Math295b Winter Term 2002

Assignment 4

1. Let A = [a_{jk}]_{1≤j,k≤n} = A^{T} ≥ α > 0 , b ∈ ℝ^{n} , p ∈ (1, ∞) and compute H^{∗} for

H(x) = \frac{1}{p}|x|^{p} and H(x) = \frac{1}{2} ∑_{j,k=1}^{n} a_{jk}x_{j}x_{k} + ∑_{i=1}^{n} b_{i}x_{i} .

2. Let H : ℝ^{n} → ℝ be convex. Its subdifferential at p ∈ ℝ^{n} is the set

given by

∂H(p) = {q ∈ ℝ^{n} | H(y) ≥ H(p) + q · (y − p) , y ∈ ℝ^{n}} . Show that q ∈ ∂H(p) ⟺ p ∈ ∂H^{∗}(q) ⟺ p · q = H(p) + H^{∗}(q) .

3. Let H ∈ C^{−1}(ℝ^{n}) be convex and satisfy lim_{|p|→∞} H(p)/|p| = ∞ and assume g ∈ C^{−1}(ℝ^{n}). Prove that

min_{y∈ℝ^{n}} \{tH^{∗}(\frac{x − y}{t} ) + g(y)\} = min_{y∈𝔹(x,Rt)}\{tH^{∗}(\frac{x − y}{t} ) + g(y)\}

for R = ∥∇H(∇g)∥_{∞}.

4. Let H ∈ C^{−1}(ℝ^{n}) be convex and satisfy lim_{|p|→∞} H(p)/|p| = ∞ and assume g_{i} ∈ C^{1−}(ℝ^{n}) , i = 1, 2. Let u_{i} be a weak solution of

\begin{cases}u_{t} + H(∇u) = 0 & in ℝ^{n} × (0, ∞) , \\ u = g_{i} & on ℝ^{n} × \{0\} . \end{cases}

Show the validity of the following contraction property

∥u_{1}(·, t) − u(·, t)∥_{∞} ≤ ∥g_{1} − g_{2}∥_{∞} .

5. Compute the unique entropy solution of

\begin{cases}u_{t} + [\frac{u^{2}}{2} ]_{x} = 0 & in ℝ × (0, ∞) \\ u = g & on ℝ × \{0\} \end{cases}

where g is given by

g(x) = \begin{cases}1 & if x ∈ (−∞, −1] \\ 0 & if x ∈ (−1, 0] \\ 2 & if x ∈ (0, 1] \\ 0 & otherwise. \end{cases}

The Homework is due on March 22 2002