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The strong maximum principle for subharmonic functions states that an upper semicontinuous subharmonic function on a connected open set which attains its finite global maximum at an interior point is constant. In particular this holds for a harmonic function; negating the function gives the corresponding minimum principle for superharmonic functions.

Let the maximum be $M$, attained at $x_0$. For every sufficiently small ball centered at $x_0$, the mean value inequality givesTherefore its average equals $M$. If some point in the ball had value less than $M$, upper semicontinuity would give an open neighborhood where $u\le M-\eta$ for some $\eta >0$. That positive-measure neighborhood would make the average strictly less than $M$, a contradiction. Thus $u=M$ throughout a neighborhood of $x_0$.

The same argument at each maximizing point shows that $\{u=M\}$ is open; it is also closed relative to the domain by upper semicontinuity. It is nonempty, so connectedness forces it to be the whole domain. Without connectedness, constancy is only guaranteed on the component containing the maximizing point.

The strong maximum principle for subharmonic functions is: on a connected open domain, the canonical upper semicontinuous representative of a subharmonic function cannot attain a finite global maximum at an interior point unless it is constant. A local maximum instead forces constancy on an appropriate neighbourhood; it alone does not assert constancy on the whole domain unless that value is a global maximum.

Suppose $u\le M$ on $\Omega$ and $u(x_0)=M$. For every sufficiently small ball centred at $x_0$, the subharmonic mean inequality givesThe nonnegative function $M-u$ has integral zero, so $u=M$ almost everywhere on that ball. Its canonical representative is then $M$ at every point of the ball, by taking still smaller local averages.

The maximum set $E=\{x:u(x)=M\}$ is consequently open by the same argument at each of its points. It is closed relative to $\Omega$, because upper semicontinuity makes its complement $\{u<M\}$ open. It is nonempty, and connectedness gives $E=\Omega$. Therefore $u\equiv M$. This proves the principle, including its representative and connectedness hypotheses. Arbitrary representatives do not satisfy it: the function equal to zero almost everywhere but assigned value one at the origin has zero distributional Laplacian and an artificial pointwise maximum.