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# PARTIAL DIFFERENTIAL EQUATIONS MTH 627

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This 5 page Class Notes was uploaded by Miss Johan Jacobson on Monday October 19, 2015. The Class Notes belongs to MTH 627 at Oregon State University taught by Staff in Fall. Since its upload, it has received 32 views. For similar materials see /class/224449/mth-627-oregon-state-university in Mathematics (M) at Oregon State University.

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Date Created: 10/19/15

35 Distributions with Compact Support 627w2004 7 Bent E Petersen 20040125 Let Q be an open subset of R and let C denote the space of in nitely differentiable functions on Q A linear functional u on C is continuous if and only if there exists a compact subset L of Q a constant C gt 0 and an integer m 2 0 such that lltuq gtl S C max sup DD q5zl L lalimxe for each 15 E C The restriction of u to C50 is then a distribution with suppu Q L It is clear that the space of distributions with compact support may be identi ed with 8 the dual space of C Note each distribution with compact support has nite order The smallest value of the integer m above is the order 0 u Theorem 1 Let u 6 89 have order m and support K Then for each relatively open subset to g Q with K g to we have a constant Cw such that lltu gtl S Cw max sup DD q5zl 6w 345mm for each 15 E C Proof By hypothesis there exists a compact subset L of Q and a constant C gt 0 such that lltu7 gtl O max sup WWI l la lSm 16L for each 15 E C Let X E C Q be chosen such that X l in a neighborhood of K Then lltu7 gtl lltu7X gtl S C max sup DD l Sm L lal l a C max sup lt gtDu i xD qb WETLme 2 5 CM max su D r l lsm cpl 15 Mth 627 Winter 2004 20040125 35 Distributions with Compact Support Note the supremum is taken over w and the constant Cw involves the derivatives of x Since X is 1 on K and has support in to we expect some of the derivatives of X to be large on to N K ifw is close to K Thus we can not expect to obtain an estimate just on K Nonetheless we can compute lt u7 15 gt in terms of the m7jet of 15 on K Theorem 2 Letu 6 89 have orderm andsupport K Ifqb E C and D I Ofor each a g mandx E K then ltuq5gt 0 See Schwartz 3 theorem XXVIH28 chapter 1113 section 7 Friedman 1 theorem 24 chapter 3 section 6 Donoghue 6 theorem in part 112 section 21 In the case the support K is a regular set in the sense of Whitney 5 we can actually obtain ltuk7 15gt 7gt 0 if qbk and its derivatives up to a certain order depending on the shape of K converge to 0 uniformly on K 7 see Schwartz 3 theorem XXX1V34 chapter 1113 section 7 7 but for a general K this result does not hold See the example below For related ideas see also Malgrange 2 especially chapter VH7 and for a monograph concerning differentiable functions see Tougeron 4 Proof Since u E 8 has order m there exists a compact set L such that lltu7 gtl O max suplDW MSTL L for each 1 E C Q Note it follows that K g L For each 6 gt 0 let K5 be the closed 6 neighborhood of K and let X5 be the characteristic function of K5 Let s gt 0 be such that 36 lt dist K7 69 Let p be a Friedrichs molli er and let X2 pa X25 so x 6 one suppltx2gt g K3 and x 7 1 on K Since X 1 in a neighborhood of K namely K5 we have ltu gtlt u x2wgt and therefore ltu1Jgt g C m alt sup 1D 4in 1 17m Lka for any 1 E C lfx 6 K35 we can choose y E K with z 7 y g 36 Let 90 D y t1 7 y 0 lttlt17 so by Taylor s theorem with remainder m7ot 1 1 Da I L k ltm17lalgt ltgt 91 go My lt0gtm17 a g t Copyright 2004 BentE Petersen 2 Oregon State University Corvallis Oregon Mth 627 Winter 2004 20040125 35 Distributions with Compact Support for some t with 0 lt t lt 1 where gk0 is a linear combination of D y for B 2 a and 1 51 g m By hypothesis then gk0 0 since y E K and W l g m Thus 1 D04 lt quot171040 t l m ltm iww 9 lt su Derl lal DD tzi 7 m17 a 0lt l z My y Since I 6 K35 and y e Kwe have DD 1 ltC max su D emlilal i r gt17 M which yields 1mm 1 g o em1lal We have obtained this inequality for all z 6 K35 a l g m and all e gt 0 with 36 lt dist K 69 Now WFWMM7WNy implies Du x zl S Ceilu l for e gt 0 Thus by Leibniz formula 1D x l g 2 lD XQ 1mm 504 S 0 Z 6m171 lelae l rise oane Hence we have 1 ltuq5gt l g Ce ifs gt 0 is small thatis ltuq5gt 0 D Now here is the example promised above It is taken from Schwartz 3 example at the end of chapter 1113 section 7 Example 3 Let uk be the distribution with compact support on R de ned by k 1 7 7 7 7 lt uk gt 7 k 0 logk a5 0 0 a5 h for 15 E C By Taylor s theorem the is 1 E C R such that we lt0gt lt0gtz am It follows k k ltukq5gt gs0 2 40 Zip hl h1 Copyright 2004 BentE Petersen 3 Oregon State University Corvallis Oregon Mth 627 Winter 2004 20040125 35 Distributions with Compact Support The coe lcient of 15 0 converges to the EuleriMascheroni constant and since i 1 l is bounded on 07 1 the last sum Euler s series converges Hence by the Banachisteinhaus theorem ltuq5gt lim ltukq5gt 1H is continuous that is de nes a distribution u Moreover it is clear that 11 70177 1 suppu I i I 7 72737 1f 15 E C R and lt15 0 on supp and 0 0 then lt u7 15 gt 0 This example illustrates theorem 1 Now suppose we choose qbk E C R such that 0 g qbk g i and i for z gt 1 M00 E 71k 0 for I g m If h 2 1 then the smoothness of qu implies Dhqbk 0 and Dhqbk 0 Thus by de nition Dhqbk 0 on suppu ifh 2 1 On the other hand 0 g qbk g i for all 1 Thus Dhqbk A Ouniformly on supp for each h 2 0 as k A 00 On the other hand h k 1 1 k lt gt139 7139 77xE Wk h12021 k J blag y Thus the qbk together with all their derivatives converge uniformly to 0 on supp but lt u7 45k gt A 00 as k A 00 Proposition 4 Leta E R u E 301 andsuppose suppu Q a Then there exist an integer m 2 0 and constants ca 6 C such that u E caDo oar iaism Proof Since u has compact support it has nite order say m If qbk E C 1R then by Taylor s theorem 1 151 2 Jowaxz 7 a W a 3m where 1 E C R and D r a 0 for 151 g m By theorem 1 it follows that lt ugly gt 0 Thus ltu7 15gt 2 a Sm caDu da where AW 11 ltuzia0 gtr D As you may expect there are similar structure theorems for multiple layers that is distributions supported by submanifolds That is true See Schwartz 3 Letme leave you with a little exercise to play with Copyright 2004 BentE Petersen 4 Oregon State University Corvallis Oregon Mth 627 Winter 2004 20040125 35 Distributions with Compact Support Exercise 1 Find all solutions u E D R ofthe equation emu 0 where m 2 0 is an integer Copyright 2004 Bent E Petersen All rights reserved Permission is granted to du plicate this document for nonipro t educational purposes provided that no alterations are made and provided that this copyright notice is preserved on all copies Bent E Petersen 24 hour phone numbers Department of Mathematics Oregon State University o ice 541 7375163 Corvallis OR 973314605 fax 541 7370517 benta1ummitedu petersenmathoregonstateedu httporegonstateedupeterseb References 1 Avner Friedman Generalized Functions and Partial Di erential Equations PrenticeHall Inc Englewood Cliffs N 1 1963 2 Bernard Malgrange Ideals of Di erentiable Functions Oxford University Press for Tata Insti tute of F undam ental Research Bombay London New York Bombay 1966 3 Laurent Schwartz Ihe orie des distributions Combined volumes I and II Hermann Paris 1966 I 1950 195711 195119591961 4 Jean Claude Tougeron Ide aux de Fonctions Di e rentiables SpringepVerlag Berlin Heidel berg New York 1972 5 Hassler Whitney Analytic extensions of differentiable functions de ned in closed sets Trans Amer Math Soc 3663789 1934 6 Jr William F Donoghue Distributions andFourier Transforms Academic Press New York and London 1969 Copyright 2004 BentE Petersen 5 Oregon State University Corvallis Oregon

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