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 $\mathcal{M}$ to a compact Riemannian manifold $\mathcal{N}$ 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 $W^{s,p} (\mathcal{M}, \mathcal{N})$, with $sp = m = \dim \mathcal{M}$. In particular, when the fundamental group $\pi_1 (\mathcal{N})$ acts trivially on the homotopy group $\pi_m (\mathcal{N})$, 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 $$\iint\limits_{\substack{(x, y) \in \mathcal{M} \times \mathcal{M} \\ d_\mathcal{N} (f (x), f (y)) \ge \varepsilon}}\frac{1}{d_\mathcal{M} (x, y)^{2 m}} \, \mathrm{d} x \, \mathrm{d} y.$$ When $m \ge 2$, the estimates scale optimally as $\varepsilon \to 0$. Linear estimates on the Hurewicz homorphism and the induced cohomology homomorphism are also obtained.Comment: 45 pages, minor correction

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

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 $$-\Delta u+u=(I_\alpha*F(u))F'(u)\qquad\text{in }\mathbb{R}^2,$$ where $I_\alpha$ is the Riesz potential of order $\alpha$ on the plane $\mathbb{R}^2$ under general nontriviality, growth and subcriticality on the nonlinearity $F \in C^1 (\mathbb{R},\mathbb{R})$.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

Nodal solutions for the Choquard equation

We consider the general Choquard equations $$-\Delta u + u = (I_\alpha \ast |u|^p) |u|^{p - 2} u$$ where $I_\alpha$ is a Riesz potential. We construct minimal action odd solutions for $p \in (\frac{N + \alpha}{N}, \frac{N + \alpha}{N - 2})$ and minimal action nodal solutions for $p \in (2,\frac{N + \alpha}{N - 2})$. We introduce a new minimax principle for least action nodal solutions and we develop new concentration-compactness lemmas for sign-changing Palais--Smale sequences. The nonlinear Schr\"odinger equation, which is the nonlocal counterpart of the Choquard equation, does not have such solutions.Comment: 23 pages, revised version with additional details and symmetry properties of odd solution