1,721,402 research outputs found
An application of the theorem on Sums to viscosity solutions of degenerate fully nonlinear equations
No full textWe prove Hölder continuous regularity of bounded, uniformly continuous, viscosity solutions of degenerate fully nonlinear equations defined in all of n space. In particular, the result applies also to some operators in Carnot groups
Weyl and Marchaud Derivatives: A Forgotten History
Full text linkIn this paper, we recall the contribution given by Hermann Weyl and André Marchaud to the notion of fractional derivative. In addition, we discuss some relationships between the fractional Laplace operator and Marchaud derivative in the perspective to generalize these objects to different fields of the mathematics
Some Nonlocal Operators in the First Heisenberg Group
Full text linkIn this paper we construct some nonlocal operators in the Heisenberg group. Specifically, starting from the Grünwald-Letnikov derivative and Marchaud derivative in the Euclidean setting, we revisit those definitions with respect to the one of the fractional Laplace operator. Then, we define some nonlocal operators in the non-commutative structure of the first Heisenberg group adapting the approach applied in the Euclidean case to the new framework
The Soap Bubble Theorem and a -Laplacian overdetermined problem
Full text linkWe consider the p-Laplacian equation -Delta(p)u = 1 for 1 < p < 2, on a regular bounded domain Omega subset of R-N, with N >= 2, under homogeneous Dirichlet boundary conditions. In the spirit of Alexandrov's Soap Bubble Theorem and of Serrin's symmetry result for the overdetermined problems, we prove that if the mean curvature H of partial derivative Omega is constant, then Omega is a ball and the unique solution of the Dirichlet p-Laplacian problem is radial. The main tools used are integral identities, the P-function, and the maximum principle
A new glance to the Alt-Caffarelli-Friedman monotonicity formula
Full text linkIn this paper we revisit the proof of the Alt-Caffarelli-Friedman monotonicity formula.
Then, in the framework of the Heisenberg group, we discuss the existence of an analogous
monotonicity formula introducing a necessary condition for its existence, recently proved in [18]
The Fractional Powers of the Sub-Laplacian in Carnot Groups Through an Analytic Continuation
Full text linkIn this paper we construct the fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation approach. In addition, we characterize the powers of the fractional sub-Laplacian in the Heisenberg group, and as a byproduct we compute the k-th order momenta with respect to the heat kernel
Recent Results on Nonlinear Elliptic Free Boundary Problems
Full text linkIn this paper we give an overview of some recent and older results concerning free boundary problems governed by elliptic operators.Fil: Ferrari, Fausto. Universidad de Bologna; ItaliaFil: Lederman, Claudia Beatriz. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Salsa, Sandro. Politecnico di Milano; Itali
Regularity of flat free boundaries for two-phase p(x)-Laplacian problems with right hand side
Full text linkWe consider viscosity solutions to two-phase free boundary problems for the -Laplacian with non-zero right hand side. We prove that flat free boundaries are . No assumption on the Lipschitz continuity of solutions is made.
These regularity results are the first ones in literature for two-phase free boundary problems for the -Laplacian and also for two-phase problems for singular/degenerate
operators with non-zero right hand side. They are new even when , i.e., for the -Laplacian.
The fact that our results hold for merely viscosity solutions allows a wide applicability
Some counterexamples to Alt–Caffarelli–Friedman monotonicity formulas in Carnot groups
Full text linkIn this paper we continue the analysis of an Alt-Caffarelli-Friedman (ACF) monotonicity formula in Carnot groups of step s >1 confirming the existence of counterexamples to the monotone increasing behavior. In particular, we provide a sufficient condition that implies the existence of some counterexamples to the monotone increasing behavior of the ACF formula in Carnot groups.
The main tool is based on the lack of orthogonality of harmonic polynomials in Carnot groups. This paper generalizes the results proved in n Ferrari and Forcillo (Atti Accad
Naz Lincei Rend Lincei Mat Appl 34(2):295–306, 2023)
Regularity of flat free boundaries for a p(x)-Laplacian problem with right hand side
No full textWe consider viscosity solutions to a one-phase free boundary problem for the p(x)-Laplacian with non-zero right hand side. We apply the tools developed in De Silva (2011) to prove that flat free boundaries are C1,α. Moreover, we obtain some new results for the operator under consideration that are of independent interest.Fil: Ferrari, Fausto. Universidad de Bologna; ItaliaFil: Lederman, Claudia Beatriz. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin
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