122 research outputs found
The Calder\'on-Zygmund theory for elliptic problems with measure data
We consider non-linear elliptic equations having a measure in the right hand
side, of the type \divo a(x,Du)=\mu, and prove differentiability and
integrability results for solutions. New estimates in Marcinkiewicz spaces are
also given, and the impact of the measure datum density properties on the
regularity of solutions is analyzed in order to build a suitable
Calder\'on-Zygmund theory for the problem. All the regularity results presented
in this paper are provided together with explicit local a priori estimates
Optimal Lipschitz criteria and local estimates for non-uniformly elliptic problems
We report on new techniques and results in the regularity theory of general
non-uniformly elliptic variational integrals. By means of a new potential
theoretic approach we reproduce, in the non-uniformly elliptic setting, the
optimal criteria for Lipschitz continuity known in the uniformly elliptic one
and provide a unified approach between non-uniformly and uniformly elliptic
problems
Manifold constrained non-uniformly elliptic problems
We consider the problem of minimizing variational integrals defined on
\cc{nonlinear} Sobolev spaces of competitors taking values into the sphere. The
main novelty is that the underlying energy features a non-uniformly elliptic
integrand exhibiting different polynomial growth conditions and no homogeneity.
We develop a few intrinsic methods aimed at proving partial regularity of
minima and providing techniques for treating larger classes of similar
constrained non-uniformly elliptic variational problems. In order to give
estimates for the singular sets we use a general family of Hausdorff type
measures following the local geometry of the integrand. A suitable comparison
is provided with respect to the naturally associated capacities.Comment: 50 page
On the regularity of minima of non-autonomous functionals
We consider regularity issues for minima of non-autonomous functionals in the
Calculus of Variations exhibiting non-uniform ellipticity features. We provide
a few sharp regularity results for local minimizers that also cover the case of
functionals with nearly linear growth. The analysis is carried out provided
certain necessary approximation-in-energy conditions are satisfied. These are
related to the occurrence of the so-called Lavrentiev phenomenon that that
non-autonomous functionals might exhibit, and which is a natural obstruction to
regularity. In the case of vector valued problems we concentrate on higher
gradient integrability of minima. Instead, in the scalar case, we prove local
Lipschitz estimates. We also present an approach via a variant of Moser's
iteration technique that allows to reduce the analysis of several non-uniformly
elliptic problems to that for uniformly elliptic ones.Comment: 32 page
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
Regularity for general functionals with double phase
We prove sharp regularity results for a general class of functionals of the
type featuring non-standard growth
conditions and non-uniform ellipticity properties. The model case is given by
the double phase integral with . This changes its ellipticity rate according to the geometry of the level set
of the modulating coefficient . We also present new
methods and proofs, that are suitable to build regularity theorems for larger
classes of non-autonomous functionals. Finally, we disclose some new
interpolation type effects that, as we conjecture, should draw a general
phenomenon in the setting of non-uniformly elliptic problems. Such effects
naturally connect with the Lavrentiev phenomenon
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
Guide to nonlinear potential estimates
One of the basic achievements in nonlinear potential theory is that the typical linear pointwise estimates via fundamental solutions find a precise analog in the case of nonlinear equations. We give a comprehensive account of this fact and prove new unifying families of potential estimates. We also describe new fine properties of solutions to measure data problems
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