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00. 25 Averaging functions in Lp(RN), 1 < p < oo. Let al : RN - ][8 have the properties and al (=- L1(RN) al = 1.

11 := II . I Lp(]RN) II, prove that If - fp II 0. 1 Linear elliptic operators of order two As always, S2 denotes an open non-empty subset of RN. 1) j=1 i,j=1 whenever u E C2(S2) and x E S2, is a linear partial differential operator, of order two. Here a = (aij) : f - RN2, b = (bj) : --+ RN, c: --+R are given measurable functions. The N x N matrix a is symmetric : a ji(x) = ai j(x) for all i, j e {1,. , N} and all x e 0. 2) i,j=1 that L is elliptic in n if it is elliptic at every x E S2; and that L is uniformly elliptic in 0 if there is a constant AO > 0 such that 2(x) > 20 for all x E S2.

5. 6 If u E C2 (92) and Du > 0 in Sl, then u cannot have a local (let alone a global) maximum at a point of 52. Proof Assume (for contradiction) that u has a local maximum at q E fl. Then for each j E {1, ... u)(q) < 0. Hence (Au)(q) < 0. 1 Introduction 24 k T1, (k) Fig. 3. 3 On reflection in hyperplanes With this method, as with maximum principles, the proof of the pudding will be in the eating. However, we can take a first step in this section, and look ahead a little. 3 illustrates the objects in the next definition; the notation will be shortened when we come to use it in earnest.

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An Introduction to Maximum Principles and Symmetry in Elliptic Problems by L. E. Fraenkel

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