12,841 research outputs found
Local Estimates for Viscosity Solutions of Neumann-type Boundary Value Problems
In this article, we prove the local regularity and provide
estimates for viscosity solutions of fully nonlinear, possibly
degenerate, elliptic equations associated to linear or nonlinear Neumann type
boundary conditions. The interest of these results comes from the fact that
they are indeed regularity results (and not only a priori estimates), from the
generality of the equations and boundary conditions we are able to handle and
the possible degeneracy of the equations we are able to take in account in the
case of linear boundary conditions
Lipschitz Regularity of Solutions for Mixed Integro-Differential Equations
We establish new Hoelder and Lipschitz estimates for viscosity solutions of a
large class of elliptic and parabolic nonlinear integro-differential equations,
by the classical Ishii-Lions's method. We thus extend the Hoelder regularity
results recently obtained by Barles, Chasseigne and Imbert (2011). In addition,
we deal with a new class of nonlocal equations that we term mixed
integro-differential equations. These equations are particularly interesting,
as they are degenerate both in the local and nonlocal term, but their overall
behavior is driven by the local-nonlocal interaction, e.g. the fractional
diffusion may give the ellipticity in one direction and the classical diffusion
in the complementary one
Nonlocal equations with measure data
We develop an existence, regularity and potential theory for nonlinear
integrodifferential equations involving measure data. The nonlocal elliptic
operators considered are possibly degenerate and cover the case of the
fractional -Laplacean operator with measurable coefficients. We introduce a
natural function class where we solve the Dirichlet problem, and prove basic
and optimal nonlinear Wolff potential estimates for solutions. These are the
exact analogs of the results valid in the case of local quasilinear degenerate
equations established by Boccardo & Gallou\"et \cite{BG1, BG2} and
Kilpel\"ainen & Mal\'y \cite{KM1, KM2}. As a consequence, we establish a number
of results which can be considered as basic building blocks for a nonlocal,
nonlinear potential theory: fine properties of solutions, Calder\'on-Zygmund
estimates, continuity and boundedness criteria are established via Wolff
potentials. %In particular, optimal Lorentz spaces continuity criteria follow.
A main tool is the introduction of a global excess functional that allows to
prove a nonlocal analog of the classical theory due to Campanato \cite{camp}.
Our results cover the case of linear nonlocal equations with measurable
coefficients, and the one of the fractional Laplacean, and are new already in
such cases
Geometric approach to nonvariational singular elliptic equations
In this work we develop a systematic geometric approach to study fully
nonlinear elliptic equations with singular absorption terms as well as their
related free boundary problems. The magnitude of the singularity is measured by
a negative parameter , for , which reflects on
lack of smoothness for an existing solution along the singular interface
between its positive and zero phases. We establish existence as well sharp
regularity properties of solutions. We further prove that minimal solutions are
non-degenerate and obtain fine geometric-measure properties of the free
boundary . In particular we show sharp
Hausdorff estimates which imply local finiteness of the perimeter of the region
and a.e. weak differentiability property of
.Comment: Paper from D. Araujo's Ph.D. thesis, distinguished at the 2013 Carlos
Gutierrez prize for best thesis, Archive for Rational Mechanics and Analysis
201
Schauder a priori estimates and regularity of solutions to boundary-degenerate elliptic linear second-order partial differential equations
We establish Schauder a priori estimates and regularity for solutions to a
class of boundary-degenerate elliptic linear second-order partial differential
equations. Furthermore, given a smooth source function, we prove regularity of
solutions up to the portion of the boundary where the operator is degenerate.
Degenerate-elliptic operators of the kind described in our article appear in a
diverse range of applications, including as generators of affine diffusion
processes employed in stochastic volatility models in mathematical finance,
generators of diffusion processes arising in mathematical biology, and the
study of porous media.Comment: 58 pages, 1 figure. To appear in the Journal of Differential
Equations. Incorporates final galley proof corrections corresponding to
published versio
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