1,721,327 research outputs found
Semilinear elliptic equations with degenerate and singular weights related to Caffarelli-Kohn-Nirenberg inequalities
No full textIn this note we give some existence and nonexistence results of solutions to a problem of the type -div(vertical bar x vertical bar(-2 gamma) del u) = lambda/vertical bar x vertical bar(2(gamma+1)) u + u(p)/vertical bar x vertical bar(alpha) + f/vertical bar x vertical bar(2 gamma) in Omega u >= 0, u not equivalent to 0 in Omega (P-t,P-p) u = 0 on partial derivative Omega, where Omega is an open bounded subset of R-N containing the origin, the constants p, t. alpha, gamma, lambda satisfy suitable conditions and f not equivalent to 0 is a nonnegative, smooth bounded function on Omega. The results that will be given generalize some known results in Brezis et al. (2005) [1] and Dupaigne (2002) [2]. (C) 2012 Elsevier Inc. All rights reserved
Symmetrization for fractional Neumann problems
No full textIn this paper we complement the program concerning the application of symmetrization methods to nonlocal PDEs by providing new estimates, in the sense of mass concentration comparison, for solutions to linear fractional elliptic and parabolic PDEs with Neumann boundary conditions. These results are achieved by employing suitable symmetrization arguments to the Stinga–Torrea local extension problems, corresponding to the fractional boundary value problems considered. Sharp estimates are obtained first for elliptic equations and a certain number of consequences in terms of regularity estimates is then established. Finally, a parabolic symmetrization result is covered as an application of the elliptic concentration estimates in the implicit time discretization scheme
Handles in graphs and sphere-bundles over S^1
No full textWe extend to dimension n the concept of "combinatorial handle". Them we study the operation of cancelling of such a handle, which always reveals a connected sum decomposition of the represented manifold
Comparison results for solutions of nonlinear parabolic equations
No full textWe prove a comparison result for the solutions of Cauchy–Dirichlet problems for a nonlinear parabolic equation. Using Schwarz spherical symmetrization, we compare the concentration of solutions to such problems with the concentration of solutions to conveniently symmetrized problems. The result takes into account, in a sharp form, the influence of the zero-order term
Bourgain-Brézis-Mironescu formula for magnetic operators
Full text linkWe prove a Bourgain-Brézis-Mironescu-type formula for a class of nonlocal magnetic spaces, which builds a bridge between a fractional magnetic operator recently introduced and the classical theory
SYMMETRIZATION FOR FRACTIONAL NONLINEAR ELLIPTIC PROBLEMS
No full textIn this note we prove a new symmetrization result, in the form of mass concentration comparison, for solutions of nonlocal nonlinear Dirichlet problems involving fractional p Laplacians. Some regularity estimates of solutions will be established as a direct application of the main result
Comparison results for solutions of parabolic equations with a singular potential
Full text linkWe consider the solution u of the Cauchy-Dirichlet problem for a class of linear parabolic equations in which the coefficient of the zero order term could have a singularity at the origin of the type 1/|x|^2 . We prove that u can be compared “in the sense of rearrangements” with the solution v of a problem whose data are radially symmetric with respect to the space variable.<br /
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