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

**Read or Download An elementary treatise on partial differential equations. Designed for the use of students in the university (2nd edition, 1873) PDF**

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

**Example text**

122) to obtain the result. 122), noting that the other terms have a similar structure and can be treated in the same way. 123) where we have used the Cauchy–Schwarz inequality. 5 Sobolev Spaces 45 In Sect. 3), we saw that in a Banach space, every sequence converging for a given norm is necessarily bounded for that norm. In other words, there exists M > 0 independent of n such that for all n ∈ N, vn L 2 (Ω) ≤ M. 123), that ∂u n ∂u vn dΩ = v dΩ. 124) lim n→+∞ Ω ∂ x i ∂ Ω xi Notes. 122) can be treated in the same way.

157) over i, we obtain n i =1 ∂v ∂ xi n 2 H 1 (Ω) = i =1 ∂v ∂ xi n 2 L 2 (Ω) ∂ 2v ∂ xi ∂ x j + i, j = 1 2 L 2 (Ω) ≤ v 2 . 6, the trace theorem in H 2 (Ω). 25 Let Ω be a regular bounded open subset of Rn . 159) where n is the outward-pointing unit normal vector on the boundary of the open set Ω, and ∂u/∂n is the projection of the gradient vector in the direction of the normal n. 5 for H 1 (Ω), we use the fact that C0∞ (Ω) is dense in H 2 (Ω), together with the continuity of the trace map γ1 . Note.

One possible answer is provided by the Cauchy–Schwarz inequality, because under the conditions of validity of this inequality, the integral of f ⊗ ϕ is bounded in the following way: f ⊗ (x)ϕ(x) dx ≤ K f ⊗2 (x) dx K 1/2 1/2 ϕ 2 (x)dx . 41) K Hence, if f ⊗ and ϕ are two square-integrable functions on K , the integral of f ⊗ ϕ over K will converge. By symmetry, if f and ϕ ⊗ are square-integrable functions on K , then the integral of f ϕ ⊗ over K will also converge. 10 If f is a real-valued square-integrable function on K , we shall say that f is weakly differentiable if there is a real-valued square-integrable function g on K such that f (x)ϕ ⊗ (x) dx = − K g(x)ϕ(x) dx, ∀ “sufficiently” well behaved ϕ.

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