Maximum principles convert the sign of an elliptic or parabolic operator into global control from boundary data. They imply uniqueness, comparison, and positivity without an explicit solution formula.
Weak elliptic maximum principle
Theorem. Let \(\Omega\) be bounded and connected. If \[ Lu=-a^{ij}u_{ij}+b^iu_i+c u\le0 \] with \(A=(a^{ij})\) uniformly positive definite and \(c\ge0\), then \[ \max_{\overline\Omega}u \le\max\{0,\max_{\partial\Omega}u\}. \]
Proof for \(c=0\). If \(u\) had a strict positive interior maximum at \(x_0\), then \(\nabla u(x_0)=0\) and \(D^2u(x_0)\preceq0\), so \[ Lu(x_0)=-a^{ij}u_{ij}(x_0)\ge0. \] To make the contradiction strict, perturb \(u_\varepsilon=u+\varepsilon e^{\alpha x_1}\) with \(\alpha\) large so \(Lu_\varepsilon>0\), apply the nonstrict argument, and let \(\varepsilon\downarrow0\). The \(c\ge0\) case applies to the positive part.
Strong maximum principle
Theorem. Under uniform ellipticity, if \(Lu\le0\) and \(u\) attains a nonnegative maximum in the interior, then \(u\) is constant.
A proof uses interior balls and a barrier to propagate the set of maximum points, or Harnack's inequality applied to the nonnegative difference from the maximum. Connectedness then forces constancy.
Hopf boundary lemma
Theorem. Suppose a nonconstant subsolution attains a strict maximum at a boundary point satisfying an interior ball condition. Then its outward normal derivative is strictly positive under the sign convention \(Lu\le0\).
An exponential radial barrier in the tangent interior ball has a known operator sign. Comparing it with the solution and differentiating at the contact point yields the strict normal derivative. Hopf's lemma is central to boundary uniqueness and moving-plane symmetry arguments.
Comparison and uniqueness
Corollary. If \[ Lu\le Lv\quad\text{in }\Omega,\qquad u\le v\quad\text{on }\partial\Omega, \] then \(u\le v\) in \(\Omega\). In particular, the Dirichlet problem has at most one classical solution.
Proof. Apply the maximum principle to \(u-v\).
Parabolic maximum principle
If \[ u_t-a^{ij}u_{ij}-b^iu_i-cu\le0,\qquad c\le0, \] on \(\Omega\times(0,T]\), then the positive maximum occurs on the parabolic boundary \[ (\overline\Omega\times\{0\}) \cup(\partial\Omega\times[0,T]). \] The terminal time slice is not prescribed: time orientation makes the initial boundary the relevant one.