1. Assume that 2 is a bounded open set of Rd with Cd+2 boundary. When u : Q - R, let us set u+ (x ) = u(x) if u(x) > 0, 0 and u_(x ) = -u(x) ifu(x) 0 the first zero-Dirichlet eigenvalue in Q. (i) Show that if u e H)(2) satisfies [4 marks] [ NumPax = 1, u(x)dx then [Vu(x). Vo(x)dx = 1, [u(x)(x)dx, Yu E H;(1). Hint: You may find useful to decompose u into an L'-orthonormal basis of eigenvectors of -A with zero Dirichlet boundary condition. Set No = u EH;(Q) : Vu(x). Vu(x)dx = 1, u(x)(x)dx, Vue H;(2). and let us consider u E Na, \\(0). (ii) Show that u e C2 (2). [4 marks] Hint: Apply the relevant regularity theorem for elliptic PDEs combined with Morrey's theorem