partial-differential-equations-by-evans · Problem 6

Nonlinear First-Order PDE

Mathematics — Calculus · harnack-inequality

Problem

Use Poisson's formula for the ball to prove rn−2r−∣x∣(r+∣x∣)n−1u(0)≤u(x)≤rn−2r+∣x∣(r−∣x∣)n−1u(0)r^{n-2}\frac{r-|x|}{(r+|x|)^{n-1}} u(0) \leq u(x) \leq r^{n-2}\frac{r+|x|}{(r-|x|)^{n-1}} u(0) whenever uu is positive and harmonic in B0(0,r)B^0(0,r). This is an explicit form of Harnack's inequality.

Browse the problem library