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**Unformatted text preview: **MATH 220: PROBLEM SET 2 DUE THURSDAY, OCTOBER 8, 2009 Problem 1. Show that the only solution u D ( R ) of u = 0 is u = c , c a constant function. Hint: u = 0 means that u ( ) = 0 for all C c ( R ). You need to show that there is a constant c such that u ( ) = integraltext c dx for all C c ( R ). To do so, consider when C c ( R ) is of the form = , C c ( R ), paying particular attention to the issue of compact supports. Then write an arbitrary as a linear combination of a fixed C c ( R ) and the derivative of some C c ( R ). Problem 2. Consider the PDE au x + u y = 0 . We already know that the C 1 solutions are of the form u ( x, y ) = f ( x- ay ), f C 1 ( R ). (1) Show that if f is merely piecewise continuous (or if you wish locally inte- grable), then the so defined u still solves the PDE in the sense of distribu- tions....

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