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

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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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