Interior regularity by difference quotients

Choose a coordinate direction and a small nonzero translation h. Define the difference quotient only where both the original and translated points remain inside the domain. If an H-one function has a weak derivative in that direction, its difference quotients are bounded locally in L-two by an appropriate gradient norm. Conversely, a locally uniform L-two bound on the difference quotients of an L-two function yields a weak derivative by testing, weak compactness, and passage to the limit. The bound must be independent of h.

For a weak solution of minus divergence of A gradient u equals f, insert a cutoff supported strictly inside the domain. A useful test is minus the backward difference quotient of the cutoff squared times the forward difference quotient of u. Discrete integration by parts transfers a translation between factors. The leading resulting term controls the squared gradient of the difference quotient, while cutoff terms are handled as in a localized energy estimate. Every translated support must remain admissible for the weak identity.

Assume A is real symmetric, uniformly elliptic, and C-one on the larger interior region, with bounded first derivatives there; assume f is L-two and u is an H-one weak solution. Differences of A are then bounded by its first-derivative bound times the translation length. This controls the commutator terms after dividing by h. Cauchy–Schwarz and Young's inequality absorb part of the unknown gradient-quotient term into ellipticity, leaving a uniform estimate in terms of known norms.

Consequently the solution belongs to H-two on every smaller compactly contained region, with an estimate depending on the ellipticity bounds, coefficient derivative bound, dimension, and separation from the outer region. The result is an interior theorem and imposes no boundary condition. Reaching H-two all the way to the boundary requires extra boundary geometry and compatible boundary data. Repeating the argument yields higher regularity only when coefficients and source supply the additional derivatives it consumes.

Across a material interface, the weak equation controls the flux A gradient u. If A jumps, a continuous flux can coexist with a jumping gradient. The weak second derivative of u then contains an interface delta, which is not an L-two function. Such a solution may be perfectly valid in H-one while failing H-two near the interface. This behavior is not a contradiction of elliptic regularity: the coefficient derivative hypothesis of this particular H-two theorem has failed.

A finite-difference computation can expose the same issue. Differences of a step-like gradient have height proportional to one over h on an interval of width h, so their squared L-two norm grows like one over h rather than remaining bounded. Refining a grid will not turn that interface delta into an ordinary square-integrable derivative. One must choose a regularity framework and numerical formulation consistent with the weak transmission condition, rather than interpret every large discrete derivative as an implementation bug.