211 research outputs found
Limiting Sobolev inequalities for vector fields and canceling linear differential operators
The estimate [\lVert D^{k-1}u\rVert_{L^{n/(n-1)}} \le \lVert A(D)u
\rVert_{L^1} ] is shown to hold if and only if (A(D)) is elliptic and
canceling. Here (A(D)) is a homogeneous linear differential operator (A(D)) of
order (k) on (\mathbb{R}^n) from a vector space (V) to a vector space (E). The
operator (A(D)) is defined to be canceling if [\bigcap_{\xi \in \mathbb{R}^n
\setminus {0}} A(\xi)[V]={0}.] This result implies in particular the classical
Gagliardo-Nirenberg-Sobolev inequality, the Korn-Sobolev inequality and
Hodge-Sobolev estimates for differential forms due to J. Bourgain and H.
Brezis. In the proof, the class of cocanceling homogeneous linear differential
operator (L(D)) of order (k) on (\mathbb{R}^n) from a vector space (E) to a
vector space (F) is introduced. It is proved that (L(D)) is cocanceling if and
only if for every (f \in L^1(\mathbb{R}^n; E)) such that (L(D)f=0), one has (f
\in \dot{W}^{-1, n/(n-1)}(\mathbb{R}^n; E)). The results extend to fractional
and Lorentz spaces and can be strengthened using some tools of J. Bourgain and
H. Brezis.Comment: 40 pages, incorporated corrections suggested by the refere
Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps
A free homotopy decomposition of any continuous map from a compact Riemmanian
manifold to a compact Riemannian manifold into a
finite number maps belonging to a finite set is constructed, in such a way that
the number of maps in this free homotopy decomposition and the number of
elements of the set to which they belong can be estimated a priori by the
critical Sobolev energy of the map in ,
with . In particular, when the fundamental group
acts trivially on the homotopy group , the number of homotopy classes to which a map can belong can be
estimated by its Sobolev energy. The estimates are particular cases of
estimates under a boundedness assumption on gap potentials of the form
When , the estimates scale
optimally as . Linear estimates on the Hurewicz homorphism
and the induced cohomology homomorphism are also obtained.Comment: 45 pages, minor correction
Interpolation inequalities between Sobolev and Morrey-Campanato spaces: A common gateway to concentration-compactness and Gagliardo-Nirenberg interpolation inequalities
We prove interpolation estimates between Morrey-Campanato spaces and Sobolev
spaces. These estimates give in particular concentration-compactness
inequalities in the translation-invariant and in the translation- and
dilation-invariant case. They also give in particular interpolation estimates
between Sobolev spaces and functions of bounded mean oscillation. The proofs
rely on Sobolev integral representation formulae and maximal function theory.
Fractional Sobolev spaces are also covered.Comment: 12 page
Explicit approximation of the symmetric rearrangement by polarizations
We give an explicit sequence of polarizations such that for every measurable
function, the sequence of iterated polarizations converge to the symmetric
rearrangement of the initial function.Comment: 10 page
Subelliptic Bourgain-Brezis Estimates on Groups
We show that divergence free vector fields which belong to L^1 on stratified,
nilpotent groups are in the dual space of functions whose sub-gradient are L^Q
integrable where Q is the homogeneous dimension of the group. This was first
obtained on Euclidean space by Bourgain and Brezis.Comment: 15 pages, v2 has some typos fixed in lemma 2.
Existence of groundstates for a class of nonlinear Choquard equations in the plane
We prove the existence of a nontrivial groundstate solution for the class of
nonlinear Choquard equation where is the Riesz potential of order on
the plane under general nontriviality, growth and subcriticality
on the nonlinearity .Comment: revised version, 16 page
Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics
We consider a semilinear elliptic problem [- \Delta u + u = (I_\alpha \ast
\abs{u}^p) \abs{u}^{p - 2} u \quad\text{in (\mathbb{R}^N),}] where (I_\alpha)
is a Riesz potential and (p>1). This family of equations includes the Choquard
or nonlinear Schr\"odinger-Newton equation. For an optimal range of parameters
we prove the existence of a positive groundstate solution of the equation. We
also establish regularity and positivity of the groundstates and prove that all
positive groundstates are radially symmetric and monotone decaying about some
point. Finally, we derive the decay asymptotics at infinity of the
groundstates.Comment: 23 pages, updated bibliograph
Nonlocal Hardy type inequalities with optimal constants and remainder terms
Using a groundstate transformation, we give a new proof of the optimal
Stein-Weiss inequality of Herbst [\int_{\R^N} \int_{\R^N} \frac{\varphi
(x)}{\abs{x}^\frac{\alpha}{2}} I_\alpha (x - y) \frac{\varphi
(y)}{\abs{y}^\frac{\alpha}{2}}\dif x \dif y \le \mathcal{C}_{N,\alpha,
0}\int_{\R^N} \abs{\varphi}^2,] and of its combinations with the Hardy
inequality by Beckner [\int_{\R^N} \int_{\R^N} \frac{\varphi
(x)}{\abs{x}^\frac{\alpha + s}{2}} I_\alpha (x - y) \frac{\varphi
(y)}{\abs{y}^\frac{\alpha + s}{2}}\dif x \dif y \le \mathcal{C}_{N, \alpha, 1}
\int_{\R^N} \abs{\nabla \varphi}^2,] and with the fractional Hardy inequality
[\int_{\R^N} \int_{\R^N} \frac{\varphi (x)}{\abs{x}^\frac{\alpha + s}{2}}
I_\alpha (x - y) \frac{\varphi (y)}{\abs{y}^\frac{\alpha + s}{2}}\dif x \dif y
\le \mathcal{C}_{N, \alpha, s} \mathcal{D}_{N, s} \int_{\R^N} \int_{\R^N}
\frac{\bigabs{\varphi (x) - \varphi (y)}^2}{\abs{x-y}^{N+s}}\dif x \dif y]
where (I_\alpha) is the Riesz potential, (0 < \alpha < N) and (0 < s < \min(N,
2)). We also prove the optimality of the constants. The method is flexible and
yields a sharp expression for the remainder terms in these inequalities.Comment: 9 page
Approximation of symmetrizations by Markov processes
Under continuity and recurrence assumptions, we prove that the iteration of
successive partial symmetrizations that form a time-homogeneous Markov process,
converges to a symmetrization. We cover several settings, including the
approximation of the spherical nonincreasing rearrangement by Steiner
symmetrizations, polarizations and cap symmetrizations. A key tool in our
analysis is a quantitative measure of the asymmetry
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