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**Unformatted text preview: **Math 425, Spring 2011 Jerry L. Kazdan Problem Set 10 DUE: Thursday April 14 [ Late papers will be accepted until 1:00 PM Friday ]. 1. This problem is to help with a computation in class today (Thursday) finding a formula for a particular solution of the inhomogeneous heat equation using an idea due to Duhamel. a) If f ( x , r ) is a smooth function of the real variables t , r , let H ( t , r ) : = Z t f ( x , r ) dx . Compute H t ( t , r ) and H r ( t , r ) . b) Let K ( t ) : = H ( t , t ) . Use the chain rule to compute dK / dt . 2. a) Let A be a positive definite n × n real matrix, b ∈ R n , and consider the quadratic polyno- mial Q ( x ) : = 1 2 h x , Ax i-h b , x i . Show that Q is bounded below, that is, there is a constant m so that Q ( x ) ≥ m for all x ∈ R n . b) If x ∈ R n minimizes Q , show that Ax = b . [Moral: One way to solve Ax = b is to minimize Q .] c) Let Ω ∈ R n be a bounded region with smooth boundary and F ( x ) a bounded continuous function. Also, let S...

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