pde7 - P DE LECTURE NOTES M ATH 237A-B 83 7 Test Functions...

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PDE LECTURE NOTES, MATH 237A-B 83 7. Test Functions and Partitions of Unity 7.1. Convolution and Young’s Inequalities. Letting δ x denote the “delta— function” at x, we wish to de f ne a product ( ) on functions on R n such that δ x δ y = δ x + y . Now formally any function f on R n is of the form f = Z R n f ( x ) δ x dx so we should have f g = Z R n × R n f ( x ) g ( y ) δ x δ y dxdy = Z R n × R n f ( x ) g ( y ) δ x + y dxdy = Z R n × R n f ( x y ) g ( y ) δ x dxdy = Z R n ·Z R n f ( x y ) g ( y ) dy ¸ δ x dx which suggests we make the following de f nition. De f nition 7.1. Let f,g : R n C be measurable functions. We de f ne f g ( x )= Z R n f ( x y ) g ( y ) dy whenever the integral is de f ned, i.e. either f ( x · ) g ( · ) L 1 ( R n ,m ) or f ( x · ) g ( · ) 0 . Notice that the condition that f ( x · ) g ( · ) L 1 ( R n ,m ) is equivalent to writing | f | | g | ( x ) < . Notation 7.2. Given a multi-index α Z n + , let | α | = α 1 + ··· + α n , x α := n Y j =1 x α j j , and α x = μ ∂x α := n Y j =1 μ ∂x j α j . Remark 7.3 (The Signi f cance of Convolution) . Suppose that L = P | α | k a α α is a constant coe cient di f erential operator and suppose that we can solve (uniquely) the equation Lu = g in the form u ( x )= Kg ( x ):= Z R n k ( x, y ) g ( y ) dy where k ( x, y ) is an “integral kernel.” (This is a natural sort of assumption since, in view of the fundamental theorem of calculus, integration is the inverse operation to di f erentiation.) Since τ z L = z for all z R n , (this is another way to characterize constant coe cient di f erential operators) and L 1 = K we should have τ z K = z . Writing out this equation then says Z R n k ( x z,y ) g ( y ) dy =( Kg )( x z )= τ z Kg ( x )=( z g )( x ) = Z R n k ( x, y ) g ( y z ) dy = Z R n k ( x, y + z ) g ( y ) dy. Since g is arbitrary we conclude that k ( x z,y )= k ( x, y + z ) . Taking y =0 then gives k ( x, z )= k ( x z, 0) =: ρ ( x z ) .
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84 BRUCE K. DRIVER We thus f nd that Kg = ρ g. Hence we expect the convolution operation to appear naturally when solving constant coe cient partial di f erential equations. More about this point later. The following proposition is an easy consequence of Minkowski’s inequality for integrals. Proposition 7.4. Suppose q [1 , ] ,f L 1 and g L q , then f g ( x ) exists for almost every x, f g L q and k f g k p k f k 1 k g k p . For z R n and f : R n C , let τ z f : R n C be de f ned by τ z f ( x )= f ( x z ) .
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This note was uploaded on 10/11/2010 for the course MATH 11 taught by Professor Smith during the Three '10 term at ADFA.

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pde7 - P DE LECTURE NOTES M ATH 237A-B 83 7 Test Functions...

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