989 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
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
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
Nonlinear porous medium flow with fractional potential pressure
We study a porous medium equation, with nonlocal diffusion effects given by
an inverse fractional Laplacian operator. We pose the problem in n-dimensional
space for all t>0 with bounded and compactly supported initial data, and prove
existence of a weak and bounded solution that propagates with finite speed, a
property that is nor shared by other fractional diffusion models.Comment: 32 pages, Late
A logarithmic epiperimetric inequality for the obstacle problem
For the general obstacle problem, we prove by direct methods an epiperimetric
inequality at regular and singular points, thus answering a question of Weiss
(Invent. Math., 138 (1999), 23--50). In particular at singular points we
introduce a new tool, which we call logarithmic epiperimetric inequality, which
yields an explicit logarithmic modulus of continuity on the regularity of
the singular set, thus improving previous results of Caffarelli and Monneau
Optimal regularity and structure of the free boundary for minimizers in cohesive zone models
We study optimal regularity and free boundary for minimizers of an energy
functional arising in cohesive zone models for fracture mechanics. Under
smoothness assumptions on the boundary conditions and on the fracture energy
density, we show that minimizers are , and that near non-degenerate
points the fracture set is , for some .Comment: 39 page
A rigorous analysis using optimal transport theory for a two-reflector design problem with a point source
We consider the following geometric optics problem: Construct a system of two
reflectors which transforms a spherical wavefront generated by a point source
into a beam of parallel rays. This beam has a prescribed intensity
distribution. We give a rigorous analysis of this problem. The reflectors we
construct are (parts of) the boundaries of convex sets. We prove existence of
solutions for a large class of input data and give a uniqueness result. To the
author's knowledge, this is the first time that a rigorous mathematical
analysis of this problem is given. The approach is based on optimal
transportation theory. It yields a practical algorithm for finding the
reflectors. Namely, the problem is equivalent to a constrained linear
optimization problem.Comment: 5 Figures - pdf files attached to submission, but not shown in
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