Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: De Giorgi-Nash-Moser and H{\"o}rmander theories: new interplay   We report on recent results and a new line of research at the crossroad of
two major theories in the analysis of partial differential equations. The
celebrated De Giorgi-Nash-Moser theory shows H{\"o}lder estimates and the
Harnack inequality for uniformly elliptic or parabolic equations with rough
coefficients in divergence form. The theory of hypoellipticity of H{\"o}rmander
shows, under general "bracket" conditions, the regularity of solutions to
partial differential equations combining first and second order derivative
operators when ellipticity fails in some directions. We discuss recent
extensions of the De Giorgi-Nash-Moser theory to hypoelliptic equations of
Kolmogorov (kinetic) type with rough coefficients. These equations combine a
first-order skew-symmetric operator with a second-order elliptic operator
involving derivatives in only certain variables, and with rough coefficients.
We then discuss applications to the Boltzmann and Landau equations in kinetic
theory and present a program of research with some open questions.

32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
15
The categories of this paper are: nucl-ex
23
The categories of this paper are: solv-int
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
12
The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph