1,720,995 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
A note on a Sobolev inequality with a remainder term for functions vanishing on some part of the boundary
No full textIn this paper we prove a Sobolev inequality with a remainder term in dimension n > 2
Long-time behavior for the porous medium equation with small initial energy
No full textWe study the long-time behavior for the solution of the Porous Medium Equation in an open bounded connected set, with smooth boundary. Homogeneous Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution converges to a nontrivial constant sign solution of a sublinear Lane-Emden equation, once suitably rescaled. We point out that the initial datum is allowed to be sign-changing.
We also give a sufficient energetic criterion on the initial datum, which permits to decide whether convergence takes place towards the positive solution or to the negative one
Symmetrization for Fractional Elliptic Problems: A Direct Approach
No full textWe provide new direct methods to establish symmetrization results in the form of a mass concentration (that is, integral) comparison for fractional elliptic equations of the type (-Δ)su=f(0<1) in a bounded domain Ω , equipped with homogeneous Dirichlet boundary conditions. The classical pointwise Talenti rearrangement inequality in [47] is recovered in the limit s→ 1. Finally, explicit counterexamples constructed for all s∈ (0 , 1) highlight that the same pointwise estimate cannot hold in a nonlocal setting, thus showing the optimality of our results
Comparison results for solutions of parabolic equations with a singular potential
No full textWe 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 of a problem whose data are radially symmetric with respect to the space variable
Optimal estimates for Fractional Fast diffusion equations
No full textWe obtain a priori estimates with best constants for the solutions of the fractional fast diffusion equation ut+(-δ)σ/2um=0, posed in the whole space with 0<σ<2, 0<m≤1. The estimates are expressed in terms of convenient norms of the initial data, the preferred norms being the L1-norm and the Marcinkiewicz norm. The estimates contain exact exponents and best constants. We also obtain optimal estimates for the extinction time of the solutions in the range m near 0 where solutions may vanish completely in finite time. Actually, our results apply to equations with a more general nonlinearity. Our main tools are symmetrization techniques and comparison of concentrations. Classical results for σ=2 are recovered in the limit
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