Math295c Spring Term 2002

Assignment 1

1. Assume that f ∈ C([0, 1]) and that

F ∈ C([0, 1] × [0, 1] × ℝ) , ∂_{u}F ∈ C([0, 1] × [0, 1] × ℝ)

and consider the integral equation

u(x) = ∫_{0}^{1} F (x, y, u(y)) dy + f(x) , 0 ≤ x ≤ 1 .

Show that, if ∥∂_{u}F ∥_{∞} < 1, the integral equation has a unique solution u ∈ C([0, 1]).

Let Ω ⊂ ℝ^{n} be open. A map f : Ω × ℝ → ℝ is called Carathéodory function whenever

(i) f(·, s) : Ω → ℝ is measurable for every s ∈ ℝ. (ii) f(x, ·) : ℝ → ℝ is continuous for almost every x ∈ Ω.

2. Let f : Ω×ℝ → ℝ be a Carathéodory function and p, q ≥ 1. Assume

that

|f(x, s)| ≤ c|s|^{p/q} + g(x)

for some g ∈ L_{q}(Ω). Prove that the Nemytzki operator (substitution operator) N_{f} : L_{p}(Ω) → L_{q}(Ω) defined through

(N_{f}u)(x) := f(x, u(x)) , x ∈ Ω

is well-defined, continuous and maps bounded sets onto bounded sets.

3. Let Ω ⊂ ℝ^{n} be open and bounded. Assume that the Carathéodory function f : Ω × ℝ → ℝ satisfies

\underline{f} ≤ \frac{f(x, u) − f(x, v)}{u − v} ≤ \overline{f} and f(·, 0) ∈ L_{2}(Ω)

with σ(−△_{D}) ∩ [\underline{f}, \overline{f}] = ∅. Show that

\begin{cases}△u = f(x, u) & in Ω , \\ u = 0 & on ∂Ω \end{cases}

possesses a unique weak solution u ∈ H^{∘}^{1}(Ω).

4. (Kolmogoroff) Show that a subset K ⊂ L_{p}(ℝ^{n}) (1 ≤ p < ∞) is compact iff

(i) K is closed and bounded. (ii) ∫_{|x|≥N} |f(x)|^{p} dx → 0 , N → ∞ , uniformly in f ∈ K. (iii) ∫_{ℝ^{n}} |f(x + h) − f(x)|^{p} dx → 0 , |h| → 0 , uniformly in f ∈ K. [Hint: Use the density of test functions in L_{p}(ℝ^{n}), the strong continuity of the translation semigroup on L_{p}(ℝ^{n}) and Arzéla-Ascoli.]

5. Let 1 ≤ p < ∞ and prove that

(∫_{ℝ^{n}} |u(x + h) − u(x)|^{p} dx)^{1/p} ≤ |h| ∥u∥_{1,p}

for u ∈ W_{p}^{1}(ℝ^{n}). Use this estimate and Kolmogoroff’s characterization of compactness to show that W_{p}^{1}(Ω) − ↪ L_{p}(Ω) for Ω ⊂ ℝ^{n} open and bounded.

The Homework is due by April 19 2002