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Jean Van Schaftingen

Publications and source records attributed to Jean Van Schaftingen.

At least 91 records · Page 5Linked to original sources
arXiv2015

Strong density for higher order Sobolev spaces into compact manifolds

Given a compact manifold $N^n$, an integer $k \in \mathbb{N}_*$ and an exponent $1 \le p < \infty$, we prove that the class $C^\infty(\overline{Q}^m; N^n)$ of smooth maps on the cube with values into $N^n$ is dense with respect to the strong topology in the Sobolev space $W^{k, p}(Q^m; N^n)$ when the homotopy group $π_{\lfloor kp \rfloor}(N^n)$ of order $\lfloor kp \rfloor$ is trivial. We also prove the density of maps that are smooth except for a set of dimension $m - \lfloor kp \rfloor - 1$, without any restriction on the homotopy group of $N^n$

math.FA↗
arXiv2015

Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field

We construct solutions to the nonlinear magnetic Schrödinger equation $$ \left\{ \begin{aligned} - \varepsilon^2 Δ_{A/\varepsilon^2} u + V u &= \lvert u\rvert^{p-2} u & &\text{in}\ Ω,\\ u &= 0 & &\text{on}\ \partialΩ, \end{aligned} \right. $$ in the semiclassical régime with strong magnetic fields. In contrast with the well-studied mild magnetic field régime, the limiting energy depends on the magnetic field allowing to recover the Lorentz force in the semi-classical limit. Our solutions concentrate around global or local minima of a limiting energy that depends on the electric potential and the magnetic field. The results cover unbounded domains, fast-decaying electric potential and unbounded electromagnetic fields. The construction is variational and is based on an asymptotic analysis of solutions to a penalized problem in the spirit of M. del Pino and P. Felmer.

math.AP↗
arXiv2015

Geometric partial differentiability on manifolds: the tangential derivative and the chain rule

We define the tangential derivative, a notion of directional derivative which is invariant under diffeomorphisms. In particular this derivative is invariant under changes of chart and is thus well-defined for functions defined on a differentiable manifold. This notion is weaker than the classical directional derivative in general and equivalent to the latter for Lipschitz continuous functions. We characterize also the pairs of tangentially differentiable functions for which the chain rule holds.

math.AP↗
arXiv2014

Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent

We consider nonlinear Choquard equation $$ - Δu + V u = \bigl(I_α\ast |u|^{\fracα{N}+1}\bigr) |u|^{\fracα{N}-1} u\quad\text{in (\mathbb{R}^N)},$$ where $N \ge 3$, $V \in L^\infty (\mathbb{R}^N)$ is an external potential and $I_α(x)$ is the Riesz potential of order $α\in (0, N)$. The power $\fracα{N}+1$ in the nonlocal part of the equation is critical with respect to the Hardy-Littlewood-Sobolev inequality. As a consequence, in the associated minimization problem a loss of compactness may occur. We prove that if $\liminf_{|x| \to \infty} \bigl(1 - V (x)\bigr)|x|^2 > \frac{N^2 (N - 2)}{4 (N + 1)}$ then the equation has a nontrivial solution. We also discuss some necessary conditions for the existence of a solution. Our considerations are based on a concentration compactness argument and a nonlocal version of Brezis-Lieb lemma.

math.AP↗
arXiv2014

Stationary solutions of the nonlinear Schrödinger equation with fast-decay potentials concentrating around local maxima

We study positive bound states for the equation $$- ε^2 Δu + Vu = u^p, \qquad \text{in $\mathbf{R}^N$}, $$ where $ε> 0$ is a real parameter, $\frac{N}{N-2} < p < \frac{N+2}{N-2}$ and $V$ is a nonnegative potential. Using purely variational techniques, we find solutions which concentrate at local maxima of the potential $V$ without any restriction on the potential.

math.AP↗
arXiv2014

Density of smooth maps for fractional Sobolev spaces $W^{s, p}$ into $\ell$ simply connected manifolds when $s \ge 1$

Given a compact manifold $N^n \subset \mathbb{R}^ν$, $s \ge 1$ and $1 \le p < \infty$, we prove that the class of smooth maps on the cube with values into $N^n$ is strongly dense in the fractional Sobolev space $W^{s, p}(Q^m; N^n)$ when $N^n$ is $\lfloor sp \rfloor$ simply connected. For $sp$ integer, we prove weak density of smooth maps with values into $N^n$ when $N^n$ is $sp - 1$ simply connected. The proofs are based on the existence of a retraction of $\mathbb{R}^ν$ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s, p} \cap W^{1, sp}$.

math.FA↗
arXiv2013

Intrinsic weak derivatives and Sobolev spaces between manifolds

We define the notion of colocally weakly differentiable maps from a manifold $M$ to a manifold $N$. If $p \ge 1$ and $M$ and $N$ are endowed with a Riemannian metric, this allows us to define intrinsically the homogeneous Sobolev space $\dot{W}^{1, p} (M, N)$. This new definition is equivalent with the definition by embedding in the Euclidean space and with the definition of Sobolev maps into a metric space. The colocal weak derivative is an approximate derivative. The colocal weak differentiability is stable under the suitable weak convergence. The Sobolev spaces can be endowed with various intrinsinc distances that induce the same topology and for which the space is complete.

math.FA↗
arXiv2013

Semi-classical states for the Choquard equation

We study the nonlocal equation $$-\varepsilon^2 Δu_\varepsilon + V u_\varepsilon = \varepsilon^{-α} \bigl(I_α\ast \lvert u_\varepsilon\rvert^p\bigr) \lvert u_\varepsilon \rvert^{p - 2} u_\varepsilon\quad\text{in \(\mathbf{R}^N\)}, $$ where $N \ge 1$, $α\in (0, N)$, $I_α(x) = A_α/\lvert x \rvert^{N - α}$ is the Riesz potential and $\varepsilon > 0$ is a small parameter. We show that if the external potential $V \in C (\mathbb{R}^N; [0, \infty))$ has a local minimum and $p \in [2, (N + α)/(N - 2)_+)$ then for all small $\varepsilon > 0$ the problem has a family of solutions concentrating to the local minimum of $V$ provided that: either $p > 1 + \max (α, \frac{α+ 2}{2})/(N - 2)_+$, or $p > 2$ and $\liminf_{\lvert x\rvert \to \infty} V (x) \lvert x \rvert^2 > 0$, or $p = 2$ and $\inf_{x \in \mathbb{R}^N} V (x) (1 + \lvert x \rvert^{N - α}) > 0$. Our assumptions on the decay of $V$ and admissible range of $p\ge 2$ are optimal. The proof uses variational methods and a novel nonlocal penalization technique that we develop in this work.

math.AP↗
arXiv2013

Strong approximation of fractional Sobolev maps

Brezis and Mironescu have announced several years ago that for a compact manifold $N^n \subset \mathbb{R}^ν$ and for real numbers $0 < s < 1$ and $1 \le p < \infty$ the class $C^\infty(\overline{Q}^m; N^n)$ of smooth maps on the cube with values into $N^n$ is dense with respect to the strong topology in the Sobolev space $W^{s, p}(Q^m; N^n)$ when the homotopy group $π_{\lfloor sp \rfloor}(N^n)$ of order $\lfloor sp \rfloor$ is trivial. The proof of this beautiful result is long and rather involved. Under the additional assumption that $N^n$ is $\lfloor sp \rfloor$ simply connected, we give a shorter proof of their result. Our proof for $sp \ge 1$ is based on the existence of a retraction of $\mathbb{R}^ν$ onto $N^n$ except for a small subset in the complement of $N^n$ and on the Gagliardo-Nirenberg interpolation inequality for maps in $W^{1, q} \cap L^\infty$. In contrast, the case $sp < 1$ relies on the density of step functions on cubes in $W^{s, p}$.

math.FA↗
arXiv2013

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.

math.AP↗
arXiv2013

Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics

We consider a semilinear elliptic problem [- Δu + u = (I_α\ast \abs{u}^p) \abs{u}^{p - 2} u \quad\text{in (\mathbb{R}^N),}] where (I_α) is a Riesz potential and (p>1). This family of equations includes the Choquard or nonlinear Schrödinger-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.

math.AP↗
arXiv2012

Existence of groundstates for a class of nonlinear Choquard equations

We prove the existence of a nontrivial solution (u \in H^1 (\R^N)) to the nonlinear Choquard equation [- Δu + u = \bigl(I_α\ast F (u)\bigr) F' (u) \quad \text{in (\R^N),}] where (I_α) is a Riesz potential, under almost necessary conditions on the nonlinearity (F) in the spirit of Berestycki and Lions. This solution is a groundstate; if moreover (F) is even and monotone on ((0,\infty)), then (u) is of constant sign and radially symmetric.

math.AP↗
arXiv2012

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{φ(x)}{\abs{x}^\fracα{2}} I_α(x - y) \frac{φ(y)}{\abs{y}^\fracα{2}}\dif x \dif y \le \mathcal{C}_{N,α, 0}\int_{\R^N} \absφ^2,] and of its combinations with the Hardy inequality by Beckner [\int_{\R^N} \int_{\R^N} \frac{φ(x)}{\abs{x}^\frac{α+ s}{2}} I_α(x - y) \frac{φ(y)}{\abs{y}^\frac{α+ s}{2}}\dif x \dif y \le \mathcal{C}_{N, α, 1} \int_{\R^N} \abs{\nabla φ}^2,] and with the fractional Hardy inequality [\int_{\R^N} \int_{\R^N} \frac{φ(x)}{\abs{x}^\frac{α+ s}{2}} I_α(x - y) \frac{φ(y)}{\abs{y}^\frac{α+ s}{2}}\dif x \dif y \le \mathcal{C}_{N, α, s} \mathcal{D}_{N, s} \int_{\R^N} \int_{\R^N} \frac{\bigabs{φ(x) - φ(y)}^2}{\abs{x-y}^{N+s}}\dif x \dif y] where (I_α) is the Riesz potential, (0 < α< 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.

math.AP↗
arXiv2012

Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains

We consider a semilinear elliptic problem with a nonlinear term which is the product of a power and the Riesz potential of a power. This family of equations includes the Choquard or nonlinear Schroedinger--Newton equation. We show that for some values of the parameters the equation does not have nontrivial nonnegative supersolutions in exterior domains. The same techniques yield optimal decay rates when supersolutions exists.

math.AP↗
arXiv2011

Finding critical points whose polarization is also a critical point

We show that near any given minimizing sequence of paths for the mountain pass lemma, there exists a critical point whose polarization is also a critical point. This is motivated by the fact that if any polarization of a critical point is also a critical point and the Euler-Lagrange equation is a second-order semi-linear elliptic problem, T. Bartsch, T. Weth and M. Willem (J. Anal. Math., 2005) have proved that the critical point is axially symmetric.

math.AP↗