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PaperIII_68 (2) - MATHEMATICAL TRIPOS Part III Monday 9...

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MATHEMATICAL TRIPOS Part III Monday, 9 June, 2014 1:30 pm to 3:30 pm PAPER 68 IMAGE PROCESSING - VARIATIONAL AND PDE METHODS Attempt no more than THREE questions. There are FOUR 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 Let 1 < q < ∞. Give a formal argument for the fact that functions u W 1,q (0, 1) are Hölder continuous. What does this imply for a function u W 1,q ((0, 1)2 )? Define the space BV ((0, 1)2 ) and for a function u BV ((0, 1)2 ) the total variation |Du|((0, 1)2 ). Give a formal derivation for the fact that for u W 1,1((0, 1)2 ) we have 1 1 |Du|((0, 1)2 ) = | u| dx dy. 0 0 For g L2((0, 1)2 ) and α > 0 consider the functional 1 1 1 1 J (u) = α 1 + |Du|2 + 1 (u − g)2 dx dy, 0 0 2 0 0 where {∫ 1 + |Du|2 = sup ( 0 + udiv ) dx ϕ ϕ | C 1c (D; R2 ), ϕ ∈ D D } 0 C 1c (D), | (x)| ≤ 1, |
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