The mixed problem for laplaces equation in the plane by Griffith C. E. R. C. Miles. R. N. Haskell. EVANS

By Griffith C. E. R. C. Miles. R. N. Haskell. EVANS

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Let S(x) denote the unit m-vector associated with the oriented tangent plane to S at x. Then for any differential m-form ϕ, define S(x), ϕ d H m . S(ϕ) = S Furthermore, we will allow S to carry a positive integer multiplicity μ(x), with m < ∞, and define S μ(x) d H S(x), ϕ μ(x) d H m . S(ϕ) = S Finally, we will require that S have compact support. Such currents are called rectifiable currents. Normal and Rectifiable Currents 41 Definitions for currents are by duality with forms. The boundary of an m-dimensional current T ∈ Dm is the (m − 1)-dimensional current ∂T ∈ Dm−1 defined by ∂T(ϕ) = T(dϕ).

Define a function g : Rm → Rn+m x → (f(x), εx). Then Jm (g) ≤ ε(Lip f + ε)m−1 . Now by Case 1, H m (f(A)) ≤ H m (g(A)) = Jm g A ≤ ε(Lip f + ε)m−1 L m (A). Therefore, the right-hand side also must vanish. 7(1) by approximating u by simple functions. The following useful formula relates integrals of a function f over a set A to the areas of the level sets A ∩ f −1 {y} of the function. 11] Consider a Lipschitz functionf : Rm → Rn with m > n. If A is an L m measurable set, then Jn f(x) dL m x = A Rn H m−n (A ∩ f −1 {y}) dL n y.

The oriented m-planes through the origin in Rn are in one-to-one correspondence with the unit, simple m-vectors in m Rn . 3]). Let Rn∗ denote the space of covectors dual to Rn , with dual orthonormal basis ∗ e1 , . . , en∗ . We remark that dxj is a common alternate notation for e∗j . The dual Normal and Rectifiable Currents 39 space to m Rn is the space m Rn ≡ m (Rn∗ ) of linear combinations of wedge products of covectors, called m-covectors. The dual basis is {e∗i1 ... im : i1 < · · · < im }. A differential m-form ϕ on Rn is an m-covectorfield; that is, a map ϕ : Rn → m Rn .

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