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2 A Two-Point Boundary Value Problem For the purpose of preparing for the treatment of boundary value problems for elliptic partial dierential...
Please solve the following problems (Chapter covering these topics attached for your convenience):Problem 2.2. Determine Green’s functions for the following problems:(a) -u'' = f in Ω = (0, 1), with u(0) = u(1) = 0,(b) − u'' + cu = f in Ω = (0, 1), with u(0) = u(1) = 0,where c is a positive constant.Problem 2.3. Consider the nonlinear boundary value problem−u'' + u = e^u in Ω = (0, 1), with u(0) = u(1) = 0.Use the maximum principle to show that all solutions are nonnegative, i.e.,u(x) ≥ 0 for all x ∈ ¯Ω. Use the strong version of the maximum principle toshow that all solutions are positive, i.e., u(x) > 0 for all x ∈ Ω.Problem 2.4. Assume that b = 0 as in Theorem 2.3 and let G(x, y) be theGreen’s function defined there.(a) Prove that G is symmetric, G(x, y) = G(y, x).(b) Prove thata(v,G(x, ·)) = v(x), ∀v ∈ H^1_0, x ∈ Ω.This means that AG(x, ·) = δ_x, where δ_x is Dirac’s delta at x, defined as the linear functional δ_x(φ) = φ(x) for all φ ∈ C^0_0"
