Distributions, Partial Differential Equations, and Harmonic by Dorina Mitrea
By Dorina Mitrea
The thought of distributions constitutes a necessary instrument within the learn of partial differential equations. This textbook would supply, in a concise, principally self-contained shape, a swift advent to the idea of distributions and its purposes to partial differential equations, together with computing primary ideas for the main easy differential operators: the Laplace, warmth, wave, Lame and Schrodinger operators.
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Extra info for Distributions, Partial Differential Equations, and Harmonic Analysis (Universitext)
Sample text
3) we see that sup |∂ α ψj (x)| < x∈K |α|≤j 1 j ∀ j ∈ N. 4) Now let α ∈ Nn0 be arbitrary. 5) D(Ω) thus ψj −−−→ 0. Since u is a distribution in Ω, the latter implies lim u, ψj = j→∞ j→∞ 0, contradicting the fact that u, ψj = 1 for each j ∈ N. This completes the proof of the proposition. 1. 5. Recall that for each compact set K ⊂ Ω we denote by DK (Ω) the vector space of functions in C ∞ (Ω) with support contained in K endowed with the topology inherited from E(Ω). 4 may be rephrased as saying that a linear map u : D(Ω) → C is a distribution in Ω if and only if u DK (Ω) is continuous for each compact set K ⊂ Ω.
E. D (Ω) is stable under the action of ∂ α for any α ∈ N0 ). (2) If u ∈ D (Ω) and k, ∈ {1, . . , n} then ∂k ∂ u = ∂ ∂k u in D (Ω). D (Ω) D (Ω) j→∞ j→∞ (3) If uj −−−−→ u and α ∈ Nn0 , then ∂ α uj −−−−→ ∂ α u. (4) For any u ∈ D (Ω) and any a ∈ C ∞ (Ω) we have ∂j (au) = (∂j a)u + a(∂j u) in D (Ω). Proof. The first property follows immediately from the definition of distributional derivatives. To prove the remaining properties, fix an arbitrary function ϕ ∈ C0∞ (Ω). 1) repeatedly and the symmetry of mixed partial derivatives for smooth functions (Schwarz’s theorem), we have ∂k ∂ u, ϕ = − ∂ u, ∂k ϕ = u, ∂ ∂k ϕ = u, ∂k ∂ ϕ = − ∂k u, ∂ ϕ = ∂ ∂k u, ϕ , ∀ k, ∈ {1, .
2. Let u : D(Ω) → C be a linear map. Then u is a distribution on Ω D(Ω) if and only if for every sequence {ϕj }j∈N ⊂ C0∞ (Ω) with ϕj −−−→ ϕ for some j→∞ ϕ ∈ C0∞ (Ω), we have lim u, ϕj = u, ϕ (where the latter limit is considered in C). 3. In general, if X, Y are topological vector spaces and Λ : X → Y is a linear map, then Λ is sequentially continuous on X if and only if Λ is sequentially continuous at the zero vector 0 ∈ X. 2, D. 1007/978-1-4614-8208-6 2, © Springer Science+Business Media New York 2013 17 CHAPTER 2.