5. Let I be a bounded open subset of R of class C1 which is symmetric with respect to R-1 x {0} i.e. if x = (x', xn) and x' = (11, ..., In-1), then (X', 2n) E N if and only if (x', -Xn) EN. Let f e C(2), g C(an) and let u C?(2) be a solution of the BVP for Poisson's equation: -u = f in u=9 We define the functions (x', en) = u(x', un), f(x, xn) = f(x, xn), and g(x', n) = g(x', -n). a) Show that is a solution of the BVP: s -A= f in 2 on a 2 = On b) Show that if f and g are even with respect to Xn, then so is u. You may use Theorem 10-Lecture 15 to show that u =. Theorem 10: Assume that I is a bounded connected open set, f e C(92) and g C(212). Then there exists at most one solution u E C?(12) n (2) of the Dirichlet problem for Poisson's equation u = f in u=g on a 2 5. Let I be a bounded open subset of R of class C1 which is symmetric with respect to R-1 x {0} i.e. if x = (x', xn) and x' = (11, ..., In-1), then (X', 2n) E N if and only if (x', -Xn) EN. Let f e C(2), g C(an) and let u C?(2) be a solution of the BVP for Poisson's equation: -u = f in u=9 We define the functions (x', en) = u(x', un), f(x, xn) = f(x, xn), and g(x', n) = g(x', -n). a) Show that is a solution of the BVP: s -A= f in 2 on a 2 = On b) Show that if f and g are even with respect to Xn, then so is u. You may use Theorem 10-Lecture 15 to show that u =. Theorem 10: Assume that I is a bounded connected open set, f e C(92) and g C(212). Then there exists at most one solution u E C?(12) n (2) of the Dirichlet problem for Poisson's equation u = f in u=g on a 2