homework3

# homework3 - Thus assume that D is an open and bounded...

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Homework Set 3: Math 716, Due: Wednesday, February 11th 1. Use energy method to prove uniqueness of classical solution to the initial value problem for the damped wave equation ( ǫ > 0): u tt + ǫu t = Δ u for x Ω R n , t > 0 , with u ( x , 0) = φ ( x ) , u t ( x , 0) = ψ ( x ) , u ( x , 0) = 0 on Ω 2. Find representation of solution to heat equation with Neumann boundary conditions on a half-line u t = u xx 0 < x < , t > 0 , with u ( x, 0) = 0 , u x (0 , t ) = sin t 3. Find a solution to the inhomogeneous wave equation u tt = c 2 u xx + f ( x, t ) , u ( x, 0) = φ ( x ) , u t ( x, 0) = ψ ( x ) for x R with c n = 0. Assume φ C 2 , ψ C 1 , and f C 0 are given bounded functions. Hint: Use Duhammel’s principle, which for ODEs, says the following: If v ( t ; τ ) is the solution to dv dt = Av, v ( τ ; τ ) = f ( τ ) then u ( t ) = i t 0 v ( t ; τ ) is a solution to du dt = Au + f ( t ), u (0) = 0. 4. Prove the weak maximum principle for Laplace’s equation.
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Unformatted text preview: Thus, assume that D is an open and bounded subset of R n with boundary ∂D . Assume also that u ( x ) is a solution to Laplace’s equation Δ u = 0 in D and that u is continuous on ¯ D , twice di±erentiable in D . Show that sup ¯ D u = sup ∂D u 5. a. Prove that if there exists a solution of the Neumann problem Δ u = f for x ∈ D ⊂ R n , ∂u ∂n = h ( x ) for x ∈ ∂D, then it is unique up to adding an arbitrary constant. b. Consider the Robin problem Δ u = f for x ∈ D , ∂u ∂n + a ( x ) u = h ( x ) for x ∈ ∂D Show that its solutions are unique. 1...
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