45 research outputs found
Partial regularity for manifold constrained p(x)-harmonic maps
We prove that manifold constrained -harmonic maps are
-regular outside a set of zero -dimensional Lebesgue's measure,
for some . We also provide an estimate from above of the
Hausdorff dimension of the singular set
Higher Integrability for Constrained Minimizers of Integral Functionals with (p,q)-Growth in low dimension
We prove higher summability for the gradient of minimizers of strongly convex
integral functionals of the Calculus of Variations with (p,q)-Growth conditions
in low dimension. Our procedure is set in the framework of Fractional Sobolev
Spaces and renders the desired regularity as the result of an approximation
technique relying on estimates obtained through a careful use of difference
quotients.Comment: 22 pages, 0 figure
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
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
Gradient bounds for solutions to irregular parabolic equations with -growth
We provide quantitative gradient bounds for solutions to certain parabolic
equations with unbalanced polynomial growth and non-smooth coefficients.Comment: 27 page
Regularity for solutions of fully nonlinear elliptic equations with non-homogeneous degeneracy
We prove that viscosity solutions to fully nonlinear elliptic equations with
degeneracy of double phase type are locally -regular.Comment: 19 page
Regularity results for a class of non-autonomous obstacle problems with -growth
We study some regularity issues for solutions of non-autonomous obstacle
problems with -growth. Under suitable assumptions, our analysis covers
the main models available in the literature.Comment: 35 page
Interpolative gap bounds for nonautonomous integrals
For nonautonomous, nonuniformly elliptic integrals with so-called
-growth conditions, we show a general interpolation property allowing to
get basic higher integrability results for H\"older continuous minimizers under
improved bounds for the gap . For this we introduce a new method, based on
approximating the original, local functional, with mixed local/nonlocal ones,
and allowing for suitable estimates in fractional Sobolev spaces.Comment: 25 page