Some Properties of Solutions to a Class of Dirichlet Boundary Value Problems

and Applied Analysis 3 We denote by ∧l ∧l R the space of l-covectors in R and the direct sum ∧ R n ⊕ l 0 ∧l R 2.1 is a graded algebra with respect to the wedge product ∧. We will make use of the exterior derivative: d : C∞ ( Ω,∧l ) −→ C∞ ( Ω,∧l 1 ) 2.2 and its formal adjoint operator d∗ −1 nl 1 ∗ d∗ : C∞ ( Ω,∧l 1 ) −→ C∞ ( Ω,∧l ) , 2.3 known as the Hodge codifferential, where the symbol ∗ denotes the Hodge star duality operator. Note that each of the operators d and d∗ applied twice gives zero. Let C∞ Ω,∧l be the class of infinitely differentiable l-forms on Ω ⊂ R. Since Ω is a smooth domain, near each boundary point one can introduce a local coordinate system x1, x2, . . . , xn such that xn 0 on ∂Ω and such that the xn-curve is orthogonal to ∂Ω. Near this boundary point, every differential form ω ∈ C∞ Ω,∧l can be decomposed as ω x ωT x ωN x , where ωT x ∑ 1≤i1<···<il<n ωi1,...,il x dxi1 ∧ · · · ∧ dxil , ωN x ∑ 1≤i1<···<il n ωi1,...,il x dxi1 ∧ · · · ∧ dxil 2.4 are called the tangential and the normal parts of ω, respectively. Now, the duality between d and d∗ is expressed by the integration by parts formula ∫ Ω 〈du, v〉 ∫ Ω 〈u, d∗v〉, 2.5 for all u ∈ C∞ Ω,∧l and v ∈ C∞ Ω,∧l 1 , provided uT 0 or vN 0. The symbol 〈·, ·〉 denotes the inner product, that is, let α ∑ I αI x dxI and β ∑ I βI x dxI , then 〈α, β〉 ∑ I αI x βI x . Due to 2.5 , extended definitions for d and d∗ can be introduced as the introduction of weak derivatives. 4 Abstract and Applied Analysis Definition 2.1 see 2 . Suppose that ω ∈ Lloc Ω,∧l and v ∈ Lloc Ω,∧l 1 . If ∫ Ω 〈 ω, d∗η 〉 ∫ Ω 〈 v, η 〉 2.6 for every test form η ∈ C∞ 0 Ω,∧l 1 , one says that ω has generalized exterior derivative v and write v d̃ω. The notion of the generalized exterior coderivative d̃∗ can be defined analogously. Definition 2.2 see 2 . Suppose that ω ∈ Lloc Ω,∧l and v ∈ Lloc Ω,∧l−1 . If ∫ Ω 〈 ω, dη 〉 ∫ Ω 〈 v, η 〉 2.7 for every test form η ∈ C∞ 0 Ω,∧l−1 , one says that ω has generalized exterior coderivative v and write v d̃∗ω. Remark 2.3. i Observe that generalized exterior derivatives have many properties similar to those of weak derivatives. For example, i1 if it exists, it is unique; i2 ifω is differentiable in the conventional sense, then its generalized exterior derivative d̃ω is identical to its the classical exterior differential dω. Analogous results hold for generalized exterior coderivative. ii If the generalized exterior derivative of ω, d̃ω, exists, then d̃ω also has its generalized exterior derivative d̃ d̃ω . Moreover, d̃ d̃ω 0. In fact, according to Definition 2.1, it holds ∫ Ω 〈 ω, d∗φ 〉 ∫ Ω 〈 d̃ω, φ 〉 2.8 for every test form φ ∈ C∞ 0 Ω,∧l 1 . Thus, for every η ∈ C∞ 0 Ω,∧l 2 , we have d∗η ∈ C∞ 0 Ω,∧l 1 and by taking φ d∗η in the above integral equality implies ∫ Ω 〈 d̃ω, d∗η 〉 ∫ Ω 〈 ω, d∗d∗φ 〉 ∫