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by: Otilia Murray I
Otilia Murray I
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This 10 page Class Notes was uploaded by Otilia Murray I on Tuesday September 8, 2015. The Class Notes belongs to MAT 201C at University of California - Davis taught by Staff in Fall. Since its upload, it has received 84 views. For similar materials see /class/187429/mat-201c-university-of-california-davis in Mathematics (M) at University of California - Davis.

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Date Created: 09/08/15
Chapter 6 Distribution 61 Test Function and Distribution De nition Let 9 Q R 9 open later 9 R De ne the space of test function 739 be in with an added notion of convergence 15m A 45 6 739 if there exists a compact set K E 9 such that suppq m 7 E K for all m7 and for every multiindex a Du qu A D0215 uniformly on K for all a as m A 00 Remark Note 739 is a linear space The derivative of 45 with multiindex a 011 7an E Z is a a 01 a an D 1 W De nition A distribution T is a continuous linear functional on That is7 Note that lal ELI ai T1 1 A2 2gt A1T 1 A2T 2 linear and qbn A ab in 739 implies that A continuous 739 is the space of distribution and it is the dual space to Notion of convergence in 73 9 We say Tj A T in 739 if Tjq5 A in C pointwise Remark This is quite a weak of notion ofconvergence When we see de nition of derivative of a distribution7 we will see that if Tj A T7 then DaTj A DD T for all derivative of Tj and differentiation is a continuous operation in 73 9 As opposed to pointwise convergence fn A f ae7 may not converge at all 62 Locally Integrable Functions How are distributions related to functions Let7s de ne a new space De nition De ne LP loc measurable functions de ned on all of 9 and K lflpdz lt 00 for every compact C We say fj A f strongly in L m9 if fj A f strongly in Lp9 for all compact K E 9 and similarly 706 is a vector space but without a simply de ned norm loc 9 be the space of locally p integrable functions We say f 6 LP if f is Borel 9 for weak convergence L Remark 1 Note if f E L1lR 7 then f E L1 Rn but the converse is not true think of loc 2 Note for 7 gt p 2 17 we have L m9 D LITOC9 by Holder because uK lt 00 for compact K E 9 We will mostly consider Llloc9 most useful for distribution Lemma Let f E L1 and de ne loc TM Q f1q51d1 for all gt 6 729 Then Tf is a distribution7 and we identify Tf with Proof Check continuity Assume 45m A Then W 7 Trlt gtl Q l m1 WM was dz Ham 7 MM K WM dz 7 0 for all compact K C 9 Similar to its derivative D This gives Tf E DQ for all f 6 L11 D Example De ne Diracdelta function 61 lt15 ME for all 45 E Note that 61 151 52 1I 152W 610 mr A MI 61015 675 is a distribution which is not given by integration like above Tf 739 is a rather restrictive class of function But it is big enough to distinguish distributions Theorem Functions are uniquely determined by their distributions Let Q Q R open7 f9 E L1 with lot Qf Qg for all 45 E Then 91 ae 1 E 9 Proof Form 12 de ne l 9m169 1969 1f m Note that 9m is open Let j E CfquotlR 7 suppj Q B1 and fan 1 Now let m jm17 and thus suppjm Q Blm Fix M7 if m 2 M7 let jm1 7 y and use fact that Qf Qg jmfr 7 Ajmltz7ygtfltygtdy7 Ajmltz7ygtgltygtdy7wgltzgt for all 45 6 739 we have on 9m Thus jm jm 91 for all 1 E QM m 2 M Since jmfquotf7 m9quot9 as m A 00 so 91 as m A 00 Take M A 00 QM A Q and complete the proof D 63 Derivatives of Distribution or Weak Derivative De nition Let T E DQ and take a any multiindex We de ne the distributional or weak derivative 0 y ltD Tgtlt gt 7 71gt Q TltDa gt Remark l The symbol BiT denotes DD T in the special case ai l aj 0 for j i 2 The symbol VT called the distributional gradient of T denotes the n tiple EAT ng BnT 3 Assume f E ClD l by integration by parts with zero boundary terms since 45 6 C we have D Tflt gt 71gt a fD dz Dw dz TDaflt gt If f has enough derivatives weak derivatives are strong derivatives and in this weak sense every distribution has in nitely many derivatives Lemma The derivative of a distribution DD T is a distribution Proof Check DaT is a continuous linear functional acting on By de nition DD T 71 lD lTDD q linearity is obvious Now we check for continuity Let 45 6 Cam qu A 45 in 739 and thus there exists a compact K E 9 such that 45139 A 45 uniformly on K Hence D qu A D qb for all derivatives D uniformly on K Applying T and the above de nition we have D T j A D Tq D Lemma Differentiation D of distribution is a continuous operator in 39D Proof If Tj A T ie Tjq A for all 45 E By de nition DD TJ39WW UlD lT Dw A Ulo lan DB TW since D0215 E So we proved that DaTj A DDT and thus D is a continuous in PKG D 64 Sobolev Spaces We begin with I Q A C f E Lied and If as a distribution in Lied for i ln Note that is a vector space but not a norm space and Winn 3 Wm for p lt r We can de ne W1gtpQ C as Wl p9 I Q A C f and If are in LPG for i ln W1gtp9 is a Banach space follow directly from completeness of L17 and de nition of distribution derivative with norm 7 117 llfllW1 PQz llfllipmleaifllipngt i1 Here is the notion of convergence fj A f in if fj A f and Bifj A ij in for all compact KQQ foril2n De nition The Sobolev space Wk are set of functions f E L17 whose distribution derivatives up to order k are in They are Banach space with norm de ned by llfllzvvam llfllipm Z llDD fllipm wk We also denote Wkgt2 HWQ H for Hilbert Same notions of convergence can be generalized to W113 Lemma lnterchanging distribution with convolution Let Q Q R be open 45 E De ne O Q R by 0 1 y 6 R I supp r y E 9 Note that O is open and nonernptyi Take T 6 779 the function y 7 Tq5z 7 y is in 000 D3Tlt ltz 7 y 7 71gt a TD z 7 w 7 WT lt ltz 7 y ltgt Let 1 E L1O with compact support Then 0 wltygtTlt ltz 7 mm 7 w lt15 ltgt Proof Observe that l 1y riyizl S Ce uniformly because 45 6 739 for any 2 suf ciently small So 7 y 7 2 7gt 7 y as 7gt 0 Thus 7 y 7 7gt i as 7gt 0 Similar for derivatives using difference quotients W Vwmw S 015 2 7 and thus by a similar argument y gt7 i is differentiable Continuing in this manner we nd that holdsi To prove WLOG assume 1 6 CC00 else approximate it by CC00 functionsi That is we have 5 6 CC Q for each 6 gt 0 so that fo W5 7 lt 6 The integrand 7 6 Cam and hence the integral can be taken as Riemann integral and thus can be approximated by nite surnsi ZwltyjgtTlt ltz 7 mm 7 wltygtTlt ltz 7 mm j1 04gt where Am 7 0 as m 7gt 00 Likewise for any rnultiindex a Zwyanz 7 yjmm 7 O wltygtDa ltz 7 my 7 DW was 11 gt as m 7gt 00 Since T is continuous so we have D 65 Fundamental theorem of Calculus for Distribution Theorem Fundamental theorem of Calculus for Distribution Let Q Q R open T E 39D Q 45 E Assume for y E R the translation 7 ty 6 739 for all 0 S t S 1 then 1 n Tlt ltz 7 ygtgt 7 Tlt ltzgtgt 7 ZyjltajTgtlt ltz 7 m at ltgt j1 As a particular case of if T Tf I f dz for f E VVIECWQ then for all y E R and for are I E R 1 fzy 7 fltzgt 7 yVfzty dt ltgt 0 for are I E Q where Vf Ej ij means distribution derivativesi Proof We prove for general distribution Let O 26 R 16739 be open and nonemptyi De ne 1 7 iww wz 7 mm and Cy T I y T IA Now our goal is to show CO F0 07 AF AG E C i Recall 2 gt gt BjTq z 7 is C00 on 045 we lt61Tlt ltz 7 z 7 7ltajTgtltaiolto 7 z Hence7 a 1 n 1 8ij 7 am 7 7 lt61Tgtltai ltz 7 W dt lt61Tgtlt ltz 7 WM since and by product rule Note that 7 lt61Tgtltai ltz 7 W dt 7 maz iajwz 7 my dt 7 7 matgm 7 m dt 7 tgmxasw 7 mm Thus we have 61F 7 1 tgwmw 7 WM 1ltaiTgtlt ltz 7 WM 7 goalWm 7tygtgtdt 7 6mm 7 y Note F 6 00 aiF Blmow 7 y F0 0 Recall G LHS Roy 7 no Bio 6iTq5z 7 y C 6 00 610 aiF F0 00 so F Gi Now we prove for T Tf f e W111 Rn Note that ij z 7 ty ajfz z 7 tydz let I 7 ty 2 5 5 z 7 Iijztydzo Assume holds Now translate this to T 7 Tf since lt15 6 00 07 ajf E Llloc7 apply Fubini ltzgt x 7 y 7 wow 7 1 lt Iijz 7 two dt lt iyjaj ertyMt dzi ltzgt x 7 y 7 mm 7 9 W lt1y Vfz tydtgt dz Now we have 5 holds for all 45 6 CC00 So by uniqueness 1 Wwi AyV wHw ae z 1 Theorem Equivalence of classical and distributional derivatives Let Q C R be open T E 39D Q and let G BIT E 39D Q i l 2 n The following are equivalent 1 T is a function f E C1Q 2 G is a function gi E C0Q In each case gi g in the classical sense Proof l 2 Integration by parts 2 l Use FTC for distribution 1 Corollary Distributions With zero derivatives are constants Let Q C R be open and connected and T E 39DQ lf BiT 0 for all i 12 n then there exists a constant C such that nwc for all 45 E Proof Proof is an exercise use equivalence of distribution and classical derivatives D 66 Multiply Distribution by 000 Function and Convolution De nition Let T E 39DQ 1 E C00 De ne the product T by its action on 45 6 739 as WT lt15 TQM for all lt15 6 0509 for all 45 E Remark The de nition of T makes sense because 45 E CC Q and then 6 C2 To differentiate T we apply the product rule 610 15 6iT 61397JT Observe that When T Tf for some f E VVIECWQ then T T1 Moreover if f E then f E Wli CpQ and the differentiation reads 94101095 f13i 1 Iaif1 for ae I Where means the distributional derivative The same holds for WkgtpQ and Wk p9 loc De nition Convolution of T With j 6 CC00 Rn is de ned by uawawToeMwT4gwwaiww for all 45 E BUR When T Tf then j Tfq Tjfq Remark Need the requirement that j Inust compact support and need T to act on CC00 functions Since 1 lt15 6 0501 T makes sense Theorem Approximate of distributions by CC00 functions Let T E DRn j E CC R Then there exists a function t E C R t depends only on T and j such that UWWPA mww for all 45 E BUR lf further fan dz 1 and let j5z 6 jze for e gt 0 then jg 9 T A T in 39DRn as e A 0 Proof With the lemma of interchanging convolutions with distribution we have 139 Tgtlt gt Tltjlt7zgt lt15 7 T lt m 7 gammy 7 My 7 zgtgt ltygtdyi Rquot Rquot De ne tz Tjy 7 and note that t E Cm thus we proved the rst part For jg 9 T A T lt1 Tgtlt gt T lt jeltygt ltz 7 my 7 we 7 mum 1Rquot Rn 9jydy by changing variables Ru 7 M AM my 7 Ms since i E Coo as a function of y and j has compact support D 67 Properties for o 1 671 0 06 ls Dense 1n VVIOCWQ Theorem Let f E for every 0 open with the property that there exists K compact O C K C 9 Then there exists C C00 0 such that fk A f in W1gtpO iiei llf 7 fklleo Z llaif aifkllem A 0 i1 askaooi Remark Later in 76 C Q is dense in H1 W1gt2Qi Q Q Rni It is very hard to approx Vf at 89 The case 9 R is proved in the book ref Sobolev space Adamsi Proof Fix 6 gt 0 consider jg f where j z e jze with j 6 Cam suppj Q B10 fRn z 1 Let 0 be given such that O C K C 9 Thus we have jg f E C O since Dwe f z ltDmgtltz 7 yfydy for 6 small enough and for any derivative of order on Furthermore since 0 C K C Q with K compact we assume that osuppltjegt H2 z z e 0 z e suppw g K for small 6 Thus we have the integral 61 m 7 mm 7 m 7 y6ifydy K K which makes sense in sense of distributions since f E if 6 LP Moreover f If E L17 and we loc use C00 approximation theorem for Lp to approximate f and Bif and thus Hf fklleO 2 Naif aifkllLuO A 0 i1 for 6 suf cient small 6 N If 9 R we can actually approximate with CC00 functions 1 672 Chain Rule Theorem Let G C31 37 G R 7gt C be differentiable function With bounded and continuous derivative lf ul denotes n functions in lVlo cp97 then we have 1 Q A C W e E n 8Gu Buk 7 7 117 7 7 vz 7 G o 7 E 71069 and 6in 7 i1 6 an E D lf uh 7 un E W1gtp97 then 1 E W1gtp9 and the chain rule holds With the assumption that 00 implies CO 0 Proof Read the textbook 1 673 Derivative 0f Absolute Value Theorem Let f E W1477 then the absolute value of f7 denoted by and de ned by is in W1gtp9 With V given by WWI 9911 RIVRIIIVII Where 131 Re 1 lm Note if 131 is real7 then WW fgt0 VlfW 7Vfz f lt 0 0 f0 for real Remark If f is complex7 then V lt ae if f is real7 then V ae Proof Note that f E L17 and E L17 obvious Assume the formula holds7 show RVR 1V1 6 L17 7 fl 1 2 m 72 R2VR2 2RIVR W 12v12 7 RQ 12VR2 v12 WE v1 e L17 Let 643132 7 m 7 6 Note if 31 7 Ra 32 7 11 we have G RI52R2P7e 7 m RVR 1V1 Note G500 07 BGE 7 Si 63139 2S S 7 Apply Chain rule7 K41 GeRIII 6 WM and for all 45 6 739 V ltIgtK5 zdz 7 IVK5 zdz Q Q 7 RltIgtVRltIgt dx 9 62 WIN Note that K5 S 6 LPG and K5 7gt provided 5 7gt 07 thus RIVRIIIVII S WW 6 We me WW and the above approaches to W as e 7 0 so by LDCT we have the result 1 Corollary min and max of W147 functions are W147 functions More specifically7 let Mr 7 minfrgr 7 g M W 7 WI 7 9W MW 7 maxfrgr 7 g 1 WW WI 7 WW Then we have Vf f gt 9 WM f gt g WWI WW 9 gt f 7 Vm 7 Vfr y gt f Vf7Vg 91 Vf7Vy 97f Proof Let hz 7 914r 7 7 positive part of 7 91 thus h hz Vhz 0 if lt But by previous fact derivative of was 7 ltvltf 7 g f 7 g and hence Vf Vg if f g 1 Corollary Let 12041 minfzoz7 13041 maxfza Then Vf iffz lta Vfltoz 7 0 W if fa gt a p 0 otherwise 7 Vfgta1 otherwise Note that for Q is unbounded7 flta E W1gtp9 only if a 2 07 and fgta E W1gtp9 only if a S 07 Remark lfu E Q7 ifa E R and a on a set ofpositive measure E7 then Vuz 0 ae z E E 68 Green s Function and Laplace Equation De nition De ne Greenls function of Laplace equation Let n 2 17 Cy R 7gt R 7 2 SW71 1 7 2 gt 2 GAE 1 1 9 n7 7 7 S1 ln z7y n2 Remark Note that CAI is symmetric7 meaning that Cy G75 In particular for y 07 n71 1 27w COW n22 7 S1 ln z n2 is radial symmetric and is in LIIOCUR since for any compact K C R 7 dz Tn ld39rdw G0zl dz 7 2 Sn71 71 TQ n39rnildew c Tdrdw lt 00 K K K Moreover7 z z am 7 m aim 7 270ml m 7 277mm So for gt 07 n Am 26101 m i1 i1 7 n lt7n zrnil i 0 2 2 7 n in zrnil Min 7 7 it Oin n z W am and thus AGO 0 if gt 0 Theorem iAGy 6y Where 6g is Dirac measure at y That is7 for all 45 6 CC00 Rn7 Aqubdz GyAqbdz 7 yi Proof We show for y 0 and show lir139r 7450 Where 17quot A G0zdzi T Mgtr Since 45 6 CC00 Rn7 there exists R such that suppq5 C BR So 17quot A G0dz rltMltR and furthermore 45 0 on Ri Let Az Tlt I ltR 8Az 1 TUI 1 Ri Integrating by parts7 we have 00mde 2 Goaiaiqsdz A i1 A 2 lt7 aiaoaiqbdz GoalMS i1 A xr 2001de GoalMS r ma Z BidCoqde 7 A a o B i1 A z Aq AGoqbdz 7 vaoqbds GOqudS A 1 7 1 7 Let 1 i be the outward normal to A point to the origin7 and thus v00 V SHH par By divergence theorern7 A HT VG0de sn1 1 H m ziiwrlrn ldw Haw S Hrln 114w 0 ltCTgt0 and B H Cow vds HWHLm H Go v waswm as 7 A 0 Thus we have f GoAqbdz 74150


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