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Theory of distributions by M.A. Al-Gwaiz

By M.A. Al-Gwaiz

A textbook for a graduate path within the thought of distributions and similar subject matters, for college kids of utilized arithmetic or theoretical physics. Introduces the speculation, explicates mathematical constructions and the Hilbert-space points, and offers functions to general boundary difficulties. Annotation

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1. ˜ ˆ ± )−1 on the elements β(p) ˆ ± and (W = We now define pairs of operators W ˆ , p) and F (p) = fˆ(· , p), p ∈ Cκ , of H1/2,k,κ (∂S) by β(· ˜ ˆ , p), ˆ ± β)(p) (W = (Wp± β)(· ˆ ± )−1 F (p) = (W ± )−1 fˆ (· , p), (W p 50 4 Boundary Integral Equations ˆ β˜ by and the double-layer potential W ˜ ˆ ˆ β)(x, (W p) = (Wp β)(x, p), x ∈ S ± , p ∈ Cκ . Returning to the spaces of originals, we define operators W ± and (W ± )−1 and the double-layer potential W β by setting ˆ ± Lβ, W ± β = L−1 W ˆ ± )−1 Lf, (W ± )−1 f = L−1 (W ˜ ˆ β)(X), (W β)(X) = (L−1 W X ∈ G+ ∪ G− .

At this stage, we can define operators Tˆ ± and (Tˆ ± )−1 on the elements F (p) = fˆ(· , p) and G(p) = gˆ(· , p) of H1/2,k,κ (∂S) and H−1/2,k,κ (∂S), k ∈ R, respectively, by setting (Tˆ ± F )(p) = (Tp± fˆ)(· , p), (Tˆ ± )−1 G (p) = (Tp± )−1 gˆ (· , p). Finally, we return to the spaces of originals and define the Poincar´e– Steklov operators T ± and (T ± )−1 on the elements f ∈ H1/2,k,κ (Γ) and g ∈ H−1/2,k,κ (Γ), k ∈ R, by means of the equalities T ± f = L−1 Tˆ ± Lf, (T ± )−1 g = L−1 (Tˆ ± )−1 Lg.

40) 32 2 Problems with Dirichlet Boundary Conditions ¯ + ) be such where Fν (p) are the Laplace transforms of fν (t). Let v ∈ C0∞ (G that γ + v = 0. 40) and find that the Laplace transform of ∂t v at the point p∗ is p∗ V (p∗ ) − V0 , and that ∞ a+ (u, v) − (B 1/2 ∂t u, B 1/2 ∂t v)0;S + − (q, v)0;S + dt 0 ∞ −1 a+ U (p), V (p∗ ) − B 1/2 pU (p), B 1/2 (p∗ V (p∗ ) − V0 ) = (2π) 0;S + −∞ − Q(p), V (p∗ ) 0;S + dτ. 4), it follows that a+ U (p), W + p2 B 1/2 U (p), B 1/2 W 0;S + = Q(p), W ˚1 (S + ). 42), we obtain a+ U (p), V (p∗ ) − B 1/2 pU (p), B 1/2 (p∗ V (p∗ ) − V0 ) 0;S + − Q(p), V (p∗ ) 0;S + = a+ U (p), (p∗ )−1 V0 − Q(p), (p∗ )−1 V0 = p−1 Q(p), V0 0;S + 0;S + − p−1 a+ U (p), V0 .

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