. Here, as in Remark 2.7(b) we have set r^{∗} := \frac{r}{r−2}. Classes of domains satisfying (3.7) are for instance domains satisfying the cone condition with the same fixed cone (e.g. [1], Lemma V.5.10). In that case r = \frac{2N}{N−2} for n ≥ 3 and arbitrary r < ∞ if N = 2. More general classes are considered in [13], where s is not necessarily the Sobolev exponent.

4. Lower bounds for the spectrum

In this section we shall first prove the existence of a uniform lower bound for the first eigenvalue of (1.2), where A(x, t) satisfies the hypotheses of Section 2 and a_{0} is nonnegative. We shall show then, that in case of a self adjoint problem (i.e. a ≡ 0) the eigenvalues of (1.2) are uniformly bounded away from zero by a constant depending only on the width of a strip containing the domain and a lower bound for α and β.

Note that it follows from the weak maximum principle (c.f. [10], Theorem 8.1 and 8.19) and the fact that b_{0} > 0 that λ_{1} > 0. It is clear that

λ_{1}^{−1} = spr(A_{p}) ≤ ∥A_{p}∥_{L(L_{p})} .

Therefore, by (2.23) and the fact that ϱ(A) = ϱ(A_{p}) by Proposition 2.8 we have the following result.

4.1 Theorem. Suppose that a_{0} is nonnegative. Then there exist a constant λ_{0} > 0 depending only on upper bounds of |Ω|, δ and γ such that the first eigenvalue λ_{1} of (1.2) satisfies λ_{1} ≥ αλ_{0}. Here, γ and δ are defined as in (2.7).

Let us now assume that a ≡ 0. In that case, using the the variational characterization of the first eigenvalue of (1.2)

(4.1)  λ_{1} = inf_{{u∈W^{1}_{2}(Ω) \atop u≠0}} \frac{a(u, u)}{∥u∥_{2}^{2}}

we prove now a more explicit estimate for the first eigenvalue. In order to do this we need an inequality of the form

(4.2)  ∥u∥_{2} ≤ c(∥∇u∥_{2}^{2} + ∥u∥_{2,∂Ω}^{2})^{\frac{1}{2}}

for all u ∈ W_{2}^{1}(Ω). This is of course well known. The proof of (4.1) is usually given by localization at the boundary. Therefore, the constant c depends on the parametrizations and hence on the geometry of the domain. Alternatively one could use (2.2) to prove the inequality estimating the L_{\frac{2N}{N−1}} -norm by the weaker L_{2}-norm. The constant then depends only on N and upper bounds for |Ω| and max{α^{−1}, β^{−1}}. We shall generalize the classical proof of Friedrichs inequality ([9]) in case of the Dirichlet problem for smooth domains, and use an approximation argument in the case of a Lipschitz domain in order to show that c in fact does only depend on the width of a strip containing Ω and a lower bound for b_{0}.

4.2 Theorem. Let Ω be a Lipschitz domain contained in a strip of width d_{0} > 0. Then,

(4.3)  ∥u∥_{2}^{2} ≤ 2d_{0}(d_{0}∥∇u∥_{2}^{2} + ∥u∥_{2,∂Ω}^{2})

for all u ∈ W_{2}^{1}(Ω).

Proof. We prove the assertion in case Ω is of class C^{1}. In the general case the result is obtained by approximating a Lipschitz domain by smooth domains in such a way that also the boundary integrals approximate the original ones. For details we refer to the appendix.

Let Ω be a bounded domain contained in a strip of width d_{0}. By changing the coordinates we may assume without loss of generality that the strip containing Ω is parallel to the hyperplane F spanned by the first N − 1 vectors of the natural basis of ℝ^{N}. Denote by P the orthogonal projection onto F. As Ω is of class C^{1} the restriction P_{∂Ω} of P to ∂Ω is a C^{1}-map from ∂Ω to F. By Sard’s Theorem (e.g. [2], Theorem 4.3.1), the set of critical values S of P_{∂Ω} has Lebesgue measure zero in F. Clearly, S is a closed set in F. Its complement, the set O of regular values of P_{∂Ω}, is an open set. Moreover, the cardinality of P_{∂Ω}^{−1}(x̂) is finite for all x̂ ∈ O and locally constant (e.g. [2], Prop. 4.1.5 and Cor. 4.1.6). Hence, for any connected component V of O the cardinality of P_{∂Ω}^{−1}(x̂) is constant for all x̂ ∈ V. Therefore we can find finitely many functions φ_{i}, ψ_{i} ∈ C^{1}(V, ∂Ω), i = 1, …, k such that φ_{i} < ψ_{i} for all i = 1, …, k and ∂Ω ∩ V × ℝ is the disjoint union of the graphs of these functions. Let now x = (x̂, x_{N}) be an arbitrary point in Ω_{V} := Ω ∩ V × ℝ. Then

φ_{i}(x̂) < x_{N} < ψ_{i}(x̂)

holds for some i ∈ \{1, …, k\}. If u ∈ C^{∞}(\overline{Ω}) is arbitrary then

u(x) = u(x̂, φ_{i}(x̂)) + ∫_{φ_{i}(x̂)}^{x_{N}} ∂_{ξ}u(x̂, ξ) dξ .

Applying Cauchy’s inequality we obtain

|u(x)| ≤ |u(x̂, φ_{i}(x̂))| + d0^{\frac{1}{2}} (∫_{φ_{i}(x̂)}^{ψ_{i}(x̂)} |∂_{ξ}u(x̂, ξ)|^{2} dξ)^{\frac{1}{2}} ,

where d_{0} is the width of the strip containing Ω. An elementary inequality yields

|u(x)|^{2} ≤ 2(|u(x̂, φ_{i}(x̂))|^{2} + d_{0} ∫_{φ_{i}(x̂)}^{ψ_{i}(x̂)} |∂_{ξ}u(x̂, ξ)|^{2} dξ) .

Integrating over Ω_{V} we obtain

∫_{Ω_{V}} |u(x)|^{2} ≤ 2(∫_{V} ∑_{i=1}^{k} ∫_{φ_{i}(x̂)}^{ψ_{i}(x̂)}|u(x̂, φ_{i}(x̂))|^{2} dx_{N} dx̂

+ d_{0} ∫_{V} ∑_{i=1}^{k} ∫_{φ_{i}(x̂)}^{ψ_{i}(x̂)} ∫_{φ_{i}(x̂)}^{ψ_{i}(x̂)} |∂_{ξ}u(x̂, ξ)|^{2} dξ dx_{N} dx̂)

≤ 2d_{0}(∑_{i=1}^{k} ∫_{V} |u(x̂, φ_{i}(x̂))|^{2}\sqrt{1 + |∇φ_{i}(x̂)|^{2}} dx̂

+ d_{0} ∑_{i=1}^{k} ∫_{V} ∫_{φ_{i}(x̂)}^{ψ_{i}(x̂)} |∂_{ξ}u(x̂, ξ)|^{2} dξ dx̂)

= 2d_{0}(∑_{i=1}^{k} ∥u∥_{L_{2}(φ_{i}(V))}^{2} + d_{0}∥∇u∥_{L_{2}(Ω_{V} )}^{2}) .

Using that the Lebesgue measure of S is zero, and adding up over all connected components of O we see that (4.3) is true for all u ∈ C^{∞}(\overline{Ω}). By density of C^{∞}(\overline{Ω}) in W_{2}^{1}(Ω) this is true for all u ∈ W_{2}^{1}(Ω). □

As an easy consequence of the above Result and (4.1) we obtain the following estimate for the first eigenvalue of (1.2).

4.3 Corollary. Suppose Ω satisfies the same conditions as in the above Theorem. Then, the first eigenvalue λ_{1} of (1.2) satisfies

λ_{1} ≥ \frac{1}{2d_{0} max\{d_{0}α^{−1}, β^{−1}\}} .

Finally we remark that an optimal estimate (i.e. an isoperimetric inequality) for domains contained in a strip are proved in Sperb [16] provided the mean curvature on the boundary is nonnegative.

Appendix

In this appendix we shall prove a technical lemma on the approximation of Lipschitz domains by smooth domains. We shall prove the following result.

A.1 Proposition. Let Ω be a bounded Lipschitz domain. Then there exists a sequence (Ω_{n}) of domains of class C^{∞} such that for any open set O containing \overline{Ω} and any compact subset K of Ω we have that

(A.1)  K ⊂ Ω_{n} ⊂ O

for all n ∈ ℕ large enough. Moreover, for any φ ∈ D(ℝ^{N})

(A.2)  lim_{n→∞} ∫_{Ω_{n}} φ(x) dx = ∫_{Ω} φ(x) dx

and

(A.3)  lim_{n→∞} ∫_{∂Ω_{n}} φ(x) dσ = ∫_{∂Ω} φ(x) dσ .

Of course the only difficulty is to prove that we can choose the approximation in such a way that (A.3) holds. The proof will be given in a sequence of lemmas. We always assume that U and V are open sets in ℝ^{k}. The proof of the following result on differentiability is elementary and left to the reader.

A.2 Lemma. Let φ ∈ C(U, V) be a homeomorphism and f: U → ℝ^{N}, g: V → ℝ^{N} be such that f ∘ φ = g. Suppose that φ and g are differentiable in x_{0} ∈ U and that Dφ(x_{0}) is invertible. Then, f is differentiable in φ(x_{0}) and

Df(φ(x_{0})) = Dg(x_{0})[Dφ(x_{0})]^{−1}.

Taking g the identity and f = φ^{−1} we obtain the following corollary.

A.3 Corollary. (a) Let φ ∈ C(U, V) be a homeomorphism. If φ is differentiable in x_{0} and Dφ(x_{0}) is invertible then φ^{−1} is differentiable in φ(x_{0}) and

Dφ^{−1}(φ(x_{0})) = [Dφ(x_{0})]^{−1}.

(b) Suppose in addition that φ and φ^{−1} are locally Lipschitz. Then φ is differentiable in x_{0} if and only if φ^{−1} is differentiable in φ(x_{0}).

The second part follows from the first because Dφ(x) is nonsingular whenever φ is differentiable in x if φ and φ^{−1} are locally Lipschitz.

A.4 Lemma. Let g_{i} ∈ C^{1−}(ℝ^{N−1}) (i=1,2) and set

(A.4)  G_{i}(x) := T_{i}\left[{x \atop g_{i}(x)}\right]

for all x ∈ R^{N−1}, where T_{i} ∈ SO(N). Suppose that U_{i} ∈ ℝ^{N} are open sets such that

M := G_{1}(U_{1}) = G_{2}(U_{2}) .

Further assume that (ψ_{ε})_{ε>0} is a mollifier and α ∈ D(ℝ^{N−1}) with 0 ≤ α ≤ 1. Define

(A.5)  g_{1ε} := α(g_{1} ∗ ψ_{ε}) + (1 − α)g_{1}  and  G_{1ε}(x) := T_{1}\left[{x \atop g_{1ε}(x)}\right] .

Then, for all ε small enough there exists a function g_{1ε} ∈ C^{1−}(ℝ^{N−1}) and an open set U_{2ε} ⊂ ℝ^{N−1} such that

M_{ε} := G_{1ε}(U_{1}) = G_{2ε}(U_{2ε}) ,

where G_{2ε} := T_{2}\left[{x \atop g_{2ε}(x)}\right].

The above lemma asserts that if M is a graph of a Lipschitz function in two different directions then also the approximation M_{ε} is a graph in the same directions.

Proof. Denote by P the projection of ℝ^{N} onto the subspace ℝ^{N−1} spanned by the first N − 1 vectors of the natural basis of ℝ^{N}. By our assumptions it is clear that φ := P T_{2}^{−1}G_{1} ∈ C^{1−}(U_{1}, U_{2}) is bijective with φ^{−1} = P T_{1}^{−1}G_{2} ∈ C^{1−}(U_{2}, U_{1}). From Lemma A.1 and Corollary A.2 we infer that g_{1} is differentiable in x ∈ U_{1} iff and only if g_{2} is differentiable in φ(x) ∈ U_{2}. By Rademacher’s theorem (e.g. [8], Theorem 3.1.6) g_{1}, g_{2} are differentiable almost everywhere. Hence, the unit normal on M is defined almost everywhere and given by

ν(x) = \frac{1}{\sqrt{1 + |∇g_{1}(x)|^{2}}} T_{1}\left[{∇g_{1}(x) \atop −1}\right] = \frac{±1}{\sqrt{1 + |∇g_{2}(φ(x))|^{2}}} T_{2}\left[{∇g_{2}(φ(x)) \atop −1}\right] .

By changing the orientation we may assume without loss of generality that the positive sign holds. From this we conclude that for almost all x ∈ U_{1}

(ν(x)|T_{2}e_{N}) = \frac{1}{\sqrt{1 + |∇g_{2}(φ(x))|^{2}}} ≥ δ

for some constant δ > 0. Hence,

ν(x) ∈ C_{δ} := {v ∈ ℝ^{N}(v|T_{2}e_{N}) ≥ δ} .

for almost all x ∈ U_{1}, where e_{N} := (0, … 0, 1) ∈ ℝ^{N}. We shall show that

(A.6)  ν_{ε}(x) ∈ C_{\frac{δ}{2}}

for almost all x ∈ U_{1} and ε small enough. Here ν_{ε}(x) is the normal unit vector on M_{ε} := G_{1ε}(U_{1}). To show this it is sufficient to show that ∇g_{1ε} lies in a suitable small neighborhood of the closed convex hull of {∇g_{1}(x): x ∈ U_{1}}. As ∇g_{1ε} is given by

∇g_{1ε} = [α(∇g_{1} ∗ ψ_{ε}) + (1 − α)∇g_{1}] + [∇α(g_{1} ∗ ψ_{ε} − g_{1})]

this follows easily. Indeed, ∇g_{1} ∗ ψ_{ε} lies in the convex hull and hence the term in the first brackets. The last term tends to zero uniformly as ε approaches zero. The next lemma shows that (A.6) implies the assertion of the lemma. □

A.5 Lemma. Let g ∈ C^{1−}(U). Denote the graph of g by M and by ν the outer unit normal on M. Suppose z is such that

(A.7)  (ν|z) ≥ δ > 0

almost everywhere. Then, M is the graph of a Lipschitz function over the hyperplane E normal to z.

Proof. E is the image of ℝ^{N−1} under an orthogonal transformation T ∈ O(N). Define

φ(x) := P T^{−1}\left[{x \atop g(x)}\right]

for all x ∈ U, where P is the projection of ℝ^{N} onto ℝ^{N−1} along the space spanned by e_{N}. Our assertion follows if we are able to show that φ is injective with Lipschitz inverse. In that case we can define g̃(y) := g(φ^{−1}(y)).

In a first step we consider the case N = 2. Then, an elementary calculation shows that

(ν(x)|z) = −\frac{φ′(x)}{\sqrt{1 + |g′(x)|^{2}}} ,

where

ν(x) := \frac{1}{\sqrt{1 + |g′(x)|^{2}}} \left[{g′(x) \atop −1}\right]

is the unit normal vector on M, which is defined almost everywhere. Using the assumption on ν, this implies that for almost all x ∈ U

−\sqrt{1 + |g′(x)|^{2}} ≤ φ′(x) ≤ −δ\sqrt{1 + |g′(x)|^{2}}

and therefore φ is a strictly decreasing continuous function. Thus φ is injective and φ^{−1} is Lipschitz with Lipschitz constant smaller than δ^{−1}.

In a second step we consider the case N > 2 and prove that M is a graph over E. Let F be the subspace of ℝ^{N} spanned by z and e_{N}. By the continuity of g it is sufficient to prove that M_{a} := M ∩ (a + F) is a graph over E_{a} := E ∩ (a + F) for almost all a ∈ F^{⊥}. As dom(Dφ) is U up to a set of measure zero, a similar assertion holds for D_{a} := dom(Dφ) ∩ (a + F) for almost all a ∈ F^{⊥} (This follows from Fubini’s Theorem).

For all x ∈ D_{a} the normal on M_{a} in (x, g(x)) is up to a scalar factor the projection of the normal on M onto a + F. From the first step we can conclude now that M_{a} is the graph of a function over E_{a} for almost every a ∈ F^{⊥}. This shows that φ is injective. It remains to show the Lipschitz continuity of φ^{−1}. To this end note that φ is differentiable almost everywhere. It follows from (A.7) and the definition of φ that Dφ(x) is injective and hence invertible whenever the derivative exists. Moreover, by Cor. A.3 and (A.7) the norm of Dφ^{−1}(φ(x)) is uniformly bounded proving that φ^{−1} is Lipschitz. This completes the proof of the lemma. □

Of course, the proof of the above lemma is obvious from the implicit function theorem if g ∈ C^{1}(U). We are now ready to give the proof of Proposition A.1

Proof of Proposition A.1. Let Ω be a Lipschitz domain. Then there exist g_{i} ∈ C^{1−}(ℝ^{N}), T_{i} ∈ SO(N) and open sets U_{i} ⊂ ℝ^{N−1} (i = 1 …, m) such that

(A.8)  ∂Ω = ⋃_{i=1}^{m} G_{i}(U_{i}) ,

where G_{i} is defined as in (A.4). Choose V_{1} ⊂ ℝ^{N−1} in such a way that its closure is contained in U_{1} and (A.8) still holds when we replace U_{1} by V_{1}. Take α ∈ D(U_{1}) with 0 ≤ α ≤ 1 and α ≡ 1 on V_{1}, and define g_{1ε} and G_{1ε} as in (A.5). Replacing g_{1} by g_{1ε} we obtain a new domain Ω_{ε}. By our choice of α the part G_{1ε}(V_{1}) of ∂Ω_{ε} is of class C^{∞}. Moreover, as

lim_{ε→0} g_{1ε} = g_{1}

uniformly,

lim_{ε→0} ∇g_{1ε} = ∇g_{1}

almost everywhere, and ∥g_{1ε}∥_{∞} ≤ ∥∇g_{1}∥_{∞} + ∥∇α∥_{∞}∥g_{1}∥_{∞} we easily see that (A.1)– (A.3) hold as ε approaches zero. By Lemma A.4 we can find Lipschitz functions g_{iε} (i = 2, … m) and open sets U_{iε} ⊂ ℝ^{N−1} such that

∂Ω_{ε} = ⋃_{i=1}^{m} G_{iε}(U_{iε}) ,

where G_{iε} are defined as in Lemma A.4. Fix now ε = ε^{(1)} > 0 small enough. Repeating the above construction using g_{2ε^{(1)}} instead of g_{1} we obtain a sequence Ω_{ε^{(1)}ε} having the required properties. Note that in each step we have to shrink slightly the open sets U_{iε}. After m steps we find domains Ω_{ε^{(1)}...ε^{(m)}} of class C^{∞}. Finally we choose sequences (ε_{n}^{(i)})_{n∈ℕ} (i = 1 … m) tending to zero as n tends to infinity, and set Ω_{n} := Ω_{ε_{n}^{(1)}...ε_{n}^{(m)}}. Then, the Ω_{n} satisfy the claimed properties, and the proof of the proposition is complete. □

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School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia