1,407 research outputs found
Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian
We use a characterization of the fractional Laplacian as a Dirichlet to
Neumann operator for an appropriate differential equation to study its obstacle
problem. We write an equivalent characterization as a thin obstacle problem. In
this way we are able to apply local type arguments to obtain sharp regularity
estimates for the solution and study the regularity of the free boundary
Traveling Wave Solutions of Advection-Diffusion Equations with Nonlinear Diffusion
We study the existence of particular traveling wave solutions of a nonlinear
parabolic degenerate diffusion equation with a shear flow. Under some
assumptions we prove that such solutions exist at least for propagation speeds
c {\in}]c*, +{\infty}, where c* > 0 is explicitly computed but may not be
optimal. We also prove that a free boundary hy- persurface separates a region
where u = 0 and a region where u > 0, and that this free boundary can be
globally parametrized as a Lipschitz continuous graph under some additional
non-degeneracy hypothesis; we investigate solutions which are, in the region u
> 0, planar and linear at infinity in the propagation direction, with slope
equal to the propagation speed.Comment: 40 pages, 1 figur
Fractional elliptic equations, Caccioppoli estimates and regularity
Let be a uniformly elliptic operator
in divergence form in a bounded domain . We consider the fractional
nonlocal equations Here , , is the fractional power of and
is the conormal derivative of with respect to the
coefficients . We reproduce Caccioppoli type estimates that allow us to
develop the regularity theory. Indeed, we prove interior and boundary Schauder
regularity estimates depending on the smoothness of the coefficients ,
the right hand side and the boundary of the domain. Moreover, we establish
estimates for fundamental solutions in the spirit of the classical result by
Littman--Stampacchia--Weinberger and we obtain nonlocal integro-differential
formulas for . Essential tools in the analysis are the semigroup
language approach and the extension problem.Comment: 37 page
Logarithmically-concave moment measures I
We discuss a certain Riemannian metric, related to the toric Kahler-Einstein
equation, that is associated in a linearly-invariant manner with a given
log-concave measure in R^n. We use this metric in order to bound the second
derivatives of the solution to the toric Kahler-Einstein equation, and in order
to obtain spectral-gap estimates similar to those of Payne and Weinberger.Comment: 27 page
Global estimates for solutions to the linearized Monge--Amp\`ere equations
In this paper, we establish global estimates for solutions to the
linearized Monge-Amp\`ere equations under natural assumptions on the domain,
Monge-Amp\`ere measures and boundary data. Our estimates are affine invariant
analogues of the global estimates of Winter for fully nonlinear,
uniformly elliptic equations, and also linearized counterparts of Savin's
global estimates for the Monge-Amp\`ere equations.Comment: v2: presentation improve
The two membranes problem for different operators
We study the two membranes problem for different operators, possibly
nonlocal. We prove a general result about the H\"older continuity of the
solutions and we develop a viscosity solution approach to this problem. Then we
obtain regularity of the solutions provided that the orders of
the two operators are different. In the special case when one operator
coincides with the fractional Laplacian, we obtain the optimal regularity and a
characterization of the free boundary
Transference of fractional Laplacian regularity
In this note we show how to obtain regularity estimates for the fractional
Laplacian on the multidimensional torus from the fractional
Laplacian on . Though at first glance this may seem quite
natural, it must be carefully precised. A reason for that is the simple fact
that functions on the torus can not be identified with functions on
. The transference is achieved through a formula that holds in
the distributional sense. Such an identity allows us to transfer Harnack
inequalities, to relate the extension problems, and to obtain pointwise
formulas and H\"older regularity estimates.Comment: 7 pages. To appear in Special Functions, Partial Differential
Equations and Harmonic Analysis. Proceedings of the conference in honor of
Calixto P. Calder\'on, Roosevelt University at Chicago, November 16-18, 2012.
C. Georgakis, A. Stokolos and W. Urbina (eds
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