1,010 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
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
Free boundary problems for Tumor Growth: a Viscosity solutions approach
The mathematical modeling of tumor growth leads to singular stiff pressure
law limits for porous medium equations with a source term. Such asymptotic
problems give rise to free boundaries, which, in the absence of active motion,
are generalized Hele-Shaw flows. In this note we use viscosity solutions
methods to study limits for porous medium-type equations with active motion. We
prove the uniform convergence of the density under fairly general assumptions
on the initial data, thus improving existing results. We also obtain some
additional information/regularity about the propagating interfaces, which, in
view of the discontinuities, can nucleate and, thus, change topological type.
The main tool is the construction of local, smooth, radial solutions which
serve as barriers for the existence and uniqueness results as well as to
quantify the speed of propagation of the free boundary propagation
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
H^s versus C^0-weighted minimizers
We study a class of semi-linear problems involving the fractional Laplacian
under subcritical or critical growth assumptions. We prove that, for the
corresponding functional, local minimizers with respect to a C^0-topology
weighted with a suitable power of the distance from the boundary are actually
local minimizers in the natural H^s-topology.Comment: 15 page
Remarks on the KLS conjecture and Hardy-type inequalities
We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary
functions on a convex body , not necessarily
vanishing on the boundary . This reduces the study of the
Neumann Poincar\'e constant on to that of the cone and Lebesgue
measures on ; these may be bounded via the curvature of
. A second reduction is obtained to the class of harmonic
functions on . We also study the relation between the Poincar\'e
constant of a log-concave measure and its associated K. Ball body
. In particular, we obtain a simple proof of a conjecture of
Kannan--Lov\'asz--Simonovits for unit-balls of , originally due to
Sodin and Lata{\l}a--Wojtaszczyk.Comment: 18 pages. Numbering of propositions, theorems, etc.. as appeared in
final form in GAFA seminar note
The one-dimensional Keller-Segel model with fractional diffusion of cells
We investigate the one-dimensional Keller-Segel model where the diffusion is
replaced by a non-local operator, namely the fractional diffusion with exponent
. We prove some features related to the classical
two-dimensional Keller-Segel system: blow-up may or may not occur depending on
the initial data. More precisely a singularity appears in finite time when
and the initial configuration of cells is sufficiently concentrated.
On the opposite, global existence holds true for if the initial
density is small enough in the sense of the norm.Comment: 12 page
Repeated games for eikonal equations, integral curvature flows and non-linear parabolic integro-differential equations
The main purpose of this paper is to approximate several non-local evolution
equations by zero-sum repeated games in the spirit of the previous works of
Kohn and the second author (2006 and 2009): general fully non-linear parabolic
integro-differential equations on the one hand, and the integral curvature flow
of an interface (Imbert, 2008) on the other hand. In order to do so, we start
by constructing such a game for eikonal equations whose speed has a
non-constant sign. This provides a (discrete) deterministic control
interpretation of these evolution equations. In all our games, two players
choose positions successively, and their final payoff is determined by their
positions and additional parameters of choice. Because of the non-locality of
the problems approximated, by contrast with local problems, their choices have
to "collect" information far from their current position. For integral
curvature flows, players choose hypersurfaces in the whole space and positions
on these hypersurfaces. For parabolic integro-differential equations, players
choose smooth functions on the whole space
Strong solutions of the thin film equation in spherical geometry
We study existence and long-time behaviour of strong solutions for the thin
film equation using a priori estimates in a weighted Sobolev space. This
equation can be classified as a doubly degenerate fourth-order parabolic and it
models coating flow on the outer surface of a sphere. It is shown that the
strong solution asymptotically decays to the flat profile
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