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Jean Van Schaftingen

Publications and source records attributed to Jean Van Schaftingen.

At least 73 records · Page 4Linked to original sources
arXiv2017

Sobolev mappings: from liquid crystals to irrigation via degree theory

Sobolev spaces are a natural framework for the analysis of problems in partial differential equations and calculus of variations. Some physical and geometric contexts, such as liquid crystals models and harmonic maps, lead to consider Sobolev maps, that is, Sobolev vector functions whose range is constrained in a surface or submanifold of the space. This additional nonlinear constraint provokes the appearance of finite-energy topological singularities. These singularities are characterized by a nontrivial topological invariant such as the topological degree, they represent an obstruction to the strong approximation by smooth maps and they become source and sink terms in an optimal transportation or irrigation problem of topological charges arising in the study of the weak approximation and of the relaxed energy.

math.AP↗
arXiv2017

Weak approximation by bounded Sobolev maps with values into complete manifolds

We have recently introduced the trimming property for a complete Riemannian manifold $N^{n}$ as a necessary and sufficient condition for bounded maps to be strongly dense in $W^{1, p}(B^m; N^{n})$ when $p \in \{1, \dotsc, m\}$. We prove in this note that even under a weaker notion of approximation, namely the weak sequential convergence, the trimming property remains necessary for the approximation in terms of bounded maps. The argument involves the construction of a Sobolev map having infinitely many analytical singularities going to infinity.

math.FA↗
arXiv2017

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

math.AP↗
arXiv2016

The logarithmic Choquard equation: sharp asymptotics and nondegeneracy of the groundstate

We derive the asymptotic decay of the unique positive, radially symmetric solution to the logarithmic Choquard equation $$ - Δu + a u = \frac{1}{2 π} \Bigl[\ln \frac{1}{|x|}* |u|^2 \Bigr] \ u \qquad \text{in $\mathbb{R}^2$} $$ and we establish its nondegeneracy. For the corresponding three-dimensional problem, the nondegeneracy property of the positive ground state to the Choquard equation was proved by E. Lenzmann (Analysis & PDE, 2009).

math.AP↗
arXiv2016

Standing waves with a critical frequency for nonlinear Choquard equations

In this paper, we study the nonlocal Choquard equation $$ -\varepsilon^2 Δu_\varepsilon + V u_\varepsilon= (I_α* |u_\varepsilon|^p)|u_\varepsilon|^{p-2}u_\varepsilon $$ where $N\geq 1$, $I_α$ is the Riesz potential of order $α\in (0, N)$ and $\varepsilon>0$ is a parameter. When the nonnegative potential $V\in C (\mathbb{R}^N)$ achieves $0$ with a homogeneous behaviour or on the closure of an open set but remains bounded away from $0$ at infinity, we show the existence of groundstate solutions for small $\varepsilon>0$ and exhibit the concentration behaviour as $\varepsilon\to 0$.

math.AP↗
arXiv2016

Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at the critical frequency

We study the nonlocal Schrödinger-Poisson-Slater type equation $$ - Δu + (I_α\ast |u|^p)|u|^{p - 2} u= |u|^{q-2}u\quad\text{in \(\mathbb{R}^N\),} $$ where $N\in\mathbb{N}$, $p>1$, $q>1$ and $I_α$ is the Riesz potential of order $α\in(0,N).$ We introduce and study the Coulomb-Sobolev function space which is natural for the energy functional of the problem and we establish a family of associated optimal interpolation inequalities. We prove existence of optimizers for the inequalities, which implies the existence of solutions to the equation for a certain range of the parameters. We also study regularity and some qualitative properties of solutions. Finally, we derive radial Strauss type estimates and use them to prove the existence of radial solutions to the equation in a range of parameters which is in general wider than the range of existence parameters obtained via interpolation inequalities.

math.AP↗
arXiv2016

An $L^1$-type estimate for Riesz potentials

In this paper we establish new $L^1$-type estimates for the classical Riesz potentials of order $α\in (0, N)$: \[ \|I_αu\|_{L^{N/(N-α)}(\mathbb{R}^N)} \leq C \|Ru\|_{L^1(\mathbb{R}^N;\mathbb{R}^N)}. \] This sharpens the result of Stein and Weiss on the mapping properties of Riesz potentials on the real Hardy space $\mathcal{H}^1(\mathbb{R}^N)$ and provides a new family of $L^1$-Sobolev inequalities for the Riesz fractional gradient.

math.FA↗
arXiv2016

Choquard equations under confining external potentials

We consider the nonlinear Choquard equation $$ -Δu+V u=(I_α\ast \vert u\vert ^p)\vert u\vert ^{p-2}u \qquad \text{ in } \mathbb{R}^N $$ where $N\geq 1$, $I_α$ is the Riesz potential integral operator of order $α\in (0, N)$ and $p > 1$. If the potential $ V \in C (\mathbb{R}^N; [0,+\infty)) $ satisfies the confining condition $$ \liminf\limits_{\vert x\vert \to +\infty}\frac{V(x)}{1+\vert x\vert ^{\frac{N+α}{p}-N}}=+\infty, $$ and $\frac{1}{p} > \frac{N - 2}{N + α}$, we show the existence of a groundstate, of an infinite sequence of solutions of unbounded energy and, when $p \ge 2$ the existence of least energy nodal solution. The constructions are based on suitable weighted compact embedding theorems. The growth assumption is sharp in view of a Pohožaev identity that we establish.

math.AP↗
arXiv2016

Properties of groundstates of nonlinear Schrödinger equations under a weak constant magnetic field

We study the qualitative properties of groundstates of the time-independent magnetic semilinear Schrödinger equation \[ - (\nabla + i A)^2 u + u = |u|^{p-2} u, \qquad \text{ in } \mathbb{R}^N, \] where the magnetic potential $A$ induces a constant magnetic field. When the latter magnetic field is small enough, we show that the groundstate solution is unique up to magnetic translations and rotations in the complex phase space, that groundstate solutions share the rotational invariance of the magnetic field and that the presence of a magnetic field induces a Gaussian decay. In this small magnetic field régime, the corresponding ground-energy is a convex differentiable function of the magnetic field.

math.AP↗
arXiv2016

Odd symmetry of least energy nodal solutions for the Choquard equation

We consider the Choquard equation (also known as stationary Hartree equation or Schrödinger--Newton equation) \[ -Δu + u = (I_α\star |u|^p) |u|^{p - 2}u. \] Here $I_α$ stands for the Riesz potential of order $α\in (0,N)$, and $\frac{N - 2}{N + α} < \frac{1}{p} \le \frac{1}{2}$. We prove that least energy nodal solutions have an odd symmetry with respect to a hyperplane when $α$ is either close to $0$ or close to $N$.

math.AP↗
arXiv2016

A guide to the Choquard equation

We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$ -Δu + V(x)u = \bigl(|x|^{-(N-α)} * |u|^p\bigr)|u|^{p - 2} u \qquad \text{in $\mathbb{R}^N$}, $$ and some of its variants and extensions.

math.AP↗
arXiv2016

Least action nodal solutions for the quadratic Choquard equation

We prove the existence of a minimal action nodal solution for the quadratic Choquard equation $$ -Δu + u = \big(I_α\ast |u|^2\big)u \quad\text{in }\; \mathbb R^N,$$ where $I_α$ is the Riesz potential of order $α\in(0,N)$. The solution is constructed as the limit of minimal action nodal solutions for the nonlinear Choquard equations $$ -Δu + u = \big(I_α\ast |u|^p\big)|u|^{p-2}u \quad\text{in }\; \mathbb R^N$$ when $p\searrow 2$. The existence of minimal action nodal solutions for $p>2$ can be proved using a variational minimax procedure over Nehari nodal set. No minimal action nodal solutions exist when $p<2$.

math.AP↗
arXiv2016

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.

math.PR↗
arXiv2016

Semiclassical Sobolev constants for the electro-magnetic Robin Laplacian

This paper is devoted to the asymptotic analysis of the optimal Sobolev constants in the semiclassical limit and in any dimension. We combine semiclassical arguments and concentration-compactness estimates to tackle the case when an electromagnetic field is added as well as a smooth boundary carrying a Robin condition. As a byproduct of the semiclassical strategy, we also get exponentially weighted localization estimates of the minimizers.

math.AP↗
arXiv2016

Nodal solutions for the Choquard equation

We consider the general Choquard equations $$ -Δu + u = (I_α\ast |u|^p) |u|^{p - 2} u $$ where $I_α$ is a Riesz potential. We construct minimal action odd solutions for $p \in (\frac{N + α}{N}, \frac{N + α}{N - 2})$ and minimal action nodal solutions for $p \in (2,\frac{N + α}{N - 2})$. 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 Schrödinger equation, which is the nonlocal counterpart of the Choquard equation, does not have such solutions.

math.AP↗
arXiv2015

Applications of Bourgain-Brezis inequalities to Fluid Mechanics and Magnetism

We apply the borderline Sobolev inequalities of Bourgain-Brezis to the vorticity equation and Navier-Stokes equation in 2D. We take the initial vorticity to be in the space of functions of Bounded variation(BV). We obtain the subsequent vorticity to be in the space of functions of bounded variation, uniformly for small time, and the velocity vector to be uniformly bounded for small time. Such a conclusion cannot follow for initial vorticity taken to be just a measure or in L^1 from the Lamb-Oseen vortex example. Secondly we apply an improved Strichartz inequality obtained earlier by the first and third authors to the Maxwell equations of Electromagnetism. In particular we estimate the size of the magnetic field vector in terms of the gradient of the current density vector. The main point is that in this inequality only the L^1 norm in space appears for the gradient of the current density vector. Such a result is only possible because of a vanishing divergence inhomogeneity in the wave equation for the Magnetic field vector stemming from the Maxwell equations. A key ingredient in the proof of the improved Strichartz inequality is the Bourgain-Brezis borderline Sobolev inequalities.

math.AP↗
arXiv2015

Controlled singular extension of critical trace Sobolev maps from spheres to compact manifolds

Given $n \in \mathbb{N}_*$, a compact Riemannian manifold $M$ and a Sobolev map $u \in W^{n/(n + 1), n + 1} (\mathbb{S}^n; M)$, we construct a map $U$ in the Sobolev-Marcinkiewicz (or Lorentz-Sobolev) space $W^{1, (n + 1, \infty)} (\mathbb{B}^{n + 1}; M)$ such that $u = U$ in the sense of traces on $\mathbb{S}^{n} = \partial \mathbb{B}^{n + 1}$ and whose derivative is controlled: for every $λ> 0$, $$ λ^{n + 1} \big\vert\big\{ x \in \mathbb{B}^{n + 1} : \vert D U (x)\vert > λ\big\}\big\vert \le γ\Big(\int_{\mathbb{S}^n}\int_{\mathbb{S}^n} \frac{\vert u (y) - u (z)\vert^{n + 1}}{\vert y - z\vert^{2 n}} \,\mathrm{d} y \,\mathrm{d} z \Bigr)\ , $$ where the function $γ: [0, \infty) \to [0, \infty)$ only depends on the dimension $n$ and on the manifold $M$. The construction of the map $U$ relies on a smoothing process by hyperharmonic extension and radial extensions on a suitable covering by balls.

math.FA↗