45 research outputs found

    Partial regularity for manifold constrained p(x)-harmonic maps

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    We prove that manifold constrained p(x)p(x)-harmonic maps are C1,βC^{1,\beta}-regular outside a set of zero nn-dimensional Lebesgue's measure, for some β∈(0,1)\beta \in (0,1). 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

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    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

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    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

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    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 (p,q)(p,q)-growth

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    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

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    We prove that viscosity solutions to fully nonlinear elliptic equations with degeneracy of double phase type are locally C1,γC^{1,\gamma}-regular.Comment: 19 page

    Regularity results for a class of non-autonomous obstacle problems with (p,q)(p,q)-growth

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    We study some regularity issues for solutions of non-autonomous obstacle problems with (p,q)(p,q)-growth. Under suitable assumptions, our analysis covers the main models available in the literature.Comment: 35 page

    Interpolative gap bounds for nonautonomous integrals

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    For nonautonomous, nonuniformly elliptic integrals with so-called (p,q)(p,q)-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 q/pq/p. 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
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