A superharmonic u ∈ C 2 ( U ¯ ) {\displaystyle u\in C^{2}({\bar {U}})} satisfies − Δ u ≥ 0 {\displaystyle -\Delta u\geq 0} in U {\displaystyle U} , where here U ⊂ R n {\displaystyle U\subset \mathbb {R^{n}} } is open, bounded.
(a) Show that if u {\displaystyle u} is superharmonic, then
u ( x ) ≥ 1 α ( n ) r n ∫ B ( x , r ) u d y 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 min U ¯ u = min ∂ U u . {\displaystyle \min _{\bar {U}}u=\min _{\partial U}u.}
(c) Suppose U {\displaystyle U} is connected. Show that if there exists x 0 ∈ U {\displaystyle x_{0}\in U} such that u ( x 0 ) = min U ¯ u {\displaystyle u(x_{0})=\min _{\bar {U}}u} then u {\displaystyle u} is constant in U {\displaystyle U} .
Test
satisfies 
in 
, where here 
is open, bounded.
is superharmonic, then
.
is superharmonic, then 
is connected. Show that if there exists 
such that 
then is constant in .
