Question

Let u satisfy the Laplace equation in a disk Ω = {(x,y) | x2 + y2 < 1} and continuous on ¯Ω. If u(cosθ,sinθ) ≤ sinθ + cos2θ, then show that
u(x,y) ≤ y +x2 −y2, ∀(x,y) ∈ Ω.

Answer 1

As per the question,

u satisfy the laplace equation

$\Omega$ ={(x,y),$x^2+y^2<1$

According to the given condition

u($cos\theta,sin\theta)\le sin\theta+cos2\theta$ ...(1)

Let for disk we assume polar coordinates

x$=cos\theta,y=sin\theta$

Since u satisfy the given disk,so u must satisfy the polar coordinates of disk

New equation became

$u(cos\theta,sin\theta)\le sin\theta+cos^2\theta-sin^2\theta$

u(x,y)$\le y+x^2-y^2$

Hence

u(x,y)$\le y+x^2-y^2 \forall(x,y)\isin Q$