Partial Differential Equations

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30 1 The Fundamental Partial Differential Equations ( ∫ ) (r n−1 U r (x; r, t)) r = 1 ∂ nα(n) u tt (y, t) dy ∂r B(x;r) ( ∫ = 1 ∂ r ∫ ) nα(n) u tt (y, t) dS ∂B(x;s) (y)ds ∂r 0 ∂B(x;s) ∫ u tt (y, t) dS ∂B(x;r) (y) Hence, we have = 1 nα(n) = rn−1 vol ∂B(x;r) = ∂2 ∂t 2 ( ∂B(x;r) ∫ ∂B(x;r) r n−1 vol ∂B(x;r) = r n−1 U tt (r). ∫ u tt (y, t) dS ∂B(x;r) (y) ∂B(x;r) u(y, t) dS ∂B(x;r) (y) U tt = 1 ( (n − 1)r n−2 r n−1 U r + r n−1 ) U rr = Urr + n − 1 U r , r and this proves the lemma since the initial conditions for t = 0 immediately follow from the definitions. The partial differential equation in (1.24) is also called the Euler–Poisson–Darboux equation. We aim for a transformation which eliminates the U r -term in the Euler-Poisson-Darboux equation.For this, the following lemma will be useful. Lemma 1.4.7. Let ϕ ∈ C k+1 (R) be a real valued function. Then, we have ( ) d ( (i) 2 1 ) d k−1 ( dr 2 r dr r 2k−1 ϕ(r) ) = ( ) 1 d k 2k dϕ r dr (r dr ). (r) (ii) ( ) 1 d k−1 ( r dr r 2k−1 ϕ(r) ) = ∑ k−1 j=0 βk j rj+1 dj ϕ dr (r), where the constants β k j j are independent of ϕ for j = 0, . . . , k − 1. (iii) β0 k = 1 · 3 · 5 · · · (2k − 1). Proof. Exercise. (Hint: This is an elementary induction.) We take up the notation of Lemma 1.4.6, and we assume that u, g and h are of class C k . Set Then, Ũ(r, 0) = ˜G(r) and Ũt(r, 0) = ˜H(r). ( ) k−1 1 ∂ ( Ũ(r, t) := r 2k−1 U(x; r, t) ) , r ∂r ( ) k−1 1 ∂ ( ˜G(r) := r 2k−1 G(x; r) ) , r ∂r ( ) k−1 1 ∂ ( ˜H(r) := r 2k−1 H(x; r) ) . r ∂r Lemma 1.4.8. For n = 2k + 1, we have ⎧ Ũ tt − ⎪⎨ Ũrr = 0 on ]0, ∞[ × ]0, ∞[, Ũ = ˜G on ]0, ∞[ ×{0}, Ũ t = ⎪⎩ ˜H on ]0, ∞[ ×{0}, Ũ = 0 on {0}× ]0, ∞[. Proof. Idea: Show by induction that for fixed j ∈ N 0, x ∈ R n , and t > 0 the terms r j ∂j U (x; r, t) and ∂r j (x; r, t) remain bounded for r ∈]0, 1[. Then the Euler–Poisson–Darboux equation yields the claim. r j ∂j U tt ∂r j )
1.4 The Wave Equation 31 The initial data for t = 0 have been verified already. To compute the boundary values for r = 0, we show by induction and by the Euler–Poisson–Darboux equation that (∣ ) ∣∣r j ∂ j U ∣ ∂r (x; r, t) ∣ + ∣r j ∂j U tt j ∂r (x; r, t) ∣ ≤ C j j,x,t (1.26) sup 0
Page 1 and 2: Partial Differential Equations Summ
Page 3: Preface Let U ⊆ R n be an open se
Page 6 and 7: 4 1 The Fundamental Partial Differe
Page 12 and 13: 10 1 The Fundamental Partial Differ
Page 43 and 44: 2 First Order Equations In this cha
Page 45 and 46: 2.1 Complete Integrals and Envelopi
Page 47 and 48: 2.2 The Method of Characteristics 4
Page 55 and 56: 2.3 Local Existence of Solutions 53
Page 57 and 58: D p F (p 0 , z 0 , x 0 ) · ν ∂U
Page 59 and 60: 2.4 Hamilton-Jacobi Equations 57 By
Page 61 and 62: and Together, we get and Finally, o
Page 63 and 64: 2.4 Hamilton-Jacobi Equations 61 Pr
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80 3 Various Solution Techniques Pr
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82 3 Various Solution Techniques Wi
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84 3 Various Solution Techniques
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86 4 Linear Differential Operators
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A Function Spaces A.1 L p -spaces a
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A.1 L p -spaces 95 By this proposit
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A.1 L p -spaces 97 Proposition A.1.
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A.1 L p -spaces 99 (ii) For p < ∞
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A.1 L p -spaces 101 Lemma A.1.16 (Y
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A.2 Topological Vector Spaces 103 R
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1. Case: p αi (x) = 0 i = 1, . . .
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A.3 Spaces of Differentiable Functi
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A.4 The Fourier Transform 113 (iii)
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A.4 The Fourier Transform 115 ( Bec
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B Distributions B.1 Tempered Distri
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B.1 Tempered Distributions 119 Proo
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B.1 Tempered Distributions 121 ξ 0
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Proof. (i) The implication “⇐
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B.2 Distributions 125 Proof. Let ϕ
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B.2 Distributions 127 where ψ ∈
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B.2 Distributions 129 and the chara
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B.3 Convergence of Distributions 13
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134 C Sobolev spaces For s ∈ R, w
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136 C Sobolev spaces Theorem C.1.8.
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138 C Sobolev spaces Proof. By Prop
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140 C Sobolev spaces Theorem C.1.16
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142 C Sobolev spaces is bounded for
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144 C Sobolev spaces Fig. C.1. Theo
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146 C Sobolev spaces V m ~ U U V 0
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148 C Sobolev spaces Theorem C.3.4.
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150 C Sobolev spaces [∆ α h, ϕ]
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152 Index of the Schwartz space, 11
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154 Index singular support, 124 smo
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Partial Differential Equations

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