Partial Differential Equations

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  • Let \Omega\subset \R^n be an open, bounded, smooth domain. Let f\colon \partial\Omega\to \R be continuous. Is there a solution to Δu = 0 in Ω, u = f on \partial\Omega? If there is a solution, explain how it is found.
  • Let {vn} be a sequence of harmonic functions in Ω. Assume v_n\to v uniformly in norm and that v_n(x)\nearrow v(x)\forall x\in \Omega. Show that v is harmonic in Ω.
  • Consider utt = uxx + uyy with u(x,y,0) = 0 in x2 + y2 < 1 and uy(x,y,0) = 0 everywhere. What is u\left(0,0,{1\over 2}\right)?
  • Consider (\ast )\ u_t=u_{xx}, u(x,0) = f(x). Do you know a solution? Suppose you don't know the fundamental solution of the heat equation, how would you derive a solution of (\ast)? What conditions would you impose on f(x) for uniqueness?
  • For one dimensional Laplace's, heat and wave equations give initial and/or boundary conditions that allow you to find solutions.
  • State the mean value property for harmonic functions and explain how you prove it.
  • Consider the inviscid Burgers' equation. Assume there is a curve across which the solution is discontinuous. State hte Rankine-Hugoniot condition. State the entropy condition analytically.
  • Follow the steps below to give a heuristic derivation of the entropy condiiont (that a shoc will occur if ur < ul):
    1. Consider (\ast)\ v_t+vv_x-\epsilon v_{xx}=0. Let w = vx and differentiate (\ast) with respect to x. write the result in terms of w.
    2. Use w^2\ge 0 and the maximum principle for subsolutions of the heat equation to conclude that w is bounded for all x and t > 0.
    3. Thus v_x\le M. Explain how this implies that if ur > ul you cannot get a shock.
  • Define the fundamental solution of the wave equation. Truncate the it for negative times, what kind of equation does the resulting distribution satisfies? Is it unique? Prove it. Derive the energy principle for a truncated cone. (B.Ettinger vs. Tataru+Zworski)
  • Why can't we multiply arbitrary distributions? (S. Dyatlov vs. Borcherds)
  • Assume that $u,v\in L^\infty$. What can we say about the wavefront set of $uv$? (S. Dyatlov vs. Zworski)
  • State the uniqueness result for the scattering problem (by an obstacle). Sketch the proof. What properties of the differential operator did we use to prove the statement? Which parts of the proof can be adapted to more general cases? Does the proof work for a Schwarz class potential? (S. Dyatlov vs. Tataru-Zworski)
  • State local energy decay for the wave equation (scattering by an obstacle). Can one obtain a bound on local energy depending only on the norms of the initial conditions? If not, what type of operator convergence does this result correspond to? (S. Dyatlov vs. Zworski)

This page was originally derived from this TeX file.

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