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220-HW2 - MATH 220 PROBLEM SET 2 DUE THURSDAY OCTOBER 8...

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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 ( φ ) = 0 for all φ ∈ C c ( R ). You need to show that there is a constant c such that u ( ψ ) = integraltext cψdx for all ψ ∈ C c ( R ). To do so, consider when ψ ∈ C c ( R ) is of the form ψ = φ , φ ∈ C c ( R ), paying particular attention to the issue of compact supports. Then write an arbitrary ψ as a linear combination of a fixed φ 0 ∈ C c ( R ) and the derivative φ of some φ ∈ 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. (2) Suppose now that f is a distribution, e.g. f = δ 0 . Can you make sense of the formula u ( x,y ) = f ( x - ay )? That is, find a procedure giving a distribution u
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