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

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1 Assume That Is A Bounded Open Set Of Rd With Cd 2 Boundary When U R Let Us Se U X U X 0 If U X 0 Othe 1 (116.82 KiB) Viewed 26 times

1. Assume that is a bounded open set of Rd with Cd+2 boundary. When u : → R, let us se u+ (x) = u(x) 0 if u(x) > 0, otherwise and u_(x) { -u(x) if u(x) < 0, 0 otherwise. b) Denote by ₁0 the first zero-Dirichlet eigenvalue in Q. Set N₁₁ = {u € H}(Q) : [ Vu(x).Vv(x)dx = À\ ▼u(x).Vv(x)dx = A₁ [_ u(x)v(x)dx, \/v © H(Q).} and let us consider u € N₁, \ {0}. (iii) Prove that u, and u_ also belong to N₁₁. Hint: Use that Vv(x)\²dx ≥ λ₁ v(x)²dx for any v € H(N). (iv) Using the strong maximum principle, show that whether u(x) > 0 for all x € or u(x) < 0 for all x € N.