773 research outputs found
Singularities of the divergence of continuous vector fields and uniform Hausdorff estimates
We prove that every closed set which is not sigma-finite with respect to the
Hausdorff measure H^{N-1} carries singularities of continuous vector fields in
the Euclidean space R^N for the divergence operator. We also show that finite
measures which do not charge sets of sigma-finite Hausdorff measure H^{N-1} can
be written as an L^1 perturbation of the divergence of a continuous vector
field. The main tool is a property of approximation of measures in terms of the
Hausdorff content
Flat solutions of the 1-Laplacian equation
For every defined in an open bounded subset of
, we prove that a solution of the
-Laplacian equation in
satisfies on a set of positive Lebesgue measure. The
same property holds if has small norm in the
Marcinkiewicz space of weak- functions or if is a BV minimizer of
the associated energy functional. The proofs rely on Stampacchia's truncation
method.Comment: Dedicated to Jean Mawhin. Revised and extended version of a note
written by the authors in 201
Strong maximum principle for Schr\"odinger operators with singular potential
We prove that for every and for every potential , any
nonnegative function satisfying in an open connected
set of is either identically zero or its level set
has zero capacity. This gives an affirmative answer to an open
problem of B\'enilan and Brezis concerning a bridge between
Serrin-Stampacchia's strong maximum principle for and
Ancona's strong maximum principle for . The proof is based on the
construction of suitable test functions depending on the level set
and on the existence of solutions of the Dirichlet problem for the
Schr\"odinger operator with diffuse measure data.Comment: 21 page
Schroedinger operators involving singular potentials and measure data
We study the existence of solutions of the Dirichlet problem for the
Schroedinger operator with measure data We characterize the finite measures
for which this problem has a solution for every nonnegative potential
in the Lebesgue space with . The full answer can
be expressed in terms of the capacity for , and the
(or Newtonian) capacity for . We then prove the existence of a solution
of the problem above when belongs to the real Hardy space and
is diffuse with respect to the capacity.Comment: Fixed a display problem in arxiv's abstract. Original tex file
unchange
A note on the fractional perimeter and interpolation
We present the fractional perimeter as a set-function interpolation between
the Lebesgue measure and the perimeter in the sense of De Giorgi. Our
motivation comes from a new fractional Boxing inequality that relates the
fractional perimeter and the Hausdorff content and implies several known
inequalities involving the Gagliardo seminorm of the Sobolev spaces of order
Limit solutions of the Chern-Simons equation
We investigate the scalar Chern-Simons equation in cases where there is no solution for a given nonnegative finite measure
. Approximating by a sequence of nonnegative functions or
finite measures for which this equation has a solution, we show that the
sequence of solutions of the Dirichlet problem converges to the solution with
largest possible datum \mu^# \le \mu and we derive an explicit formula of
\mu^# in terms of . The counterpart for the Chern-Simons system with
datum behaves differently and the conclusion depends on how much
the measures and charge singletons
Strong density for higher order Sobolev spaces into compact manifolds
Given a compact manifold , an integer and an
exponent , we prove that the class of smooth maps on the cube with values into is dense with respect
to the strong topology in the Sobolev space when the
homotopy group of order is
trivial. We also prove the density of maps that are smooth except for a set of
dimension , without any restriction on the homotopy
group of $N^n
The role of interplay between coefficients in the -convergence of some elliptic equations
We study the behavior of the solutions of the linear Dirichlet problems
with respect to perturbations
of the matrix (with respect to the -convergence) and with respect to
perturbations of the nonnegative coefficient and of the right hand side
satisfying the condition
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