An elementary treatise on partial differential equations. by George Biddell Airy, K.C.B., M.A., LL.D., D.C.L.

By George Biddell Airy, K.C.B., M.A., LL.D., D.C.L.

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Extra info for An elementary treatise on partial differential equations. Designed for the use of students in the university (2nd edition, 1873)

Sample text

Thus we can identify H0m (G) with a space of distributions on G, and those distributions are characterized as follows. 42 CHAPTER II. 2 H0m (G) is (identified with) the space of distributions on G which are the linear span of the set {∂ α f : |α| ≤ m , f ∈ L2 (G)} . Proof : If f ∈ L2 (G) and |α| ≤ m, then |∂ α f (ϕ)| ≤ f L2 (G) ϕ H0m (G) , ϕ ∈ C0∞ (G) , so ∂ α f has a (unique) continuous extension to H0m (G). Conversely, if T ∈ H0m (G) , there is an h ∈ H0m (G) such that ϕ ∈ C0∞ (G) . T (ϕ) = (h, ϕ)H m (G) , But this implies T = |α|≤m (−1)|α| ∂ α (∂ α h) and, hence, the desired result, since each ∂ α h ∈ L2 (G).

This defines a function H m (G) → H0m (G) × {H m (G ∩ Gj ) : 1 ≤ j ≤ N }, where u → (β0 u, β1 u, . . , βN u). This function is clearly linear, and from βj = 1 it follows that it is an injection. Also, since each βj u belongs to H m (G ∩ Gj ) with support in Gj for each 1 ≤ j ≤ N , it follows that the composite function (βj u) ◦ ϕj belongs to H m (Q+ ) with support in Q. Thus, we have defined a linear injection Λ : H m (G) −→ H0m (G) × [H m (Q+ )]N , u −→ (β0 u, (β1 u) ◦ ϕ1 , . . , (βN u) ◦ ϕN ) .

Dxn , a distribution whose value is determined by the restriction of ϕ to {0}× Rn−1 , ∞ ∂2 ∂1 H(ϕ) = 0 ∞ ... 0 ϕ(0, ¯ 0, x3 , . . , xn ) dx3 . . ,1) H(ϕ) = ϕ(0) = δ(ϕ) , where δ is the Dirac functional which evaluates at the origin. (d) Let S be an (n − 1)-dimensional C 1 manifold (cf. 3) in Rn and suppose f ∈ C ∞ (Rn ∼ S) with f having at each point of S a limit from each side of S. For each j, 1 ≤ j ≤ n, we denote by σj (f ) the jump in f at 1. DISTRIBUTIONS 39 the surface S in the direction of increasing xj .

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