 Solution

Apply the volume mean value property to the two balls and use nonnegativity together with inclusion:For any $w,z\in B(x,r)$, the triangle inequality gives $B(z,r)\subset B(w,3r)\subset B(x,4r)\subset \Omega$. Taking $R=3r$ in the previous bound therefore gives $u(w)\ge u(z)/27$. Take the infimum over $w$ and supremum over $z$ to obtainNo positivity stronger than nonnegativity is needed; if a value vanishes, this argument forces every value in the smaller ball to vanish too.