149 research outputs found
Smooth approximations for fully nonlinear nonlocal elliptic equations
We show that any viscosity solution to a general fully nonlinear nonlocal
elliptic equation can be approximated by smooth () solutions
Schauder and Cordes-Nirenberg estimates for nonlocal elliptic equations with singular kernels
We study integro-differential elliptic equations (of order ) with
variable coefficients, and prove the natural and most general Schauder-type
estimates that can hold in this setting, both in divergence and non-divergence
form. Furthermore, we also establish H\"older estimates for general elliptic
equations with no regularity assumption on , including for the first time
operators like , provided that
the coefficients have ``small oscillation''
On global solutions to semilinear elliptic equations related to the one-phase free boundary problem
Motivated by its relation to models of flame propagation, we study globally
Lipschitz solutions of in , where is smooth,
non-negative, with support in the interval . In such setting, any
"blow-down" of the solution will converge to a global solution to the
classical one-phase free boundary problem of Alt-Caffarelli.
In analogy to a famous theorem of Savin for the Allen-Cahn equation, we study
here the 1D symmetry of solutions that are energy minimizers. Our main
result establishes that, in dimensions , if is axially symmetric and
stable then it is 1D
Free boundary regularity for almost every solution to the Signorini problem
We investigate the regularity of the free boundary for the Signorini problem
in . It is known that regular points are -dimensional
and . However, even for obstacles , the set of
non-regular (or degenerate) points could be very large, e.g. with infinite
measure.
The only two assumptions under which a nice structure result for degenerate
points has been established are: when is analytic, and when
. However, even in these cases, the set of degenerate points
is in general -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
obstacle, for "almost every" solution the non-regular part of the free boundary
is at most -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 , 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 -dimensional for
almost all times , for some .
Finally, we construct some new examples of free boundaries with degenerate
points
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