155 research outputs found
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
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
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
Geometric approach to nonvariational singular elliptic equations
In this work we develop a systematic geometric approach to study fully
nonlinear elliptic equations with singular absorption terms as well as their
related free boundary problems. The magnitude of the singularity is measured by
a negative parameter , for , which reflects on
lack of smoothness for an existing solution along the singular interface
between its positive and zero phases. We establish existence as well sharp
regularity properties of solutions. We further prove that minimal solutions are
non-degenerate and obtain fine geometric-measure properties of the free
boundary . In particular we show sharp
Hausdorff estimates which imply local finiteness of the perimeter of the region
and a.e. weak differentiability property of
.Comment: Paper from D. Araujo's Ph.D. thesis, distinguished at the 2013 Carlos
Gutierrez prize for best thesis, Archive for Rational Mechanics and Analysis
201
Mass transportation with LQ cost functions
We study the optimal transport problem in the Euclidean space where the cost
function is given by the value function associated with a Linear Quadratic
minimization problem. Under appropriate assumptions, we generalize Brenier's
Theorem proving existence and uniqueness of an optimal transport map. In the
controllable case, we show that the optimal transport map has to be the
gradient of a convex function up to a linear change of coordinates. We give
regularity results and also investigate the non-controllable case
Partial Schauder estimates for second-order elliptic and parabolic equations
We establish Schauder estimates for both divergence and non-divergence form
second-order elliptic and parabolic equations involving H\"older semi-norms not
with respect to all, but only with respect to some of the independent
variables.Comment: CVPDE, accepted (2010)
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
Stability of flows associated to gradient vector fields and convergence of iterated transport maps
In this paper we address the problem of stability of flows
associated to a sequence of vector fields under minimal regularity requirements
on the limit vector field, that is supposed to be a gradient. We apply this
stability result to show the convergence of iterated compositions of optimal
transport maps arising in the implicit time discretization (with respect to the
Wasserstein distance) of nonlinear evolution equations of a diffusion type.
Finally, we use these convergence results to study the gradient flow of a
particular class of polyconvex functionals recently considered by Gangbo, Evans
ans Savin. We solve some open problems raised in their paper and obtain
existence and uniqueness of solutions under weaker regularity requirements and
with no upper bound on the jacobian determinant of the initial datum
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
Positive Least Energy Solutions and Phase Separation for Coupled Schrodinger Equations with Critical Exponent: Higher Dimensional Case
We study the following nonlinear Schr\"{o}dinger system which is related to
Bose-Einstein condensate: {displaymath} {cases}-\Delta u +\la_1 u = \mu_1
u^{2^\ast-1}+\beta u^{\frac{2^\ast}{2}-1}v^{\frac{2^\ast}{2}}, \quad x\in
\Omega, -\Delta v +\la_2 v =\mu_2 v^{2^\ast-1}+\beta v^{\frac{2^\ast}{2}-1}
u^{\frac{2^\ast}{2}}, \quad x\in \om, u\ge 0, v\ge 0 \,\,\hbox{in \om},\quad
u=v=0 \,\,\hbox{on \partial\om}.{cases}{displaymath} Here \om\subset \R^N
is a smooth bounded domain, is the Sobolev critical
exponent, -\la_1(\om)0 and , where
\lambda_1(\om) is the first eigenvalue of with the Dirichlet
boundary condition. When \bb=0, this is just the well-known Brezis-Nirenberg
problem. The special case N=4 was studied by the authors in (Arch. Ration.
Mech. Anal. 205: 515-551, 2012). In this paper we consider {\it the higher
dimensional case }. It is interesting that we can prove the existence
of a positive least energy solution (u_\bb, v_\bb) {\it for any } (which can not hold in the special case N=4). We also study the limit
behavior of (u_\bb, v_\bb) as and phase separation is
expected. In particular, u_\bb-v_\bb will converge to {\it sign-changing
solutions} of the Brezis-Nirenberg problem, provided . In case
\la_1=\la_2, the classification of the least energy solutions is also
studied. It turns out that some quite different phenomena appear comparing to
the special case N=4.Comment: 48 pages. This is a revised version of arXiv:1209.2522v1 [math.AP
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