interior regularity
You feed a PDE some rough data and ask: how nice is the solution in the middle of the domain, away from the edges? For elliptic and parabolic equations the answer is delightful — the solution is smoother in the interior than you had any right to expect from the data, often perfectly smooth even if the data is barely continuous. The equation polishes its own solutions on the inside. Interior regularity is the study of exactly how much smoother, and why.
Precisely: if Laplacian u = f on a region, then u is two derivatives smoother than f, measured in the right scale. If f is Holder continuous, the Schauder estimates say u has Holder-continuous second derivatives, with an interior bound controlling those derivatives on any compact subset by the data on a slightly larger set. If f is in an L^p space, the Calderon-Zygmund estimates put u in the Sobolev space with two more derivatives in L^p. The crowning fact: a harmonic function (f = 0) is real-analytic in the interior — infinitely differentiable, equal to its own Taylor series. The word interior is doing real work: these are estimates on compact pieces strictly inside the domain. The boundary needs separate treatment, because the solution can be rough right up against the edge even while it is smooth inside.
This matters because it lets you bootstrap: prove a weak solution exists in some rough space, then use interior regularity repeatedly to upgrade it to a genuine classical solution where the equation holds pointwise. It also explains a clean dichotomy. Elliptic and parabolic equations smooth — singularities in the data are erased in the interior. Hyperbolic equations like the wave equation do NOT smooth — a kink in the initial data travels along characteristics and stays a kink forever. So interior regularity is a hallmark of diffusion-like, not wave-like, behaviour.
Solve Laplacian u = 0 on a disk with boundary values given by a merely continuous, jagged function. Inside the disk u is nonetheless infinitely smooth — real-analytic — and you can differentiate it as many times as you like, even though the boundary data has corners.
Elliptic equations smooth in the interior; the boundary is where roughness can survive.
Interior is not optional wording: regularity up to the boundary is a separate, harder question and can genuinely fail at corners or with rough boundary data, even when the interior is perfectly smooth. And it is an elliptic/parabolic gift — do not expect any interior smoothing from hyperbolic equations.