A superharmonic u∈C2(U¯){\displaystyle u\in C^{2}({\bar {U}})} satisfies −Δu≥0{\displaystyle -\Delta u\geq 0} in U{\displaystyle U} , where here U⊂Rn{\displaystyle U\subset \mathbb {R^{n}} } is open, bounded.
(a) Show that if u{\displaystyle u} is superharmonic, then
u(x)≥1α(n)rn∫B(x,r)udy for all B(x,r)⊂U{\displaystyle u(x)\geq {\frac {1}{\alpha (n)r^{n}}}\int _{B(x,r)}u\,dy\quad {\text{ for all }}B(x,r)\subset U} .
(b) Prove that if u{\displaystyle u} is superharmonic, then minU¯u=min∂Uu.{\displaystyle \min _{\bar {U}}u=\min _{\partial U}u.}
(c) Suppose U{\displaystyle U} is connected. Show that if there exists x0∈U{\displaystyle x_{0}\in U} such that u(x0)=minU¯u{\displaystyle u(x_{0})=\min _{\bar {U}}u} then u{\displaystyle u} is constant in U{\displaystyle U} .
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