 Solution

ID: past-exam-of-the-functional-analytic-methods-for-partial-differential-equations-course-of-the-university-of-oxford/2023/2/a/i/solution

u

g_{i}

in direction

i

when

g_{i}

is locally integrable and

\int_{B_{1}} u \partial_{i} ϕ d x = - \int_{B_{1}} g_{i} ϕ d x for every ϕ \in C_{c}^{\infty} (B_{1}) .

The test functions have compact support inside the domain, so no boundary values enter this definition. Such

g_{i}

is unique almost everywhere and is denoted

\partial_{i} u

For

1 \leq p < \infty

, the first-order Sobolev space consists of the Lp space classes whose first weak derivatives are also in that Lp space:

W^{1, p} (B_{1}) = {u \in L^{p} (B_{1}) : \partial_{i} u \in L^{p} (B_{1}), i = 1, \dots, n}, ∥ u ∥_{W^{1, p}} = (∥ u ∥_{p}^{p} + i = 1 \sum n ∥ \partial_{i} u ∥_{p}^{p})^{1/ p} .

This Sobolev norm is equivalent to

∥ u ∥_{p} + ∥\nabla u ∥_{p}

; either convention is acceptable. When

p = 2

, this is the Hilbert space

H^{1} (B_{1})

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