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6. Elliptic Ordinary Differential Operators Let o Rn be a bounded connected open region. A function u C 2 () is said to satisfy Laplaces equation if More generally if f C () is given we say u solves the Poisson equation if In order to
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5. A very short introduction to generalized functions Let U be an open subset of Rn and (5.1) denote the set of smooth functions on U with compact support in U. Denition 5.1. A sequence cfw_k D(U ) converges to D(U ), i there is a k=1
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4. Cauchy Kovalevskaya Theorem As a warm up we will start with the corresponding result for ordinary dierential equations. Theorem 4.1 (ODE Version of Cauchy Kovalevskaya, I.). Suppose a > 0 and f : (a, a) R is real analytic near 0 and
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3. Fully nonlinear first order PDE In this section let U o Rn be an open subset of Rn and (x, z, p) U Rn R F (x, z, p) R be a C 2 function. Actually to simplify notation let us suppose U =Rn . We are now looking for a solution u : Rn R
PDE LECTURE NOTES, MATH 237A-B
BRUCE K. DRIVER Abstract. These are lecture notes from Math 237A-B. See C:\driverdat\Bruce\CLASSFIL \257AF94 \course.tex for notes on contraction semi-groups. Need to add examples of using the Hille Yoshida theorem in PDE. S