Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Scale invariant regularity estimates for second order elliptic equations
  with lower order coefficients in optimal spaces   We show local and global scale invariant regularity estimates for
subsolutions and supersolutions to the equation $-{\rm div}(A\nabla
u+bu)+c\nabla u+du=-{\rm div}f+g$, assuming that $A$ is elliptic and bounded.
In the setting of Lorentz spaces, under the assumptions $b,f\in L^{n,1}$,
$d,g\in L^{\frac{n}{2},1}$ and $c\in L^{n,q}$ for $q\leq\infty$, we show that,
with the surprising exception of the reverse Moser estimate, scale invariant
estimates with "good" constants (that is, depending only on the norms of the
coefficients) do not hold in general. On the other hand, assuming a necessary
smallness condition on $b,d$ or $c,d$, we show a maximum principle and Moser's
estimate for subsolutions with "good" constants. We also show the reverse Moser
estimate for nonnegative supersolutions with "good" constants, under no
smallness assumptions when $q<\infty$, leading to the Harnack inequality for
nonnegative solutions and local continuity of solutions. Finally, we show that,
in the setting of Lorentz spaces, our assumptions are the sharp ones to
guarantee these estimates.

9
The categories of this paper are: nlin
7
The categories of this paper are: hep-th
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs
24
The categories of this paper are: dg-ga
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
36
The categories of this paper are: cmp-lg
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int
19
The categories of this paper are: eess
13
The categories of this paper are: quant-ph