This uniqueness result for the Dirichlet problem asserts that if two continuous functions on the closure of a bounded domain $D$ are both harmonic on $D$ and agree on the boundary $\partial D$, then they agree on all of $\overline{D}$. The proof is a one-line application of the maximum principle for harmonic functions: the difference $w = u - v$ is harmonic with $w = 0$ on $\partial D$, so $\max_{\overline{D}} w \leq \max_{\partial D} w = 0$ and $\max_{\overline{D}} (-w) \leq 0$, forcing $w \equiv 0$. The maximum principle used here — that a harmonic function on a bounded domain attains its maximum on the boundary — is itself a consequence of the mean value property: if $w$ achieved a strict interior maximum at $x_0$, then the spherical average of $w$ over a small sphere centred at $x_0$ would be strictly less than $w(x_0)$, contradicting harmonicity. This uniqueness result complements the existence result provided by the [Brownian Dirichlet Solution](/theorems/1186): the probabilistic formula $u(x) = \mathbb{E}_x[\varphi(B_\tau)]$ produces a harmonic function, and the present theorem guarantees that it is the ONLY harmonic function with the given boundary data (assuming continuity up to the boundary). The [Boundary Continuity under the Cone Condition](/theorems/1188) then provides a sufficient geometric condition on $D$ for this continuity to hold, completing the picture: existence via probability, uniqueness via the maximum principle, and regularity via domain geometry.