A result in potential theory giving comparative Harnack-type estimates for positive harmonic functions that vanish on a portion of the boundary of a domain, describing how their values compare near the boundary.

Principle

If two nonnegative harmonic functions in a domain vanish continuously on a common piece of boundary, then their ratio remains bounded (and often Hölder-continuous) up to the boundary under suitable geometric conditions on the domain and operator.

Demonstration

In a Lipschitz or NTA domain in R^n, two positive harmonic functions that vanish on the same boundary patch satisfy that u/v is bounded near that patch; concretely, for solutions of the Laplace equation with zero trace on a flat portion of boundary one constructs barrier functions and uses Harnack chains to obtain the comparison.

Misapplication

Applying the principle to sign-changing functions, to functions that are merely subharmonic, or to domains with fractal or highly irregular boundary without verifying geometric hypotheses leads to false conclusions about boundary ratios.

Consequence

When valid, the principle yields precise relative boundary behaviour, control of boundary limits, and is a key tool to identify Martin boundary points and to deduce boundary regularity and uniqueness statements for boundary-value problems.

Reversal

If the conditions fail (e.g., functions do not vanish on the same boundary portion or the domain lacks the required regularity), there is no uniform bound on the ratio and one can observe arbitrarily large relative oscillation near the boundary.

Boundary

Applies to positive harmonic (or more generally, positive solutions of uniformly elliptic equations) that vanish on a common boundary portion in domains satisfying geometric regularity (Lipschitz, NTA, non-tangential accessibility); excludes sign-changing solutions, general parabolic problems without modification, and severely rough boundaries.

Semantic Tension

This principle is often conflated with Hopf boundary lemmas (which give pointwise nonvanishing normal derivative information) or with interior Harnack inequalities; the boundary Harnack principle is comparative and global near the boundary whereas Hopf is local directional.

Synthesis

The Boundary Harnack Principle encapsulates a comparative regularity statement: under domain and operator regularity, positive harmonic functions that vanish together on part of the boundary must behave proportionally near that boundary, giving bounded, often Hölder, ratios that underpin fine boundary analysis.