Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quantitative a Priori Estimates for Fast Diffusion Equations with
  Caffarelli-Kohn-Nirenberg weights. Harnack inequalities and H\"older
  continuity   We study a priori estimates for a class of non-negative local weak solution
to the weighted fast diffusion equation $u_t = |x|^{\gamma} \nabla\cdot
(|x|^{-\beta} \nabla u^m)$, with $0 < m <1$ posed on cylinders of
$(0,T)\times{\mathbb R}^N$. The weights $|x|^{\gamma}$ and $|x|^{-\beta}$, with
$\gamma < N$ and $\gamma -2 < \beta \leq \gamma(N-2)/N$ can be both degenerate
and singular and need not belong to the class $\mathcal{A}_2$, a typical
assumption for this kind of problems. This range of parameters is optimal for
the validity of a class of Caffarelli-Kohn-Nirenberg inequalities, which play
the role of the standard Sobolev inequalities in this more complicated weighted
setting.
  The weights that we consider are not translation invariant and this causes a
number of extra difficulties and a variety of scenarios: for instance, the
scaling properties of the equation change when considering the problem around
the origin or far from it. We therefore prove quantitative - with computable
constants - upper and lower estimates for local weak solutions, focussing our
attention where a change of geometry appears. Such estimates fairly combine
into forms of Harnack inequalities of forward, backward and elliptic type. As a
consequence, we obtain H\"older continuity of the solutions, with a
quantitative (even if non-optimal) exponent. Our results apply to a quite large
variety of solutions and problems. The proof of the positivity estimates
requires a new method and represents the main technical novelty of this paper.
  Our techniques are flexible and can be adapted to more general settings, for
instance to a wider class of weights or to similar problems posed on Riemannian
manifolds, possibly with unbounded curvature. In the linear case, $m=1$, we
also prove quantitative estimates, recovering known results in some cases and
extending such results to a wider class of weights.

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The categories of this paper are: econ
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The categories of this paper are: hep-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: physics
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: astro-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: chao-dyn
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ex
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The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
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The categories of this paper are: math-ph
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The categories of this paper are: gr-qc