1,556 research outputs found
Bounded correctors in almost periodic homogenization
We show that certain linear elliptic equations (and systems) in divergence
form with almost periodic coefficients have bounded, almost periodic
correctors. This is proved under a new condition we introduce which quantifies
the almost periodic assumption and includes (but is not restricted to) the
class of smooth, quasiperiodic coefficient fields which satisfy a
Diophantine-type condition previously considered by Kozlov. The proof is based
on a quantitative ergodic theorem for almost periodic functions combined with
the new regularity theory recently introduced by the first author and Shen for
equations with almost periodic coefficients. This yields control on spatial
averages of the gradient of the corrector, which is converted into estimates on
the size of the corrector itself via a multiscale Poincar\'e-type inequality.Comment: 30 pages, minor revision. To appear in Arch. Ration. Mech. Ana
Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations
We deal with a class of equations driven by nonlocal, possibly degenerate,
integro-differential operators of differentiability order and
summability growth , whose model is the fractional -Laplacian with
measurable coefficients. We state and prove several results for the
corresponding weak supersolutions, as comparison principles, a priori bounds,
lower semicontinuity, and many others. We then discuss the good definition of
-superharmonic functions, by also proving some related properties. We
finally introduce the nonlocal counterpart of the celebrated Perron method in
nonlinear Potential Theory.Comment: To appear in Math. An
Borderline gradient continuity of minima
The gradient of any local minimiser of functionals of the type where has -growth,
, and , is continuous provided the optimal
Lorentz space condition is satisfied and is
suitably Dini-continuous.Comment: 30 page
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
Mesoscopic higher regularity and subadditivity in elliptic homogenization
We introduce a new method for obtaining quantitative results in stochastic
homogenization for linear elliptic equations in divergence form. Unlike
previous works on the topic, our method does not use concentration inequalities
(such as Poincar\'e or logarithmic Sobolev inequalities in the probability
space) and relies instead on a higher (, ) regularity theory
for solutions of the heterogeneous equation, which is valid on length scales
larger than a certain specified mesoscopic scale. This regularity theory, which
is of independent interest, allows us to, in effect, localize the dependence of
the solutions on the coefficients and thereby accelerate the rate of
convergence of the expected energy of the cell problem by a bootstrap argument.
The fluctuations of the energy are then tightly controlled using subadditivity.
The convergence of the energy gives control of the scaling of the spatial
averages of gradients and fluxes (that is, it quantifies the weak convergence
of these quantities) which yields, by a new "multiscale" Poincar\'e inequality,
quantitative estimates on the sublinearity of the corrector.Comment: 44 pages, revised version, to appear in Comm. Math. Phy
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