TY - JOUR
T1 - Green functions and smooth distances
AU - Feneuil, Joseph
AU - Li, Linhan
AU - Mayboroda, Svitlana
N1 - Publisher Copyright:
© 2023, The Author(s).
PY - 2023
Y1 - 2023
N2 - In the present paper, we show that for an optimal class of elliptic operators with non-smooth coefficients on a 1-sided Chord-Arc domain, the boundary of the domain is uniformly rectifiable if and only if the Green function G behaves like a distance function to the boundary, in the sense that |∇G(X)G(X)-∇D(X)D(X)|2D(X)dX is the density of a Carleson measure, where D is a regularized distance adapted to the boundary of the domain. The main ingredient in our proof is a corona decomposition that is compatible with Tolsa’s α -number of uniformly rectifiable sets. We believe that the method can be applied to many other problems at the intersection of PDE and geometric measure theory, and in particular, we are able to derive a generalization of the classical F. and M. Riesz theorem to the same class of elliptic operators as above.
AB - In the present paper, we show that for an optimal class of elliptic operators with non-smooth coefficients on a 1-sided Chord-Arc domain, the boundary of the domain is uniformly rectifiable if and only if the Green function G behaves like a distance function to the boundary, in the sense that |∇G(X)G(X)-∇D(X)D(X)|2D(X)dX is the density of a Carleson measure, where D is a regularized distance adapted to the boundary of the domain. The main ingredient in our proof is a corona decomposition that is compatible with Tolsa’s α -number of uniformly rectifiable sets. We believe that the method can be applied to many other problems at the intersection of PDE and geometric measure theory, and in particular, we are able to derive a generalization of the classical F. and M. Riesz theorem to the same class of elliptic operators as above.
UR - http://www.scopus.com/inward/record.url?scp=85169833159&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85169833159&partnerID=8YFLogxK
U2 - 10.1007/s00208-023-02715-6
DO - 10.1007/s00208-023-02715-6
M3 - Article
AN - SCOPUS:85169833159
SN - 0025-5831
JO - Mathematische Annalen
JF - Mathematische Annalen
ER -