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13. Abstract Wave Equation In the next section we consider (13.1) Before working with this explicit equation we will work out an abstract Hilbert space theory rst. Theorem 13.1 (Existence). Suppose A : H H is a self-ad
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11. Introduction to the Spectral Theorem The following spectral theorem is a minor variant of the usual spectral theorem for matrices. This reformulation has the virtue of carrying over to general (unbounded) self adjo
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9. Poisson and Laplaces Equation For the majority of this section we will assume Rn is a compact manifold with C 2 boundary. Let us record a few consequences of the divergence theorem R in Proposition 8.28 in this context. If u, v C
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8. Surfaces, Surface Integrals and Integration by Parts Denition 8.1. A subset M Rn is a n 1 dimensional C k -Hypersurface if for all x0 M there exists > 0 an open set 0 D Rn and a C k -dieomorphism : D B (x0 , ) such that (D cfw_xn =
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7. Test Functions and Partitions of Unity 7.1. Convolution and Youngs Inequalities. Letting x denote the delta function at x, we wish to dene a product () on functions on Rn such that x y = x+y . Now formally any functi
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12. Heat Equation The heat equation for a function u : R+ Rn C is the partial dierential equation 1 (12.1) t u = 0 with u(0, x) = f (x), 2
and hence that u(t, ) = et| /2 f ( ). Inverting the Fourier transform then sho
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20. Fourier Transform The underlying space in this section is Rn with Lebesgue measure. The Fourier inversion formula is going to state that n Z Z 1 (20.1) f (x) = deix dyf (y )eiy . 2 Rn Rn If we let = 2, this may b
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21. Constant Coefficient partial differential equations P Suppose that p( ) = |k a with a C and 1 (21.1) L = p(Dx ) := |N a Dx = |N a . x i Then for f S c Lf ( ) = p( )f ( ),
that is to say the Fourier transform take
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35. Compact and Fredholm Operators and the Spectral Theorem In this section H and B will be Hilbert spaces. Typically H and B will be separable, but we will not assume this until it is needed later. 35.1. Compact Ope
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14. Wave Equation on Rn (Ref Courant & Hilbert Vol II, Chap VI 12.) We now consider the wave equation (14.1) According to Section 13, the solution (in the L2 sense) is given by p sin(t 4) g. (14.2) u(t, ) = (cos(t 4)f