By Zhuoqun Wu

This ebook presents an advent to elliptic and parabolic equations. whereas there are many monographs focusing individually on every one form of equations, there are only a few books treating those varieties of equations together. This publication offers the similar simple theories and strategies to permit readers to understand the commonalities among those types of equations in addition to distinction the similarities and modifications among them.

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**Example text**

N ) ; Hi) The difference of the product can be expressed as AUf(x)g(x)) = Ay(x)n9(x) + f(x)Aig(x), where T£ is the translation operator in the direction Xi, defined by Tlu{x) = u(x + hei). We leave the proof to the reader. The following important properties on the difference of functions in Sobolev spaces are needed for our argument. 2 Let fi C R™ be a domain and i = 1, • • • , n. ' CC 0 , then for sufficiently small \h\ > 0, A ' ^ u e i 2 ^ ) and \\&huh*(n') < \\Diu\\L2{n). ii) IfuG L 2 (Q) and for fi' CC fi and sufficiently small \h\ > 0, ||AU||L2(n0 < K with constant K independent of h, then DiU €• L 2 (Q') and II Au||i*(n') < K.

Proof. Uniqueness is obvious. 3, there exists um G C°°(Q ), such that lim ||u m -ti|| i f i(Q+) = 0 . v m—>oo ' Let 0 < S < 1. Choose a smooth function r](xn) G C J [0,1], such that v(xn) = 1 for 0 < xn < 6 and r)(xn) = 0 for xn less than or equal to 1 but close to 1. Clearly lim \\r}Um-r)u\\Hi{Q+) =0. 2) imply that, for arbitrary fixed small e G [0,5], {um(-, s)} is a Cauchy sequence in L2(T); we denote its limit function by v(-,e). e. x' S T. 3) <\\rjum -r]u\\HHQ+). 4) In addition, clearly / |u m (a; / ,e) - u m (x',0)| 2 dx' < e 1 Jr Jo du„ 2 dx.

2. 7. 10. 8. Let u G W 1 , p ((0,1)) with p > 1. Prove \u(x)-u(y)\ < |x-y|1_1/p( / |i/(*)|pcft) ", for almost all x,y G [0,1]. 9. Let fi C R" be an open set, 1 < p < +oo and u G W 1 ' p (fi). e. x G {x G Q,; u(x) = 0}. 10. 6. 11. 3. 12. 3. Chapter 2 L2 Theory of Linear Elliptic Equations This chapter is devoted to the L2 theory of linear elliptic equations. e. Poisson's equation thoroughly and then turn to the general equations in divergence form. 1 Weak Solutions of Poisson's Equation Let fl C i n be a bounded domain with piecewise smooth boundary dtt.