Daniel Faraco

Lack of Uniqueness for Weak Solutions of the Incompressible Porous Media Equation

In this work we consider weak solutions of the incompressible two-dimensional porous media (IPM) equation. By using the approach of De Lellis–Székelyhidi, we prove non-uniqueness for solutions in L∞ in space and time.

Milton's conjecture on the regularity of solutions to isotropic equations

Abstract We present examples showing that the threshold for the integrability of the gradient of solutions to isotropic equations is 2K/(K−1). The main tools are p-laminates and Beltrami Operators.

Beltrami equations with coefficient in the Sobolev space W1,p

We study the removable singularities for solutions to the Beltrami equation Əƒ = μ Əƒ, where μ is a bounded function, ||μ|| ∞ ≤ k-1/k+1 < 1, and such that μ ∈ W1,p for some p ≤ 2. Our results are based on an extended version of the well known Weyl’s lemma, asserting that distributional solutions are actually true solutions. Our main result is that quasiconformal mappings with compactly supported Beltrami coefficient μ ∈ W1,p, 2k2/k2+1 < p ≤ 2, preserve compact sets of σ-finite length and vanishing analytic capacity, even though they need not be bilipschitz.

A short proof of the self-improving regularity of quasiregular mappings

. The theoryof quasiregular mappings is a central topic in modern analysis withimportant connections to a variety of topics as elliptic partial differen-tial equations, complex dynamics, differential geometry and calculus ofvariations [13] [10].A remarkable feature of quasiregular mappings is the self-improvingregularity. In 1957 [2], Bojarski proved that for planar quasiregularmappings, there exists an exponent

Obstructions to the existence of limiting Carleman weights

We give a necessary condition for a Riemannian manifold to admit limiting Carleman weights in terms of the Weyl tensor (in dimensions 4 and higher) and the Cotton-York tensor in dimension 3. As an application we provide explicit examples of manifolds without limiting Carleman weights and show that the set of such metrics on a given manifold contains an open and dense set.

RECONSTRUCTION OF DISCONTINUITIES FROM BACKSCATTERING DATA IN TWO DIMENSIONS

We prove that the nonsmooth part of a noncompactly supported potential

On Plancherel's identity for a two-dimensional scattering transform

q

Nonlinear Beltrami operators, Schauder estimates and bounds for the Jacobian☆

in the Schrodinger Hamiltonian

A new approach to counterexamples to L 1 estimates: Korn's inequality, geometric rigidity, and regularity for gradients of separately convex functions

-\Delta+q

Mappings of finite distortion:The degree of regularity.

, in dimension