Homework

Homework is due every Wednesday at 11:00 AM on [Gradescope], one hour before class starts. (Use this invite code if you’re not signed up). Please read the homework policy

Homework 1 (due 2026-09-02)

  1. Find the discussion board, and post your favorite math fact. (Include a screenshot when you turn in your homework.)
  2. Find the exponential of Jordan blocks
  3. Do any two case of this problem.
  4. Solve higher order equations using this reduction.
  5. Prove Duhamel’s principle.
  6. Prove Liouville’s formula
  7. Prove Duhamel’s principle in the time inhomogeneous case.

Homework 2 (due 2026-09-09)

  1. Prove spatial differentiability of the flow.
  2. Prove higher order differentiability of the flow.
  3. Prove Liouville Formula for the determinant of the flow map.
  4. Prove no global existence for equations with super-linear growth.
  5. Prove global existence under this condition.
  6. Prove non-uniqueness for rough coefficients.

Homework 3 (not due)

No homework this week, in light of the midterm. However, the midterm will cover Lyapunov functions, and Lyapunov’s Stability Theorem, the proofs of which were not done completely in class. (The midterm will not include Linearized stability.)

Homework 4 (due 2026-09-23)

  1. Prove stability of this second order system
  2. Prove Chetaev’s instability theorem.
    1. Prove this linear algebra lemma
    2. Prove the linear instability theorem
  4. Prove instability under this ray condition.
  5. Deterimine stability of gradient flows.

Homework 5 (due 2026-09-30)

  1. Find the minimal surface equation
  2. This problem on the method of characteristics for quasi-linear equations.
  3. Prove the boundary flattening lemma
  4. Prove this theorem on the method of characteristics.
  5. Find the Newton kernel

Homework 6 (due 2026-10-07)

Update 2026-10-02 I reordered questions after the homework was first uploaded.

  1. Prove this generalization of the mean value property.
  2. Prove these properties about sub-harmonic functions (3 parts)
    1. Prove the Hopf Lemma.
    2. Show the Hopf lemma fails when the domain is not $C^1$.
    3. Prove uniqueness for the Neumann problem
    1. Prove these derivative bounds (we only did $k=0, 1$ in class).
    2. Prove real analyticity of harmonic functions.
  5. Show these two functions are sub-harmonic.

Homework 7 (not due)

  1. This problem bounding solutions to the Dirichlet problem
  2. Harnack convergence theorem
    1. Prove that the Poisson kernel in a ball is an approximate identity.
    2. Prove the Poisson formula in a ball.
    1. This problem about harmonic functions blowing up everywhere on the boundary.
    2. This problem about harmonic functions blowing up on a dense set on the boundary.

Homework 8 (due 2026-10-28)

  1. Prove monotonicity of the Greens function
  2. Prove the poisson kernel in half space is an approximate identity.
  3. Prove $L^p$ convergence of the Poisson formula in half space
  4. Compute the Dirichlet to Neumann map on half space.
    1. Prove boundary values are attained at regular points
    2. Prove boundary regularity under the exterior sphere condition.
    3. Prove boundary regularity for $C^2$ domains.

Homework 9 (due 2026-11-04)

  1. Prove uniqueness on the punctured ball without an interior boundary condition.
  2. Prove locally subharmonic functions are subharmnonic.
  3. Prove existence of Greens functions.
  4. Prove that the Poisson kernel is an approximate identity (in general domains).
    1. Prove the Hopf lemma for elliptic equations
    2. Prove the Strong maximum principle for elliptic equations

Homework 10 (due 2026-11-11)

  1. Finish the proof of Lemma 13 (which we had unfinished in class).
  2. Find all self-similar solutions to the heat equation.
  3. Prove Duhamel’s principle for the heat equation.
  4. Write a complete proof of the mean value property (several details were skipped in class).
    1. Prove the parabolic Hopf lemma
    2. Prove the parabolic strong maximum principle

Homework 11 (not due)

  1. Verify Tychonoff’s example.
    1. Prove the Schwartz reflection principle
    2. Prove uniqueness of the Dirichlet problem
  3. Verify smoothness (last step in the proof of Proposition 24), which we only outlined in class.
  4. Prove interior derivative estimates
  5. Prove the Li Yau inequality