890 citations · 3.5k across the 69 of their papers we have counts for
45 papers · 1 filter
Singular extension of critical Sobolev mappings under an exponential weak-type estimate
Bohdan Bulanyi, Jean Van Schaftingen
Given and a compact Riemannian manifold , we construct for every map in the critical Sobolev space $W^{m/(m + 1), m + 1} (\mathb…
Endpoint Sobolev inequalities for vector fields and cancelling operators
Jean Van Schaftingen
The injectively elliptic vector differential operators from to on such that the estimate \[ \Vert D^\ell u\Vert_{L^{n/(n - \ell)} (\mathbb{R…
Limiting Sobolev estimates for vector fields and cancelling differential operators
Jean Van Schaftingen
These notes present Sobolev-Gagliardo-Nirenberg endpoint estimates for classes of homogeneous vector differential operators. Away of the endpoint cases, the classical Calderón-Zygm…
Injective ellipticity, cancelling operators, and endpoint Gagliardo-Nirenberg-Sobolev inequalities for vector fields
Jean Van Schaftingen
Although Ornstein's nonestimate entails the impossibility to control in general all the -norm of derivatives of a function by the -norm of a constant coefficient homogene…
Lifting of fractional Sobolev mappings to noncompact covering space
Jean Van Schaftingen
Given compact Riemannian manifolds and , a Riemannian covering by a noncompact covering space $\smas…
Asymptotic behavior of minimizing -harmonic maps when in dimension 2
Jean Van Schaftingen, Benoît Van Vaerenbergh
We study --harmonic maps with Dirichlet boundary conditions from a planar domain into a general compact Riemannian manifold. We show that as approaches from below, they…
Boundary ellipticity and limiting -estimates on halfspaces
Franz Gmeineder, Bogdan Raiţă, Jean Van Schaftingen
We identify necessary and sufficient conditions on th order differential operators in terms of a fixed halfspace such that the Gagliardo--N…
Reverse superposition estimates in Sobolev spaces
Jean Van Schaftingen
We study when and how the norm of a function in the homogeneous Sobolev spaces , with and either or , is…
Trace theory for Sobolev mappings into a manifold
Petru Mironescu, Jean Van Schaftingen
We review the current state of the art concerning the characterization of traces of the spaces of Sobolev mappings with value…
Lifting in compact covering spaces for fractional Sobolev mappings
Petru Mironescu, Jean Van Schaftingen
Let be a Riemannian covering, with , smooth compact connected Riemannian manifolds. If $\mathcal…
An estimate of the Hopf degree of fractional Sobolev mappings
Armin Schikorra, Jean Van Schaftingen
We estimate the Hopf degree for smooth maps from to in the fractional Sobolev space. Namely we show that for …
Metric characterization of the sum of fractional Sobolev spaces
Rémy Rodiac, Jean Van Schaftingen
We introduce a non-linear criterion which allows us to determine when a function can be written as a sum of functions belonging to homogeneous fractional spaces: for $\ell \in \mat…
On limiting trace inequalities for vectorial differential operators
Franz Gmeineder, Bogdan Raita, Jean Van Schaftingen
We establish that trace inequalities $$\|D^{k-1}u\|_{L^{\frac{n-s}{n-1}}(\mathbb{R}^{n},dμ)} \leq c \|μ\|_{L^{1,n-s}(\mathbb{R}^{n})}^{\frac{n-1}{n-s}}\|\mathbb{A}[D]u\|_{L^{1}(\ma…
Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo's lemma
Daniel Spector, Jean Van Schaftingen
We prove a family of Sobolev inequalities of the form where $A (D) : C^\i…
Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent
Jean Van Schaftingen, Jiankang Xia
We prove the existence of ground state solutions by variational methods to the nonlinear Choquard equations with a nonlinear perturbation \[ -Δu+ u=\big(I_α*|u|^{\fracα{N}+1}\big)|…
Groundstates of the Choquard equations with a sign-changing self-interaction potential
Luca Battaglia, Jean Van Schaftingen
We consider a nonlinear Choquard equation when the self-interaction potential is unbounded from below. Under…
Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent
Daniele Cassani, Jean Van Schaftingen, Jianjun Zhang
For the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation, , in where $N\g…
Sobolev mappings: from liquid crystals to irrigation via degree theory
Jean Van Schaftingen
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 liqu…
The logarithmic Choquard equation: sharp asymptotics and nondegeneracy of the groundstate
Denis Bonheure, Silvia Cingolani, Jean Van Schaftingen
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…
Standing waves with a critical frequency for nonlinear Choquard equations
Jean Van Schaftingen, Jiankang Xia
In this paper, we study the nonlocal Choquard equation where $N\ge…
Bourgain-Brezis Estimates on Symmetric Spaces of Non-compact Type
Sagun Chanillo, Jean Van Schaftingen, Po-lam Yung
Let M be a globally Riemannian symmetric space. We prove a duality estimate between pairings of vector fields with divergence zero and and in L^1 with vector fields in a critical S…
Properties of groundstates of nonlinear Schrödinger equations under a weak constant magnetic field
Denis Bonheure, Manon Nys, Jean Van Schaftingen
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 } \…
Choquard equations under confining external potentials
Jean Van Schaftingen, Jiankang Xia
We consider the nonlinear Choquard equation where , is the Riesz potentia…
Odd symmetry of least energy nodal solutions for the Choquard equation
David Ruiz, Jean Van Schaftingen
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 stands for t…