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PaperIII_64 (2) - MATHEMATICAL TRIPOS Part III Friday 30...

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MATHEMATICAL TRIPOS Part III Friday, 30 May, 2014 1:30 pm to 3:30 pm PAPER 64 MEASURE AND IMAGE Attempt no more than TWO questions. There are THREE questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Cover sheet None Treasury Tag Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 1 Answer the following questions concerning weak* convergence in BV(Ω), and the direct method in the calculus of variations. (i) Let Ω Rn be an open and bounded set with Lipschitz boundary. Characterise weak* convergence in BV(Ω), and state the criterion for a sequence in BV(Ω) to admit a weak* convergent subsequence. (ii) Provide a proof of the above result (existence of a weak* convergent subsequence). The following Lemma from the lectures can be helpful: Lemma. Let } >0 be a family of mollifiers, and w BV(Rn) with ϵ ϵ compact support. Then |(w ρ )(x) − w(x)| dx ≤ |Dw|(Ω). ϵ ϵ Rn (iii) Let f L1 (Ω)
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