Value Problems for Differential Forms on C^{1}-Domains

R. Selvaggi and I. Sisto

Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003)

Abstract

Existence and uniqueness value boundary problems for differential forms in C^{1}-domains are investigated.

Mathemaics Subject Classification: 58A10, 45E05 Key words: differential form, homology group, cohomology group, harmonic form

Introduction

In this paper we resolve boundary value problems for differential forms in a bounded C^{1}-domain Ω of R^{n}, 3 ≤ n. The obtained results are an extension of some theorems established by C. Miranda in [8] for differential forms in C^{2,α}-domains of R^{n}.

The fundamental result is Theorem 2.1:

Let F_{s−1} ∈ C_{s−1}^{1}(Ω) be a closed form with interior nontangential trace F_{s−1}^{−} ∈ L_{s−1}^{p}(∂Ω). If

∫_{τ_{s−1}^{i}} F_{s−1}^{−} = 0, i = 1, ..., R_{s−1}^{−},

where ([τ_{s−1}^{i}])_{1≤i≤R_{s−1}^{−}} is a base of C^{1}-differentiable singular homology space H_{s−1}(∂Ω)

and R_{s−1}^{−} is the (s−1)-th Betti number of Ω, then there exists a form U_{s−2} ∈ C_{s−2}^{1}(Ω), whose coefficients are in W^{1,p}(Ω), such that

(∗)  dU_{s−2} = F_{s−1}  in Ω.

This result is obtained using a formula of Bidal-de Rham (see (1.10)). The use of this formula changes the existence problem of a differential form that satisfies (∗) in a suitable Neumann problem for harmonic forms already studied in [15].

Using a regularity result for the solutions of the homogeneous Neumann problem for harmonic forms established in [15] (see Theorem 2.1) and the continuity hypothesis for F_{s−1} in \overline{Ω}, we obtain a uniqueness theorem (see Theorem 2.4). With this aim, first we prove an extension theorem (see Theorem 1.2) and then, as a consequence, we deduce a Stokes’ Theorem (see Theorem 1.3) for differential s-forms of C̃_{s}^{1}(\overline{Ω}) (see Preliminaries).

∗ Balkan Journal of Geometry and Its Applications, Vol.8, No.2, 2003, pp. 81-96.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2003.

Throughout this work we use the definitions and the results of [13] concerning differential forms and singular homology and cohomology groups of the C^{1}-manifold ∂Ω.

1 Preliminaries

In this section we summarize basic concepts, notations, and results that will be used throughout the paper.

We assume that Ω is a bounded and connected C^{1}-domain of R^{n}, 3 ≤ n. Thus, (see [16]), there exist an increasing sequence (Ω_{h})_{h∈N} of C^{∞}-domains, Ω_{h} ⊂ Ω, such that Ω_{h} → Ω in C^{1} according to Nečas (see [7] p. 85) and a sequence (Λ_{h})_{h∈N} of C^{1}- diffeomorphisms Λ_{h} : ∂Ω → ∂Ω_{h} such that

(1.1)  lim_{h} sup_{Q∈∂Ω} |Q − Λ_{h}(Q)| = 0.

Furthermore, there is a finite covering (B_{r})_{1≤r≤m} of ∂Ω by open spheres B_{r} = B(Q_{r}, δ) with center Q_{r} ∈ ∂Ω and radius δ, such that, for r = 1, …, m

B(Q_{r}, 2δ) ∩ ∂Ω = {(x, x_{n}) ∈ R^{n−1} × R : x_{n} = ξ_{r}(x)} ∩ B(Q_{r}, 2δ), (1.2)

and

B(Q_{r}, 2δ) ∩ ∂Ω_{h} = {(x, x_{n}) ∈ R^{n−1} × R : x_{n} = ξ_{rh}(x)} ∩ B(Q_{r}, 2δ), (1.3)

where ξ_{r} ∈ C_{0}^{1}(R^{n−1}), ξ_{r}(0) = \frac{∂ξ_{r}}{∂x_{l}} (0) = 0 (l = 1, …, n − 1), ξ_{rh} ∈ C_{0}^{∞}(R^{n−1}), and

lim_{h} ∥ξ_{rh} − ξ_{r}∥_{C_{0}^{1}(R^{n−1})} = 0. (1.4)

Let now x̃_{r} = (x, ξ_{r}(x)) ∈ ∂Ω ∩ B(Q_{r}, 2δ) → x ∈ R^{n−1} (1.5)

and

x̃_{hr} = (x, ξ_{hr}(x)) ∈ ∂Ω_{h} ∩ B(Q_{r}, 2δ) → x ∈ R^{n−1}. (1.6)

For l, i = 1, …, n − 1,

lim_{h} \frac{∂(x̃_{rh} ∘ Λ_{h} ∘ x_{r}^{−1})_{i}}{∂x_{l}} (x) = δ_{il} (1.7)

uniformly in U_{r} = x̃_{r}(∂Ω ∩ B(Q_{r}, 2δ)), where (x̃_{rh} ∘ Λ_{h} ∘ x_{r}^{−1})_{i} is the i-th coordinate of the function x̃_{rh} ∘ Λ_{h} ∘ x_{r}^{−1}.

Let U_{s} = ∑_{i∈N_{s}^{n}}a_{i}dX_{i} be a form defined in Ω (respectively in R^{n} \ \overline{Ω}). ^{1}

Value Problems for Differential Forms on C^{1}-Domains 83

If U_{s} ∈ C_{s}^{2}(Ω) we set δU_{s} = (dU_{s})^{∗} and ΔU_{s} = dδU_{s}, (1.8)

1 N_{s}^{n} = {i = (i_{1}, …, i_{s}) ∈ N^{s} : 1 ≤ i_{1} < … < i_{s} ≤ n}; if i = (i_{1}, …, i_{s}) ∈ N_{s}^{n}, dX_{i} = dX_{i_{1}} ∧ … ∧ X_{i_{s}}.

where dU_{s} is the exterior derivative of U_{s} and ∗ is the Hodge’s operator. U_{s} is said to be closed (harmonic, respectively) in Ω iff dU_{s} = 0 (ΔU_{s} = 0, respectively) in Ω. Furthermore U_{s} is said to be derived in Ω iff there exists a form V_{s−1} such that

dV_{s−1} = U_{s} in Ω. (1.9)

and V_{s−1} is called the primitive of U_{s}.

Thus we have the following identity of Bidal-de Rham (see (60) in [8])

dδU_{s}^{∗} + (−1)^{n}δδU_{s} = (−1)^{n(s+1)} ∑_{i∈N_{s}^{n}} Δa_{i} dX_{i}, (1.10)

where Δa_{i} = ∑_{l=1}^{n} \frac{∂^{2}a_{i}}{∂x_{l}^{2}} .

We denote with C̃_{s}^{1}(\overline{Ω}) the space of the forms U_{s} ∈ C_{s}^{1}(Ω) such that each coefficient of U_{s} and of dU_{s} is in C^{0}(\overline{Ω}).

Given k ∈ N and 1 < p < ∞, we denote with D_{s}^{k,p}(Ω) the space of the forms U_{s} such that each coefficient of U_{s} is in W^{k,p}(Ω).

We say that U_{s} has interior (exterior, respectively) nontangential trace in L_{s}^{p}(∂Ω) iff, for any i ∈ N_{s}^{n}, a_{i} has interior nontangential trace a_{i}^{−} (exterior nontangential trace a_{i}^{+}, respectively) in L^{p}(∂Ω). The form ^{2}

U_{s}^{−} = ∑_{i∈N_{s}^{n}} a_{i}^{−}dX_{i}(Q) (1.11)

(respectively the form

U_{s}^{+} = ∑_{i∈N_{s}^{n}} a_{i}^{+}dX_{i}(Q)) (1.12)

is called interior (exterior, respectively) nontangential trace of U_{s}.

If U_{s} ∈ D^{1,p}(Ω), the form

Tr(U_{s}) = ∑_{i∈N_{s}^{n}} Tr(a_{i})dX_{i}(Q) (1.13)

is considered, where the mapping Tr : W^{1,p}(Ω) → L^{p}(∂Ω) is the continuous extension to W^{1,p}(Ω) of the mapping restriction defined initially on C^{∞}(\overline{Ω}).

If U_{s} ∈ C_{s}^{0}(Ω) and U_{s} has interior nontangential trace in L_{s}^{p}(∂Ω), then (see Theorem 2.3 in [13])

lim_{h} Λ_{h}^{∗}(U_{sh}) = U_{s}^{−} in L_{s}^{p}(∂Ω), (1.14)

where U_{sh} is the restriction of U_{s} on ∂Ω_{h}.

We denote with N_{s}^{1,p}(Ω) the space of the forms U_{s} ∈ C_{s}^{1}(Ω) such that each coefficient of U_{s} and dU_{s} has interior nontangential trace in L^{p}(∂Ω).

The following result holds:

2 dX_{i}(Q) is the restriction to ∂Ω of dX_{i}, hence dX_{i}(Q) = j^{∗}dX_{i}, where j : ∂Ω → R^{n} is the inclusion map.

Theorem 1.1. If U_{s} ∈ C_{s}^{0}(Ω) ∩ D_{s}^{1,p}(Ω) and Φ_{n−s−1} ∈ C_{n−s−1}^{0}(∂Ω), then

(1.15)  lim_{h} ∫_{∂Ω_{h}} U_{s} ∧ Λ_{h}^{−1∗}(Φ_{n−s−1}) = ∫_{∂Ω} Tr(U_{s}) ∧ Φ_{n−s−1}.

Proof. By using a partition of unity (φ_{r})_{1≤r≤m}, corresponding to the covering (B_{r})_{1≤r≤m} of ∂Ω described above, in order to prove (1.15) it will suffice to show that

(1.16) lim_{h} ∫_{R^{n−1}} x_{rh}^{−1∗}(φ_{r}U_{sh} ∧ Λ_{h}^{−1∗}(Φ_{n−s−1})) = ∫_{R^{n−1}} x_{r}^{−1∗}(φ_{r}Tr(U_{s}) ∧ Φ_{n−s−1}).

for all r = 1, …, m. For simplicity of notation in the following we omit the index r. Let

U_{s} = ∑_{i∈N_{s}^{n−1}} a_{i}dX_{i} + ∑_{i∈N_{s−1}^{n−1}} a_{in}dX_{i} ∧ dX_{n} and x^{−1∗}(φΦ_{n−s−1})) = ∑_{i∈N_{n−s−1}^{n−1}} b_{i}dx_{i}.

Thus x_{h}^{−1∗}(φU_{sh} ∧ Λ_{h}^{−1∗}(Φ_{n−s−1})) is a (n − 1)-form on R^{n−1}. Its coefficients are the sum of a finite number of terms like

φ(x, ξ_{h}(x))a_{i}(x, ξ_{h}(x))b_{j}(f_{h}(x))|\frac{∂f_{hj}}{∂x_{j}} (x)|

or terms like

φ(x, ξ_{h}(x))a_{in}(x, ξ_{h}(x))b_{j}(f_{h}(x))|\frac{∂f_{hj}}{∂x_{j}} (x)|\frac{∂ξ_{h}}{∂x_{l}} (x),

where

f_{h} = x̃ ∘ Λ_{h}^{−1} ∘ x_{h}^{−1} = (f_{h1}, …, f_{h(n−1)}), and |\frac{∂f_{hj}}{∂x_{j}} | = det\frac{∂(f_{hj_{1}}, ⋯, f_{hj_{n−s−1}})}{∂(x_{j_{1}}, ⋯, x_{j_{n−s−1}})}.

Hence, since spt(φ) ⊂ B(Q, δ), the integrals in (1.16) are integrals on the compact set

K = {x ∈ R^{n−1} : |x| ≤ δ}.

We observe that using Theorem 4.5, p. 85 in [10] we have

lim_{h} a_{i}(x, ξ_{h}(x)) = Tr(a_{i})(x, ξ(x)) in L^{p}(K),

while, using (1.1) and (1.7), we have

(1.17) lim_{h} φ(x, ξ_{h}(x))b_{j}(f_{h}(x))|\frac{∂f_{hj}}{∂x_{j}} (x)| = φ(x, ξ(x))b_{j}(x) and lim_{h} \frac{∂ξ_{h}}{∂x_{j}} (x) = \frac{∂ξ}{∂x_{j}} (x)

uniformly in K. Therefore, since the sequence of functions that appear in the left side of (1.17) is uniformly bounded in K, by the Dominated Convergence Theorem we obtain the proof .

□

Corollary. Assume U_{s} ∈ C_{s}^{1}(Ω) ∩ D_{s}^{1,p}(Ω). If dU_{s} has interior nontangential trace in L_{s+1}^{p}(∂Ω), then Tr(U_{s}) ∈ W_{s}^{1,p}(∂Ω) and

dTr(U_{s}) = (dU_{s})^{−} a.e. on ∂Ω. (1.18)

Proof. It is sufficient to show that

∫_{∂Ω}(dU_{s})^{−} ∧ Φ_{n−s−2} = (−1)^{s} ∫_{∂Ω} Tr(U_{s}) ∧ dΦ_{n−s−2}

for all Φ_{n−s−2} ∈ C̃_{n−s−2}^{1}(∂Ω) (see n.1 in [13]), that is to say, because of (1.14) and since (dU_{s})_{h} = d(U_{sh}),

lim_{h} ∫_{∂Ω} Λ_{h}^{∗}(dU_{sh}) ∧ Φ_{n−s−2} = (−1)^{s} ∫_{∂Ω} Tr(U_{s}) ∧ dΦ_{n−s−2}.

Hence, it is enough to observe that

∫_{∂Ω} Λ_{h}^{∗}(dU_{sh}) ∧ Φ_{n−s−2} = ∫_{∂Ω_{h}} dU_{s} ∧ Λ_{h}^{−1∗}(Φ_{n−s−2})
= (−1)^{s} ∫_{∂Ω_{h}} U_{s} ∧ Λ_{h}^{−1∗}(dΦ_{n−s−2}),

and to apply Theorem 1.1.

□

We obtain also the following Theorems Theorem 1.2. If U_{s} ∈ C̃_{s}^{1}(\overline{Ω}), then there exists an open set Ω′ of R^{n} such that \overline{Ω} ⊂ Ω′ and there exists a form \overline{U}_{s} ∈ C̃_{0,s}^{1}(Ω′) such that \overline{U}_{s} and d\overline{U}_{s} are extensions of U_{s} and dU_{s} respectively.

Proof. Let U_{s} = ∑_{i∈N_{s}^{n}} a_{i}dX_{i} ∈ C̃_{s}^{1}(\overline{Ω}) and let dU_{s} = ∑_{j∈N_{s+1}^{n}} b_{j}dX_{j}. Then we have

(∀i ∈ N_{s}^{n}) (a_{i} ∈ C^{0}(\overline{Ω}) ∩ C^{1}(Ω)) and (∀j ∈ N_{s+1}^{n}) (b_{j} ∈ C^{0}(\overline{Ω})).

In Ω it results

(∀j ∈ N_{s+1}^{n}) (b_{j} = ∑_{k=1}^{s+1}(−1)^{k} \frac{∂a_{j^{k}}}{∂X_{j_{k}}}),

where ĵ^{k} = (j_{1}, …, j_{k−1}, j_{k+1}, …, j_{s+1}) when j = (j_{1}, …, j_{s+1}).

Arguing in a manner similar to the proof of Theorem 54 XV of [11], we prove the existence of an open set Ω′ of R^{n} such that \overline{Ω} ⊂ Ω′ and, for i ∈ N_{s}^{n} and j ∈ N_{s+1}^{n}, the existence of functions \overline{a}_{i} ∈ C_{0}^{0}(Ω′) ∩ C^{1}(Ω′ \ ∂Ω) and \overline{b}_{j} ∈ C_{0}^{0}(Ω′) extensions of a_{i} and b_{j}, respectively, such that

\overline{b}_{j} = ∑_{k=1}^{s+1}(−1)^{k} \frac{∂\overline{a}_{j^{k}}}{∂X_{j_{k}}} in Ω′ \ ∂Ω. (1.19)

Let, then

\overline{U}_{s} = ∑_{i∈N_{s}^{n}} \overline{a}_{i}dX_{i} and \overline{V}_{s+1} = ∑_{j∈N_{s+1}^{n}} \overline{b}_{j}dX_{j}.

We have that \overline{U}_{s} ∈ C_{0,s}^{0}(Ω′) and \overline{V}_{s+1} ∈ C_{0,s+1}^{0}(Ω′). In order to prove that \overline{U}_{s} is regular in Ω′, using the Lemma 16.d p. 105 in [17], it suffices to show the existence of a sequence (\overline{U}_{s}^{p})_{p∈N} in C_{s}^{∞}(Ω′) such that

\overline{U}_{s}^{p} → \overline{U}_{s} and d\overline{U}_{s}^{p} → \overline{V}_{s+1} (1.20)

uniformly in every compact subset of Ω′. Clearly we may suppose \overline{U}_{s} and \overline{V}_{s+1} defined in R^{n}. Let now (ρ_{p})_{p∈N} be a sequence of mollifiers and let

\overline{U}_{s}^{p} = ρ_{p} ∗ \overline{U}_{s} = ∑_{i∈N_{s}^{n}} ρ_{p} ∗ \overline{a}_{i}dX_{i}

and

\overline{V} _{s+1}^{p} = ρ_{p} ∗ \overline{V}_{s+1} = ∑_{j∈N_{s+1}^{n}} ρ_{p} ∗ \overline{b}_{j}dX_{j},

where ∗ is the usual convolution product between functions.

First we obtain that \overline{U}_{s}^{p} ∈ C_{0,s}^{∞}(R^{n}) and \overline{V} _{s+1}^{p} ∈ C_{0,s+1}^{∞}(R^{n}), and from Proposition IV.21, in [1], it follows that

\overline{U}_{s}^{p} → \overline{U}_{s} and \overline{V} _{s+1}^{p} → \overline{V}_{s+1}

uniformly on compact sets of R^{n}. We finish by proving the second formula in (1.20). For this it is sufficient to show that

∑_{k=1}^{s+1}(−1)^{k} \frac{∂}{∂X_{j_{k}}} (ρ_{p} ∗ \overline{a}_{j^{k}}) = ρ_{p} ∗ \overline{b}_{j}

or, for Proposition IV.2 in [1], that

∑_{k=1}^{s+1}(−1)^{k}(\frac{∂}{∂X_{j_{k}}} ρ_{p}) ∗ \overline{a}_{j^{k}} = ρ_{p} ∗ \overline{b}_{j}

for all p ∈ N and j ∈ N_{s+1}^{n}. We observe that, as a consequence of (1.19), we have

\begin{aligned}∑_{k=1}^{s+1}(−1)^{k} \frac{∂}{∂Y_{j_{k}}} (ρ_{p}(X − Y) · \overline{a}_{j^{k}}(Y)) & = ∑_{k=1}^{s+1}(−1)^{k+1}(\frac{∂}{∂X_{j_{k}}} ρ_{p})(X − Y) · \overline{a}_{j^{k}}(Y) \\ & +ρ_{p}(X − Y) · \overline{b}_{j}(Y) \end{aligned}

for X ∈ R^{n} and Y ∈ R^{n} \∂Ω. Thus ∑_{k=1}^{s+1}(−1)^{k} \frac{∂}{∂Y_{j_{k}}} (ρ_{p}(X − Y) · \overline{a}_{j^{k}}(Y)) is in L^{1}(R^{n}).

Since

\begin{matrix}∫_{R^{n}} ∑_{k=1}^{s+1}(−1)^{k} \frac{∂}{∂Y_{j_{k}}} (ρ_{p}(X − Y) · \overline{a}_{j^{k}}(Y))dY = ∫_{Ω} ∑_{k=1}^{s+1}(−1)^{k} \frac{∂}{∂Y_{j_{k}}} (ρ_{p}(X − Y) · \overline{a}_{j^{k}}(Y))dY \\ + ∫_{R^{n}\Ω} ∑_{k=1}^{s+1}(−1)^{k} \frac{∂}{∂Y_{j_{k}}} (ρ_{p}(X − Y) · \overline{a}_{j^{k}}(Y))dY = I_{1} + I_{2}, \end{matrix}

to obtain (1.20) we need to prove

I_{1} + I_{2} = 0. (1.21)

Now, since (see Preliminares)

I_{1} = lim_{h} ∑_{k=1}^{s+1}(−1)^{k} ∫_{Ω_{h}} \frac{∂}{∂Y_{j_{k}}} (ρ_{p}(X − Y) · \overline{a}_{j^{k}}(Y))dY

= ∑_{k=1}^{s+1}(−1)^{k} lim_{h} ∫_{∂Ω_{h}} ρ_{p}(X − Y) · \overline{a}_{j^{k}}(Y)dY

= ∑_{k=1}^{s+1}(−1)^{k} lim_{h} ∫_{∂Ω} ρ_{p}(X − Λ_{h}(Y)) · \overline{a}_{j^{k}}(Λ_{h}(Y))ω_{h}(Y)dY,

where ω_{h} : ∂Ω → R_{+} is the function of [17]: Theorem 1.12, by Dominated Convergence Theorem it follows that

I_{1} = ∑_{k=1}^{s+1}(−1)^{k} ∫_{∂Ω} ρ_{p}(X − Y) · \overline{a}_{j^{k}}(Y)dY. (1.22)

In the same manner we prove that

I_{2} = − ∑_{k=1}^{s+1}(−1)^{k} ∫_{∂Ω} ρ_{p}(X − Y) · \overline{a}_{j^{k}}(Y)dY (1.23)

taking a sequence (Ω′_{h})_{n∈N} of C^{∞}-domains of R^{n} with Ω ⊂ Ω′_{h} and a sequence of diffeomorphisms Λ′_{h} : ∂Ω → ∂Ω′_{h} such that Ω → Ω′_{h} in C^{1} according to Nečas (see [10] and [16]) and lim_{h} sup_{Q∈∂Ω} |Q−Λ′_{h}(Q)| = 0. From this and (1.22) we have (1.21). Hence the thesis.

□

Theorem (Stokes) 1.3. If U_{s} ∈ C̃_{s}^{1}(\overline{Ω}), then

∫_{∂Γ_{s+1}} U_{s} = ∫_{Γ_{s+1}} dU_{s} (1.24)

for all chain Γ_{s+1} ⊂ \overline{Ω}. Proof. Let Ω′ be an open set of R^{n} such that \overline{Ω} ⊂ Ω′ and let \overline{U}_{s}, d\overline{U}_{s} be regular forms in Ω′ such that the coefficients of \overline{U}_{s} and of d\overline{U}_{s} are respectively the extensions of the corresponding coefficients of U_{s} and dU_{s} (see Theorem 1.2). Then, by the definition of regular forms, (1.24) is obtained.

□

2 Existence and Uniqueness Theorems

Let 1 < s < n. Following [8] we introduce the form

ω_{n−s}(X, Y) = ∑_{i∈N_{n−s}^{n}} \frac{1}{|X − Y |^{n−2}} dX_{i}dY_{i} (2.1)

in two variables (X, Y) ∈ R^{n} ×R^{n} (see [4], Section. 7). By (2.1), if X, Y ∈ R^{n}, X ≠ Y we obtain

d_{X}δ_{X}ω_{n−s}(X, Y) = (−1)^{n−s}d_{Y} δ_{X}ω_{n−s−1}(X, Y) (2.2)

and

δ_{X}ω_{s−1}(X, Y) = (−1)^{n(s−1)−1}δ_{Y} ω_{n−s}(X, Y). (2.3)

Let ([τ_{s−1}^{j}], [γ_{s−1}^{l}])_{{1≤j≤R{− \atop s−1} \atop 1≤l≤R{+ \atop s−1}}} , ([t_{s−1}^{j}])_{1≤j≤R_{s−1}^{−}} and let ([c_{s−1}^{l}])_{1≤l≤R_{s−1}^{+}} be bases of

C^{1}-differentiable singular homology spaces H_{s−1}(∂Ω), H_{s−1}(Ω), and H_{s−1}(\overline{R}^{n} \ \overline{Ω}) respectively, verifying the following conditions (see n. 5 in [13])

τ_{s−1}^{j} ∼ 0 in \overline{R}^{n} \ Ω and γ_{s−1}^{l} ∼ 0 in \overline{Ω}

and

t_{s−1}^{j} ∼ τ_{s−1}^{j} in \overline{Ω} and c_{s−1}^{l} ∼ γ_{s−1}^{l} in \overline{R}^{n} \ Ω,

for l = 1, …, R_{s−1}^{+} and j = 1, …, R_{s−1}^{−} (see n. 5 in [13]).

Let now 1 < p < ∞. The following results hold:

Theorem 2.1. Let F_{s−1} ∈ C_{s−1}^{1}(Ω) be a closed form with interior nontangential trace in L_{s−1}^{p}(∂Ω) such that

∫_{τ_{s−1}^{i}} F_{s−1}^{−} = 0 i = 1, ..., R_{s−1}^{−}. (2.4)

Then there exists U_{s−2} ∈ C_{s−2}^{1}(Ω) ∩ D_{s−2}^{1,p}(Ω) verifying

i) dU_{s−2} = F_{s−1} in Ω, ii) Tr(U_{s−2}) ∈ W_{s−2}^{1,p}(∂Ω) and dTr(U_{s−2}) = F_{s−1}^{−} a.e. on ∂Ω, iii) dU_{s−2}^{∗} = 0 in Ω.

Proof. Let B_{n−s} be a generic harmonic form in Ω, such that δB_{n−s} has interior nontangential trace in L_{s−1}^{p}(∂Ω). Since F_{s−1}^{∗} and dB_{n−s} are forms of C_{n−s+1}^{1}(Ω) with interior nontangential trace in L_{n−s+1}^{p}(∂Ω), from Theorem 2.7 in [13] it follows that they are in L_{n−s+1}^{p}(Ω). Then we can put

G_{s−1}(X) = \frac{(−1)^{ns−1}}{k_{n}} ∫_{Ω} ω_{s−1}(X, Y) ∧ (F_{s−1}^{∗} + (−1)^{(n−1)(s−1)}dB_{n−s})(Y). (2.5)

According to Theorem 77.VI in [9] this form is in C_{s−1}^{2}(Ω) ∩ D_{s−1}^{2,p}(Ω). Let then

U_{s−2} = δG_{s−1}^{∗} and V_{n−s} = (−1)^{n}δG_{s−1}. (2.6)

From (1.10), using Theorem 77.VI in [9], we obtain that

dU_{s−2} + δV_{n−s} = F_{s−1} + δB_{n−s} a.e. in Ω. (2.7)

Then in order to prove the thesis it is sufficient to show that there exists B_{n−s} such that Value Problems for Differential Forms on C^{1}-Domains 89 δV_{n−s} = δB_{n−s} a.e. in Ω. (2.8)

Now (2.3) and (2.6) imply that

V_{n−s}(X) = \frac{1}{k_{n}} ∫_{Ω} δ_{Y} ω_{n−s}(X, Y) ∧ (F_{s−1}^{∗} + (−1)^{(n−1)(s−1)}dB_{n−s})(Y)

= \frac{1}{k_{n}} ∫_{Ω} d_{Y} ω_{n−s}(X, Y) ∧ (F_{s−1} + δB_{n−s})(Y)

= \frac{1}{k_{n}} lim_{h→∞} ∫_{Ω_{h}} d_{Y} ω_{n−s}(X, Y) ∧ (F_{s−1} + δB_{n−s})(Y).

Hence, since F_{s−1} and δB_{n−s} are closed forms in Ω, from Theorem 11, p.121 in [10] we have

V_{n−s}(X) = \frac{1}{k_{n}} lim_{h→∞} ∫_{∂Ω_{h}} ω_{n−s}(X, Y) ∧ (F_{s−1} + δB_{n−s})(Y).

Moreover, because F_{s−1} and δB_{n−s} have interior nontangential trace in L_{s−1}^{p}(∂Ω), by (1.14) it follows that

V_{n−s}(X) = H_{n−s}(X) + \frac{1}{k_{n}} ∫_{∂Ω} ω_{n−s}(X, Y) ∧ Φ_{s−1}(Y), (2.9)

where

H_{n−s}(X) = \frac{1}{k_{n}} ∫_{∂Ω} ω_{n−s}(X, Y) ∧ F_{s−1}^{−}(Y) (2.10)

and Φ_{s−1} = (δB_{n−s})^{−}. (2.11) If we require that the form V_{n−s} satisfies (2.8), it results that

δB_{n−s}(X) = δH_{n−s}(X) + \frac{1}{k_{n}} ∫_{∂Ω} δ_{X}ω_{n−s}(X, Y) ∧ Φ_{s−1}(Y). (2.12)

From this, if we take into account Theorem 1 in [14], it follows that a.e. on ∂Ω

(δH_{n−s})^{−} = (\frac{1}{2}I − \frac{1}{k_{n}} T_{s−1})(Φ_{s−1}), (2.13)

where I is the identity operator on L_{s−1}^{p}(∂Ω) and T_{s−1} is a compact operator on the same space. Consider now the the homogeneous transposed equation of (2.13)

T̃(ψ_{n−s}) = \frac{k_{n}}{2} Ψ_{n−s} + T_{n−s}(Ψ_{n−s}) = 0. (2.14)

In order to show that (2.13) has solution in L_{s−1}^{p}(∂Ω), we prove that for all i = 1, …, R_{s−1}^{−} ∫_{∂Ω} Ψ_{n−s}^{i} ∧ (δH_{n−s})^{−} = 0, (2.15) where (Ψ_{n−s}^{i})_{1≤i≤R_{n−s}^{+}} is a base of Ker(T̃_{n−s}) verifying the conditions (see Theorem 7 in [14]):

∫_{\overline{τ}{l \atop n−s}} Ψ_{n−s}^{i} = 0 and ∫_{\overline{γ}{j \atop n−s}} Ψ_{n−s}^{i} = δ_{ij} (2.16)

for l = 1, …, R_{n−s}^{−} and for i, j = 1, …, R_{n−s}^{+}. Here (\overline{τ}_{n−s}^{l}, \overline{γ}_{n−s}^{j})_{{1≤l≤R{− \atop n−s} \atop 1≤j≤R{+ \atop n−s}}} is the dual

fundamental system of the fundamental system (τ_{s−1}^{j}, γ_{s−1}^{l})_{{1≤j≤R{− \atop s−1} \atop 1≤l≤R{+ \atop s−1}}} . We now observe

that using Theorem 1 in [14] we obtain

(δH_{n−s})^{−} = \frac{1}{2}F_{s−1}^{−} + \frac{1}{k_{n}} T_{s−1}(F_{s−1}^{−}). (2.17)

Hence from (2.14) we have

∫_{∂Ω} Ψ_{n−s}^{i} ∧ (δH_{n−s})^{−} = \frac{1}{2} ∫_{∂Ω} Ψ_{n−s}^{i} ∧ F_{s−1}^{−} + \frac{1}{k_{n}} ∫_{∂Ω} Ψ_{n−s}^{i} ∧ T_{s−1}(F_{s−1}^{−})

= \frac{1}{2} ∫_{∂Ω} Ψ_{n−s}^{i} ∧ F_{s−1}^{−} − \frac{1}{k_{n}} ∫_{∂Ω} T_{n−s}(Ψ_{n−s}^{i}) ∧ (F_{s−1}^{−}) = ∫_{∂Ω} Ψ_{n−s}^{i} ∧ F_{s−1}^{−}.

(2.18)

Thus, since Ψ_{n−s}^{i} and F_{s−1}^{−} are closed forms in W_{n−s}^{1,r}(∂Ω) (for all r > 1) and in W_{s−1}^{1,p}(∂Ω) respectively, by applying Theorem 1.7 in [14], using (2.4) and (2.16), it follows that

\begin{aligned}∫_{∂Ω} Ψ_{n−s}^{i} ∧ F_{s−1}^{−} & = (−1)^{(s−1)(n−s)} ∑_{j=1}^{R_{s−1}^{−}} ∫_{τ_{s−1}^{j}} F_{s−1}^{−} ∫_{\overline{γ}{j \atop n−s}} Ψ_{n−s}^{i} \\ & + ∑_{l=1}^{R_{s−1}^{+}} ∫_{γ_{s−1}^{l}} F_{s−1}^{−} ∫_{\overline{τ}{l \atop n−s}} Ψ_{n−s}^{i} = 0. \end{aligned}

Furthermore, using (2.18) we obtain that (2.15) is verified.

Let now Φ_{s−1} be a solution of (2.13) in L_{s−1}^{p}(∂Ω). Thus, from (2.17) and Theorem 2 in [14], since F_{s−1}^{−} ∈ W_{s−1}^{1,p}(∂Ω) is a closed form, it results that also δH_{n−s} is a closed form in W_{s−1}^{1,p}(∂Ω). Hence, arguing in a manner similar to the proof of the Theorem 1.4 in [15], we have that Φ_{s−1} is a closed form in W_{s−1}^{1,p}(∂Ω). Let finally

B_{n−s}(X) = H_{n−s} + \frac{1}{k_{n}} ∫_{∂Ω} ω_{n−s}(X, Y) ∧ Φ_{s−1}(Y). (2.19)

Thus using (11) in [14], we have that B_{n−s} is a harmonic form. Moreover in Ω it is

δB_{n−s}(X) = δH_{n−s}(X) + \frac{1}{k_{n}} ∫_{∂Ω} δ_{X}ω_{n−s}(X, Y) ∧ Φ_{s−1}(Y).

Furthermore by Theorem 1 in [14] δB_{n−s} has interior nontangential trace in L_{s−1}^{p}(∂Ω) and

(δB_{n−s})^{−} = (δH_{n−s})^{−} + \frac{1}{2}Φ_{s−1} + \frac{1}{k_{n}} T_{s−1}(Φ_{s−1}).

From this, being Φ_{s−1} a solution of (2.13), we deduce that (δB_{n−s})^{−} = Φ_{s−1} and i) is satisfied. ii) and iii) follow from Corollary of Theorem 1.1 and from (2.6) respectively.

□

By applying Lemma 2.1 of [15], from (2.6) we obtain the following Corollary. If F_{s−1} is a form that satisfies the hypotheses of Theorem 2.1, then the form

U_{s−2}(X) = \frac{(−1)^{n+s}}{k_{n}} ∫_{Ω} δ_{X}ω_{n−s+1}(X, Y) ∧ (F_{s−1}(Y) + δB_{n−s}(Y)) (2.20)

is a primitive of F_{s−1}, where B_{n−s} is the form defined in (2.19) and Φ_{s−1} is a solution of (2.13). Theorem 2.2. If F_{s−1} is a form that satisfies the hypotheses of Theorem 2.1, and dF_{s−1}^{∗} ∈ L_{n−s+2}^{p}(Ω), then there exists a primitive U_{s−2} of F_{s−1} in C_{s−2}^{2}(Ω)∩D_{s−2}^{1,p}(Ω).

Proof. Arguing in a manner similar to the proof of (2.9), from (2.20) it follows that

\begin{aligned}U_{s−2}(X) = & −\frac{1}{k_{n}} ∫_{Ω} ω_{s−2}(X, Y) ∧ dF_{s−1}^{∗}(Y) \\ & +\frac{1}{k_{n}} ∫_{∂Ω} ω_{s−2}(X, Y) ∧ ((F_{s−1}^{∗})^{−}(Y) + (−1)^{(n−1)(s−1)}(dB_{n−s})^{−}(Y)) \end{aligned}

and applying Theorem 1.2 of [15] and Theorem 77.VI of [9], we obtain our claim.

□

Corollary. If F_{s−1} satisfies the hypotheses of Theorem 2.1 and F_{s−1}^{∗} is a closed form in Ω, then there exists a primitive U_{s−2} of F_{s−1} in C_{s−2}^{∞}(Ω) ∩ N_{s−2}^{1,p}(Ω). Theorem 2.3. If F_{s−1} ∈ C_{s−1}^{0}(\overline{Ω}) is a closed form and satisfies (2.4), then for all p > n there exists a primitive U_{s−2} of F_{s−1} in C_{s−2}^{0}(\overline{Ω}) ∩ D_{s−2}^{1,p}(Ω).

Proof. Let p > n. Theorem 2.1 implies that there exists a primitive U_{s−2} of F_{s−1} in D_{s−2}^{1,p}(Ω). By Theorem 77.VI of [9] it follows that U_{s−2} ∈ C^{0,μ}(\overline{Ω}), where μ = 1 − \frac{n}{p}.

□

Our purpose is to obtain the following uniqueness Theorem:

Theorem 2.4. Let F_{s−1} and G_{n−s+3} be closed forms in C_{s−1}^{0}(\overline{Ω}) ∩ C_{s−1}^{1}(Ω) and in C_{n−s+3}^{0}(\overline{Ω}) ∩ C_{n−s+3}^{1}(Ω), respectively, such that

∫_{τ_{s−1}^{l}} F_{s−1} = 0 and ∫_{τ_{n−s+3}^{j}} G_{n−s+3} = 0, (2.21)

for all l = 1, …, R_{s−1}^{−} and for all j = 1, …, R_{n−s+3}^{−}. Let L_{s−2} ∈ C̃_{s−2}^{1}(∂Ω) be such that dL_{s−2} = F_{s−1} on ∂Ω (2.22)

and

∫_{γ_{s−2}^{i}} L_{s−2} = ∫_{Γ_{s−1}^{i}} F_{s−1} (2.23)

for i = 1, …, R_{s−2}^{+}, where γ_{s−2}^{i} = ∂Γ_{s−1}^{i}. Then we have

i) if R_{n−s+2}^{−} = 0, there exists a unique form U_{s−2} ∈ C_{s−2}^{1}(Ω) with an interior nontangential trace in L_{s−2}^{p}(∂Ω) such that dU_{s−2} = F_{s−1} in Ω, dU_{s−2}^{∗} = G_{n−s+3} in Ω (2.24)

and

U_{s−2}^{−} = L_{s−2} on ∂Ω; (2.25)

ii) if R_{n−s+2}^{−} > 0, then for any sequence (α_{i})_{1≤i≤R_{n−s+2}^{−}} in R there exists a unique

form U_{s−2} ∈ C_{s−2}^{1}(Ω) with interior nontangential trace in L_{s−2}^{p}(∂Ω) satisfying (2.24), (2.25), and

∫_{t_{n−s+2}^{i}} U_{s−2}^{∗} = α_{i}, (2.26)

for all i = 1, …, R_{n−s+2}^{−}.

Proof. By applying Theorems 2.1 and 2.3 to forms F_{s−1} and G_{n−s+3}, it results that there exist two forms \overline{U}_{s−2} ∈ C_{s−2}^{0}(\overline{Ω}) ∩ C_{s−2}^{1}(Ω) and \overline{V}_{n−s+2} ∈ C_{n−s+2}^{0}(\overline{Ω}) ∩ C_{n−s+2}^{1}(Ω), such that in Ω

d\overline{U}_{s−2} = F_{s−1}, d\overline{U}_{s−2}^{∗} = 0 (2.27)

and

d\overline{V}_{n−s+2} = G_{n−s+3}, d\overline{V} _{n−s+2}^{∗} = 0. (2.28)

Let K_{n−s+1} ∈ C_{n−s+1}^{2}(Ω) ∩ N_{n−s+1}^{1,p}(Ω) and M_{s−3} ∈ C_{s−3}^{2}(Ω) ∩ N_{s−3}^{1,p}(Ω) be two arbitrary harmonic forms in Ω. Thus if we let

\overline{W}_{s−2} = \overline{U}_{s−2} + (−1)^{s(n−1)}\overline{V} _{n−s+2}^{∗} (2.29)

and

U_{s−2} = \overline{W}_{s−2} + δK_{n−s+1} + dM_{s−3}, (2.30)

in Ω, it follows that \overline{W}_{s−2} ∈ C̃_{s}^{1}(\overline{Ω}), dU_{s−2} = F_{s−1} and dU_{s−2}^{∗} = G_{n−s+3} in Ω.

We want to show that there exist K_{n−s+1} and M_{s−3} such that U_{s−2} satisfies (2.25) and (2.26). In order to do this let

M_{s−3}(X) = ∑_{k=1}^{R_{n−s+2}^{−}} β_{k} ∫_{∂Ω} ω_{s−3}(X, Y) ∧ (Ψ_{n−s+2}^{k})(Y), (2.31)

where (β_{k})_{1≤k≤R_{n−s+2}^{−}} is an arbitrary sequence in R and (Ψ_{n−s+2}^{k})_{1≤k≤R_{n−s+2}^{−}} is a

base of Ker(T̃′_{n−s+2}) such that for h, i = 1, …, R_{n−s+2}^{−} and for l = 1, …, R_{n−s+2}^{+} we have (see: Theorem 8 in [14])

∫_{τ_{n−s+2}^{h}} Ψ_{n−s+2}^{i} = δ_{ih} and ∫_{γ_{n−s+2}^{l}} Ψ_{n−s+2}^{i} = 0. (2.32)

Since Ψ_{n−s+2}^{i} belongs to Ker(T̃′_{n−s+2}), Theorem 4 in [14] implies that Ψ_{n−s+2}^{i} is a closed form in W_{n−s+2}^{1,q}(∂Ω) for any q > 1. Using Theorem 72.V in [9] we have that M_{s−3} is a continuous form in R^{n}. Moreover, by Theorems 1 and 2 in [14], it follows that M_{s−3} is harmonic form in R^{n} \∂Ω with interior and exterior nontangential trace in L_{n−s+2}^{q}(∂Ω) for any q > 1, and (δM_{s−3})^{+} = 0. Furthermore, from Theorem 2.1 in [13] it follows that δM_{s−3} ∈ L_{n−s+2}^{q}(R^{n} \ Ω) for any q > 1. By applying Theorem 11, p. 121 in [10], we have

∫_{R^{n}\Ω} dM_{s−3} ∧ δM_{s−3} = ∫_{∂Ω} M_{s−3} ∧ (δM_{s−3})^{+} = 0.

We deduce that dM_{s−3} = 0 in R^{n} \ Ω. Thus, for all Φ_{n−s+1} ∈ C̃_{n−s+1}(∂Ω), it results that

∫_{∂Ω}(dM_{s−3})^{−} ∧ Φ_{n−s+3} = ∫_{∂Ω} M_{s−3} ∧ dΦ_{n−s+3} = ∫_{∂Ω}(dM_{s−3})^{+} ∧ Φ_{n−s+3} = 0.

Hence, by Lemma 3.1 in [13], we obtain (dM_{s−3})^{−} = 0. From this, if we require that the form U_{s−2}, defined in (2.30), satisfies the boundary condition (2.25), we have

(δK_{n−s+1})^{−} = L_{s−2} − \overline{W}_{s−2} on ∂Ω. (2.33)

Let 1 < p < ∞. From (2.22), (2.27), and (2.28) it follows that L_{s−2} −\overline{W}_{s−2} is a closed form of W_{s−2}^{1,p}(∂Ω), instead from Theorem 1.3 and (2.23) we have

∫_{γ_{s−2}^{i}}(L_{s−2} − \overline{W}_{s−2}) = 0

for any i = 1, …, R_{s−2}^{+}. Hence, using Theorem 1.5 in [15] we obtain that there exists a form K_{n−s+1} ∈ C_{n−s+1}^{2}(Ω)∩N_{n−s+1}^{1,p}(Ω), harmonic in Ω and satisfying (2.33). Now, since using (2.30) and (2.31)

\begin{aligned}U_{s−2}^{∗}(X) & = \overline{W} _{s−2}^{∗}(X) + (−1)^{ns−1}dK_{n−s+1}(X) \\ & + ∑_{k=1}^{R_{n−s+2}^{+}} β_{k} ∫_{∂Ω} δ_{X}ω_{s−3}(X, Y) ∧ (Ψ_{n−s+2}^{k})(Y), \end{aligned}

we prove the existence of a sequence (β_{k})_{1≤k≤R_{n−s+3}^{+}} in R such that U_{s−2}^{∗} satisfies (2.26). For this purpose, arguing in a manner similar to the proof of the Theorem (2.2) in [15], it is sufficient to show the existence of a sequence (β_{k})_{1≤k≤R_{n−s+3}^{+}} in R such that

α_{i} = ∫_{t_{n−s+2}^{i}}(\overline{W} _{s−2}^{∗} + (−1)^{(n−1)s}dK_{n−s+1}) + β_{i} ∫_{Y ∈c_{s−2}^{i}} δ_{Y} ∫_{X∈t_{n−s+2}^{i}} ω_{n−s+2} (2.34)

for all i = 1, …, R_{n−s+2}^{−}. This is easily seen to be true, because by (27) in [8], it is

∫_{Y ∈c_{s−2}^{i}} δ_{Y} ∫_{X∈t_{n−s+2}^{i}} ω_{n−s+2} ≠ 0.

Thus the existence of a solution of the problem is obtained. Finally to show the uniqueness of the solution it is sufficient to prove that the homogeneous problem associated to the above problem is the zero form. Thus, let U_{s−2} be a solution of the homogeneous problem. Hence U_{s−2} and U_{s−2}^{∗} are closed forms in Ω, with interior nontangential trace of class L_{s−2}^{p}(∂Ω), U_{s−2}^{−} = 0, and

∫_{t_{n−s+2}^{i}} U_{s−2}^{∗} = 0 (2.35)

for all i = 1, …, R_{n−s+2}^{−}. We now prove that for i = 1, …, R_{n−s+2}^{−}

∫_{τ_{n−s+2}^{i}} U_{s−2}^{∗} = 0. (2.36)

Let l = 1, …, R_{n−s+2}^{−}. If Φ_{s−3} ∈ C̃_{s−3}^{1}(∂Ω) is a closed form associated to τ_{n−s+2}^{i}. Using Definition 5.3 in [13] and (1.14) we have

∫_{τ_{n−s+2}^{l}} U_{s−2}^{∗} = ∫_{∂Ω} U_{s−2}^{∗} ∧ Φ_{s−3} = lim_{h} ∫_{∂Ω} Λ_{h}^{∗}(U_{s−2}^{∗}) ∧ Φ_{s−3}

= lim_{h} ∫_{∂Ω_{h}} U_{s−2}^{∗} ∧ Λ_{h}^{−1∗}(Φ_{s−3}) = lim_{h} ∫_{τ_{(n−s+2)h}^{l}} U_{s−2}^{∗}

(2.37)

where τ_{(n−s+2)h}^{l} = Λ_{h}(τ_{n−s+2}^{l}). Since τ_{(n−s+2)h}^{l} is a C^{1}-differentiable cycle on ∂Ω_{h}, \overline{Ω}_{h} ⊂ Ω and ([t_{n−s+2}^{i}])_{1≤i≤R_{n−s+2}^{−}} is a base of H_{n−s+2}(Ω), there exists a sequence

(α_{i}^{h})_{1≤i≤R_{n−s+2}^{−}} in R and a cycle c_{(n−s+3)h}^{l} C^{1}-differentiable in Ω, such that

τ_{(n−s+2)h}^{l} = ∑_{i=1}^{R_{n−s+2}^{−}} α_{i}^{h}t_{n−s+2}^{i} + ∂c_{(n−s+3)h}^{l}.

Thus, using (2.35), since U_{s−2}^{∗} is a closed form in Ω, we have that

∫_{τ_{(n−s+2)h}^{l}} U_{s−2}^{∗} = ∑_{i=1}^{R_{n−s+2}^{−}} α_{i}^{h} ∫_{t_{n−s+2}^{i}} U_{s−2}^{∗} + ∫_{∂c{l \atop (n−s+2)h}} U_{s−2}^{∗} = 0.

Moreover using (2.37), it follows (2.36). Hence U_{s−2}^{∗} verifies the hypotheses of Corollary of Theorem 2.2. Thus there exists a form

P_{n−s+1} ∈ C_{n−s+1}^{0}(\overline{Ω}) ∩ N_{n−s+1}^{1,p}(Ω)

such that

dP_{n−s+1} = U_{s−2}^{∗}. (2.38)

From this, since dU_{s−2} = 0 in Ω and U_{s−2}^{−} = 0 on ∂Ω, we obtain that P_{n−s+1} is a solution of the homogeneous Neumann problem. Thus dP_{n−s+1} = 0 in Ω (see Theorem 2.1 in [15]) and, hence, from (2.38), U_{s−2} = 0 in Ω.

□

Acknowledgement. This Work was partially supported by Universitá degli Studi Bari, Universitá degli Studi di Lecce and by M.I.U.R

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