 Solution

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

w = (u - k)^{+}

u - k

. The allowed Sobolev chain rule gives

\nabla w = 1_{{u > k}} \nabla u almost everywhere .

Use the admissible zero-boundary Sobolev space function

φ^{2} w

in the weak formulation, with a valid cutoff function as explained in (ii). Its gradient is

φ^{2} \nabla w + 2 φw \nabla φ

, so

0 = \int φ^{2} A \nabla w \cdot \nabla w + 2 \int φw A \nabla w \cdot \nabla φ .

Indeed the integrands vanish off

{u > k}

, and on that set

\nabla w = \nabla u

. The uniform ellipticity and Cauchy-Schwarz inequality estimates from (ii) now apply to

w

and give

\int_{B_{1/2} \cap {u > k}} ∣\nabla u ∣^{2} d x \leq \frac{4 Λ ^{2} M ^{2}}{λ ^{2}} \int_{B_{1} \cap {u > k}} (u - k)^{2} d x, M = ∥\nabla φ ∥_{\infty} .

Fixing the cutoff function gives the required constant independent of

k

. This is a superlevel Caccioppoli inequality; it uses the positive part as a test and does not require the truncated function itself to solve the original equation.

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