1. Assume that 12 is a bounded open set of Rd with Cd+2 boundary. When u : 1 → R, let us set u+(x) = { u(x) if u(x) > 0,

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1 Assume That 12 Is A Bounded Open Set Of Rd With Cd 2 Boundary When U 1 R Let Us Set U X U X If U X 0 1 (218.49 KiB) Viewed 26 times

1. Assume that 12 is a bounded open set of Rd with Cd+2 boundary. When u : 1 → R, let us set u+(x) = { u(x) if u(x) > 0, 0 otherwise and u_(x) = {quand otherwise. a) For any p € [1, +) and u € W 1,P(12). (i) Show that ut € W1.P(12) and that for all 1 sis d, the weak di-partial derivatives of u+ [4 marks] is dju(x) for all x € 2 with u(x) > 0, au+(x) = 0 for all x € 12 with u(x) < 0. = {v+-for x SO Hint: You may find useful to consider Ff(x) and the = 0 composition ug = Fc(u(x)) and then apply the relevant chain rule in W1.P(12). (ii) Show that if u € W/P(12) then ut, u_ and Jul belong also to W.,P(S2). Hint: You may find useful to write ut – (4n)+ = (1uso - 14->0)u + 14m>o(u – 4n), where (An)nz1 is a relevant sequence in C?(12). W 71p [3 marks] 0