Author Archives: Ricardo A. Sáenz

Problem set 12, PDE

Problem 1 for any . Hint: For each N, calculate the contour integral over the contour around the rectangle with vertices . Take . Problem 2 for every . Problem 3 satisfies the heat equation. , for any , also satisfies the heat equation. Problem 4 Let be bounded and continuous, with . Find an integral representation for […]

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Homework 12, Real Analysis

Due November 16 Problem 1 If , then, for any , . Problem 2 Let . Then, for all , . Problem 3 The function on is uniformly continuous but not Lipschitz. Problem 4 Consider the operator given by for any . Starting from the constant function , verify explicitly that the nth iteration of is […]

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Problem set 11, PDE

In all problems, is a bounded domain, and derivatives are understood as weak derivatives. Problem 1 The space is a Hilbert space with respect to the inner product Problem 2 The and norms are equivalent in the space . Problem 3 The restriction is bounded from into . (Hint: Extend the normal field from to […]

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Homework 11, Real Analysis

Due November 9 Problem 1 The fixed points of a continuous might not be interior. Problem 2 The Brouwer fixed point theorem is false for the open ball. Problem 3 Let be compact and convex, and continuous. Then f has a fixed point. Problem 4 Let  be compact and convex with boundary, , and given by […]

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Problem set 10, PDE

Problem 1 Identify with . The 2-dimensional zonal harmonics are then given by Verify explicitly the properties of zonal harmonics seen in class. The polynomial given by is equal to for , where is the Chebyshev polynomial given by . Problem 2 The Gegenbauer polynomials are given by the generating function The polynomials , for […]

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