Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Free boundary regularity for almost every solution to the Signorini
  problem   We investigate the regularity of the free boundary for the Signorini problem
in $\mathbb{R}^{n+1}$. It is known that regular points are $(n-1)$-dimensional
and $C^\infty$. However, even for $C^\infty$ obstacles $\varphi$, the set of
non-regular (or degenerate) points could be very large, e.g. with infinite
$\mathcal{H}^{n-1}$ measure.
  The only two assumptions under which a nice structure result for degenerate
points has been established are: when $\varphi$ is analytic, and when
$\Delta\varphi < 0$. However, even in these cases, the set of degenerate points
is in general $(n-1)$-dimensional (as large as the set of regular points).
  In this work, we show for the first time that, "usually", the set of
degenerate points is small. Namely, we prove that, given any $C^\infty$
obstacle, for "almost every" solution the non-regular part of the free boundary
is at most $(n-2)$-dimensional. This is the first result in this direction for
the Signorini problem.
  Furthermore, we prove analogous results for the obstacle problem for the
fractional Laplacian $(-\Delta)^s$, and for the parabolic Signorini problem. In
the parabolic Signorini problem, our main result establishes that the
non-regular part of the free boundary is $(n-1-\alpha_\circ)$-dimensional for
almost all times $t$, for some $\alpha_\circ > 0$.
  Finally, we construct some new examples of free boundaries with degenerate
points.

19
The categories of this paper are: eess
27
The categories of this paper are: comp-gas
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
34
The categories of this paper are: chem-ph
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The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
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The categories of this paper are: math-ph
12
The categories of this paper are: nucl-th
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The categories of this paper are: dg-ga
26
The categories of this paper are: bayes-an