Harnack inequality and positive solutions

Let A be real symmetric, bounded measurable and uniformly elliptic on a ball of radius two R. Let u be a nonnegative H-one-local weak solution of minus divergence of A gradient u equals zero there. The interior Harnack inequality bounds the essential supremum on the concentric radius-R ball by a constant times the essential infimum on that ball. The constant depends on dimension and ellipticity bounds, and the fixed ratio of the two radii; it does not depend on the size of u.

Essential extrema are the appropriate starting language for weak functions defined only almost everywhere. De Giorgi–Nash–Moser theory also supplies a locally Hölder-continuous representative for this scalar homogeneous divergence equation with measurable coefficients. With that representative, the same inequality becomes a pointwise interior comparison. This conclusion does not upgrade a rough coefficient field to differentiability or promise continuous gradients. A weak solution can have substantially less derivative regularity than a harmonic function with constant coefficients.

For harmonic functions the mean-value property provides a direct route to local comparison. With measurable coefficients there is generally no ordinary spherical mean-value formula. Moser's approach instead tests with cutoff-weighted powers, uses energy and Sobolev inequalities, and iterates integrability bounds. Positive and negative powers address opposite sides of the comparison. Replacing u by u plus a small positive number permits the negative-power steps before taking a limit. This describes the strategy rather than supplying the full proof and all exponent choices.

Overlapping interior balls transmit the ratio estimate along a finite path in a connected domain. This gives a constant depending on the chain, not a universal comparison for points arbitrarily near every boundary feature. Together with the continuous representative, the inequality shows that a nonnegative homogeneous solution which vanishes at an interior point must vanish on that ball and then throughout its connected component. The conclusion uses nonnegativity: sign-changing harmonic functions can certainly vanish without being identically zero.

A nonzero source can create an interior minimum even for a strictly positive function, so a homogeneous ratio bound need not survive. Inhomogeneous Harnack estimates include an additive source contribution in an appropriate scaled norm and impose suitable integrability assumptions. One cannot absorb that contribution into a universal multiplicative constant when the minimum approaches zero. The example in the practice question isolates this failure while preserving smoothness and the ordinary Laplacian.

Nor is a global supremum-to-infimum ratio independent of distance to the boundary. A positive harmonic function may approach zero at one boundary point and remain large elsewhere. Interior balls avoid that conflict by staying inside a larger region where the equation is known. This makes Harnack useful for compact interior subsets, local positivity, and oscillation control, while boundary Harnack principles form a different theory with their own domain and vanishing-boundary hypotheses.