380 research outputs found
Extremal Lipschitz functions in the deviation inequalities from the mean
We obtain an optimal deviation from the mean upper bound \begin{equation}
D(x)\=\sup_{f\in \F}\mu\{f-\E_{\mu} f\geq x\},\qquad\ \text{for}\
x\in\R\label{abstr} \end{equation} where \F is the class of the integrable,
Lipschitz functions on probability metric (product) spaces. As corollaries we
get exact solutions of \eqref{abstr} for Euclidean unit sphere with
a geodesic distance and a normalized Haar measure, for equipped with a
Gaussian measure and for the multidimensional cube, rectangle, torus or Diamond
graph equipped with uniform measure and Hamming distance. We also prove that in
general probability metric spaces the in \eqref{abstr} is achieved on
a family of distance functions.Comment: 7 page
Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes
We study energy functionals obtained by adding a possibly discontinuous
potential to an interaction term modeled upon a Gagliardo-type fractional
seminorm. We prove that minimizers of such non-differentiable functionals are
locally bounded, H\"older continuous, and that they satisfy a suitable Harnack
inequality. Hence, we provide an extension of celebrated results of M.
Giaquinta and E. Giusti to the nonlocal setting. To do this, we introduce a
particular class of fractional Sobolev functions, reminiscent of that
considered by E. De Giorgi in his seminal paper of 1957. The flexibility of
these classes allows us to also establish regularity of solutions to rather
general nonlinear integral equations.Comment: 59 page
Minimizers for nonlocal perimeters of Minkowski type
We study a nonlocal perimeter functional inspired by the Minkowski content,
whose main feature is that it interpolates between the classical perimeter and
the volume functional. This problem is related by a generalized coarea formula
to a Dirichlet energy functional in which the energy density is the local
oscillation of a function.
These two nonlocal functionals arise in concrete applications, since the
nonlocal character of the problems and the different behaviors of the energy at
different scales allow the preservation of details and irregularities of the
image in the process of removing white noises, thus improving the quality of
the image without losing relevant features.
In this paper, we provide a series of results concerning existence, rigidity
and classification of minimizers, compactness results, isoperimetric
inequalities, Poincar\'e-Wirtinger inequalities and density estimates.
Furthermore, we provide the construction of planelike minimizers for this
generalized perimeter under a small and periodic volume perturbation.Comment: To appear in Calc. Var. Partial Differential Equation
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