If u solves ∇²u = 0, in D ⊆ Rn then,
(Here ∂D denotes the boundary of D and D̅ = D ∪ ∂D)

Concept:

If u is a harmonic function on a bounded domain D in Rⁿ, then u attains its maximum value on the boundary of D.

Maximum principle:

\(\mathop {\max }\limits_{\bar D} u\left( x \right) = \mathop {\max }\limits_{\partial D} u\left( x \right)\) 

Analysis:

∵ u solves ∇² u = 0

∴ u is a harmonic function.

Hence, u will satisfy the maximum principle theorem and

\(\mathop {\max }\limits_{\bar D} u\left( x \right) = \mathop {\max }\limits_{\partial D} u\left( x \right)\)