The issue of regularity has obviously played a central role in the theory of Partial Differential Equations, almost since its inception, and despite the tremendous development, it still remains a very fruitful research field.
Regularity estimates for degenerate and singular elliptic and parabolic equations have developed considerably in recent years, in many unexpected and challenging directions.
Let us mention only few of the main contributions.
The celebrated Wiener criterion for quasi-linear elliptic equations has recently been given a new proof based on a sharp Harnack inequality for supersolutions, which allows for a straightforward generalizations to sub-elliptic equations.
An intrinsic version of the parabolic Harnack inequality for non-negative solutions of quasi-linear degenerate and singular parabolic equations, a well-known and long-standing open problem, has recently been established, opening the way to a thorough understanding of the structure of solutions of quasi-linear degenerate parabolic equations.
Uniqueness, symmetry and uniform rectifiability in some free boundary problems for elliptic equations of p-laplacian type have been recently considered by Lewis and Vogel, showing that they are essential tools in solving difficult regularity problems.
Regularity for solutions of Δ∞ is a very active research field. A full proof of the C1 regularity has been given in the two-dimensional case, and Evans and Savin have recently announced the extension to C1,α, but the issue remains open for higher dimension, together with closely connected research topics. Moreover it is an outstanding open problem to understand how much of this theory can be carried over to the sub-elliptic framework. If in the case of Carnot groups preliminary results are promising, the case of general Hörmander vector fields is quite more challenging and remains open.
Boundary Harnack inequality are a fundamental tool in studying regularity; the situation for the p≠2 case is a lot less understood than for the p=2 case, but recent progress has been made, that sheds new light on the phenomena involved. The characterization of p-harmonic measure is closely connected; under this point of view, just to quote an example, Manfredi recently solved a 25-year-old conjecture.
New game theoretic interpretations of the elliptic Δp operator with 1≤ p≤∞, have been given in the last years; this point of view is at its very beginning, but promises to be really enlightening, with particular regards to the study of p-harmonic measure and p-capacity.
When structural estimates for elliptic and parabolic Partial Differential Equations are at disposal, then they can be fruitfully exploited in exploring the local structure of free boundaries.
A free boundary problem that has recently seen a new interest is the boundary obstacle problem (the so-called Signorini problem), which is also linked to the obstacle problem for the fractional laplacian or similar integral--differential operators; in Mathematical Finance, it is a model for perpetual american options, which are governed by jump processes.
Because of all these recent developments, and many other we did not mention here, it is timely to trace an overview that would highlight emerging trends and issues of this fascinating research topic in a proper and effective way.
