PARTIAL DIFFERENTIAL EQUATIONS ASSIGNMENT 1 (1) Verify that u(x, t) = (c2 t2 |x|2 )1 satises the three-dimensional wave equation except on the light cone. (2) Solve u = 0 in the spherical shell 0 < a < |x| < b in R3 with the boundary conditions u = A on |x| = a and u = B on |x| = b. (Hint: Look for a radial solution.) (3) (Uniqueness for the Poisson problem) Consider Poisson's equation u(x) = f (x) , x , with boundary conditions x , u(x) = g(x) , where Rn is a bounded open connected set with smooth boundary, and f and g are continuous functions. Use the energy method to show uniqueness for this problem, i.e., that there can be at most one solution. Hint: Given two solutions v, w, set u = v w and consider the energy E = | u|2 dx. (4) Use the maximum principle to give another proof of uniqueness for the Poisson problem. (5) (a) (The real and imaginary parts of holomorphic functions are harmonic) Let u, v be two real-valued functions in two variables. Assume that (u, v) satisfy the Cauchy-Riemann system ux = v y uy = vx . Show that u and v satisfy Laplace's equation. (b) (Construction of the conjugate harmonic function) Conversely, assume that u : R2 R satises Laplace's equation. Show that there exists a function v such that the Cauchy-Riemann differential equations hold. (Hint: The vector eld (vx , vy ) is exact.) (c) Find the conjugate harmonic function for u1 (x, y) = x2 y 2 . Also nd the the conjugate harmonic function for u2 (x, y) = 1 log (x2 + y 2 ). 2 Remark: If u, v satisfy the Cauchy-Riemann system, we can dene a function f : C C by f (x + iy) = u(x, y) + iv(x, y) . The Cauchy-Riemann system guarantees that the function is complex-differentiable, that is, df f (z) f (z0 ) (z0 ) = lim zz0 ,z=z0 dz z z0 exists at every point z0 . Such a function is called holomorphic. Holomorphic functions have many magical properties and are widely used, from Number Theory to Fluid Dynamics and Electrical Engineering. They include polynomials, exponentials and trigonometric functions, as well as Gamma and Zeta functions. 1