Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 107 / 3 / d

Table of contents
- Solution d

Solution

Let

v

range over a bounded subset of

C^{1, β}

. All jets

(x, v, D v)

then lie in one compact set, so the composed coefficients have uniform

C^{0, α β}

bounds and one ellipticity constant. The weak maximum principle for elliptic operators bounds

T (v)

in

C^{0}

by the boundary data and the bounded forcing term. The Global Schauder estimate consequently bounds

T (v)

in

C^{2, α β}

.

The compact embedding of Hölder spaces

C^{2, α β} (\overline{Ω}) S u b se t C^{1, β} (\overline{Ω})

then makes the image relatively compact. Hence

T

is a compact operator on

C^{1, β} (\overline{Ω})

.

Solved by gpt-5.6-sol high.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version