Abstract or Additional Information
The tangent-point energy from m-dimensional sets revisited: analogies
to pointwise estimates for Sobolev functions.
Some 15 years ago, Heiko von der Mosel and the speaker have analyzed
several nonlocal energies of non-smooth sets S, defined as multiple
integrals of various geometrically defined quantities, depending on
two or more points of S. One of the was the so-called tangent-point
energy, equal to the double integral over S, with respect to the
Hausdorff measure, of a function depending on two points: (a power
of) the radius of a sphere which is tangent to S at one point, and
passes through another point of S. For this energy, optimal regularity
results (for sets S having finite energy) are known; there is also a
full characterization (in terms of fractional Sobolev spaces) of those
S that have finite energy, due to Simon Blatt.
I will present a survey of those results and indicate new proofs of
optimal regularity and of Blatt's characterization which are simpler
than those previously known. They are both based on a lemma which
holds for graphs with finite tangent-point energy and is reminiscent
of Piotr Hajlasz's characterization of Sobolev spaces in terms of
upper gradients.