UMD PDE Qualifying Exams/Jan2005PDE

Problem 1 Let be a harmonic function on and suppose that Show that is a constant function. Solution Let If is harmonic (i.e. ) then so must (surely, ). Then since the absolute value as an operator is convex, we have that is a subharmonic function on . Then by the mean value property of subharmonic functions, for any we have where the second inequality is due to Cauchy-Schwarz (Hölder) inequality. This estimate hold for all . Therefore if we send we see that for all which gives us that is constant. Problem 2 Let be a piecewise smooth weak solution of the conservation law a) Derive the Rankine-Hugoniot conditions at a discontinuity of the solution. b)Find a piecewise smooth solution to the IVP Solution a) When we solve the PDE by methods of characteristics, the characteristic curves can cross, causing a shock, or discontinuity. The task at hand, is to find the curve of discontinuity, call it . Multiply the PDE by , a smooth test function with compact support in . Then by an integration by parts: Let denote the open region in to the left of and similarly denotes the region to the right of . If the support of lies entirely in either of these two regions, then all of the above boundary terms vanish and we get Now suppose the support of intersects the discontinuity . b) We can calculate . Therefore, the shock wave extends vertically from the origin. That is, Problem 3 Consider the evolution equation with initial data a) What energy quantity is appropriate for this equation? Is it conserved or dissipated? b) Show that solutions of this problem are unique. Solution 3a Consider the energy . Then . Integrate by parts to get . The boundary terms vanish since implies (similarly at ). Then by the original PDE we get where the last equality is another integration by parts. The boundary terms vanish again by the same argument. Therefore, for all ; that is, energy is dissipated. 3b Suppose are two distinct solutions to the system. Then is a solution to This tells us that at , . Therefore, . Since then for all . This implies . That is, . Problem 4 Let be a bounded open set with smooth boundary . Consider the initial boundary value problem for : where is the exterior normal derivative. Assume that and that for . Show that smooth solutions of this problem are unique. Solution Suppose are two distinct solutions. Then is a smooth solution to Consider the energy . It is easy to verify that . Then Therefore implies for all . Thus, for all which implies Problem 5 Let be a bounded open set with smooth boundary. Let . Let and define the functional . Show that is a minimizer of over if and only if satisfies the variational inequality for all . Solution Suppose minimizes , i.e. . Then for any fixed , if we let then . Let ; then we can say that . Now we must compute . We have Since we know then as desired. Conversely suppose Then Therefore, for all . That is, for all , as desired.