Jean Van Schaftingen — Research topics

Estimates for vector Sobolev spaces (2003– )

En 2004, J. Bourgain, H. Brezis et P. Mironescu have shown that if \(\Gamma \subset \mathbf{R}^n\) is a closed rectifiable curve with tangent vector \(t\) and \(\varphi : \mathbf{R}^n \to \mathbf{R}^n\) is a vector field, then \[ \int_{\Gamma} \varphi \cdot t \le \lvert \Gamma \rvert\, \lVert D \varphi \rVert_{L^n}. \] J. Bourgain and H. Brezis have generalized this to \[ \int_{\mathbf{R}^n} \varphi \cdot f \le \lVert f \rVert_{L^1} \, \lVert D \varphi \rVert_{L^n} \] whenever \(f\) is a divergence-free vector field. These inequalities are surprising since the quantity \(\lVert D \varphi \rVert_{L^n}\) does not control \(\lVert \varphi \rVert_{L^\infty} \). These inequalities have consequences in the theory of regularity of elliptic systems with \(L^1\) data.

In this domain,

I have given elementary proofs of the circulation integral inequality and the inequality for divergence-free vector-fields,
I have obtained inequalities when the divergence is replaced by a general higher-order operator,
I have studied the relationship between functions satisfying this kind of estimates and the space of functions of bounded mean oscillation \(BMO\),
with H. Brezis, I have studied the corresponding boundary estimates and posed some problems of optimal constants,
with S. Chanillo, I have obtained corresponding inequalities on stratified homogeneous groups, including for example the Heisenberg group.

Jean Van Schaftingen, Limiting Sobolev inequalities for vector fields and canceling linear differential operators, accepted in J. Eur. Math. Soc. (JEMS).

Preprint: [arXiv:1104.0192]

Jean Van Schaftingen, Limiting fractional and Lorentz spaces estimates of differential forms, Proc. Amer. Math. Soc. 138 (2010), no. 1, 235-240.

[doi:10.1090/S0002-9939-09-10005-9][pdf]

Preprint: [arXiv:0903.2182]

Sagun Chanillo and Jean Van Schaftingen, Subelliptic Bourgain-Brezis estimates on groups, Math. Res. Lett. 16 (2009), no. 3, 487–501.

Preprint: [arXiv:0712.3730]

Haïm Brezis and Jean Van Schaftingen, Circulation integrals and critical Sobolev spaces: problems of optimal constants, in Perspectives in Partial Differential Equations, Harmonic Analysis and Applications, Amer. Math. Soc., Proc. Sympos. Pure Math., No. 79, 2008, 33–47.

Jean Van Schaftingen, Estimates for \(\mathrm{L}^1\) vector fields under higher-order differential conditions, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 4, 867–882.

Preprint: [dvi] [ps][pdf]

Haïm Brezis and Jean Van Schaftingen, Boundary estimates for elliptic systems with \(L^1\)-data, Calc. Var. Partial Differential Equations 30 (2007), no. 3, 369–388.

[doi:10.1007/s00526-007-0094-9]

Preprint: [dvi] [ps][pdf]

Jean Van Schaftingen, Function spaces between BMO and critical Sobolev spaces, J. Funct. Anal. 236 (2006), no. 2, 490–516.

[doi:10.1016/j.jfa.2006.03.011][MR:2240172]

Preprint: [dvi] [ps][pdf]

Jean Van Schaftingen, Estimates for \(L^1\) vector fields with a second order condition, Acad. Roy. Belg. Bull. Cl. Sci. (6) 15 (2004), no. 1-6, 103–112.

Preprint: [dvi] [ps][pdf]

Jean Van Schaftingen, Estimates for \(L^1\)-vector fields, C. R. Math. Acad. Sci. Paris 339 (2004), no. 3, 181–186.

[doi:10.1016/j.crma.2004.05.013][MR:20708071][Zbl 1049.35069]

Jean Van Schaftingen, A simple proof of an inequality of Bourgain, Brezis and Mironescu, C. R. Math. Acad. Sci. Paris 338 (2004), no. 1, 23–26.

[doi:10.1016/j.crma.2003.10.036][MR:2038078]

Preprint: [dvi] [ps][pdf]

Sobolev spaces of maps into manifolds (2005– )

If \(M\) is a compact manifold and \(N \subset \mathbf{R}^\nu\) is an imbedded compact manifold, one considers for \(k \in \mathbf{N}_*\) and \(p \ge 1\) the Sobolev space \[ W^{k, p}(M; N)=\{ u \in W^{k, p}(M; \mathbf{R}^\nu)\;:\; u \in N \text{ almost everywhere} \}. \] One wondeers whether the class of smooth functions \(C^\infty(M; N)\) is dense in \(W^{k, p}(M; N)\). In general, the answer is negative. For \(k=1\), Bethuel, and Hang and Lin have proved that the answer is positive if and only if \(M\) and \(N\) satisfy some topological condition.

With Pierre Bousquet and Augusto Ponce, we are studying the corresponding problem for \(k \ge 2\).

Augusto C. Ponce and Jean Van Schaftingen, Closure of Smooth Maps in \(W^{1,p}(B^3;S^2)\), Differential Integral Equations 22 (2009), no. 9-10, 881-900.

Preprint: [arXiv:0901.4491]

Pierre Bousquet, Augusto C. Ponce and Jean Van Schaftingen, A case of density in \(W^{2,p}(M;N)\), C. R. Math. Acad. Sci. Paris 346 (2008), no. 13-14, 735–740.

[doi:10.1016/j.crma.2008.05.006][MR:2427072]

Desingularization of vortices for the Euler equation of incompressible flows (2006– )

The Euler equation of incompressible flows \[ \left\{ \begin{aligned} \nabla \cdot \mathbf{v} &= 0\\ \mathbf{v}_t + \mathbf{v} \cdot \nabla \mathbf{v} &= -\nabla p, \end{aligned} \right. \] has in two dimensions singular solutions, where the vorticity is the sum of Dirac’s masses. The position of the vortices is governed by a Hamiltonian system. The solutions are solutions in the sense of distributions.

With Didier Smets, we have shown that some solutions could be approximated by stationnary classical solutions. We construct the velocity field as \(\nabla \psi^\perp\), where \[ \left\{ \begin{aligned} -\varepsilon^2 \Delta \psi & = \psi_+^p & & \text{dans \(\Omega\)},\\ \psi&=\psi_0-\frac{\kappa}{2\pi} \ln \frac{1}{\varepsilon} & & \text{sur \(\partial \Omega\)}. \end{aligned} \right. \] and \(\epsilon \to 0\). We treat the case of a single stationnary vortex and a pair of stationnary vortices in a bounded domain and also the case of a pair of vortices in translation in \(\mathbf{R}^2\).

Didier Smets and Jean Van Schaftingen, Desingularization of vortices for the Euler equation, Arch. Rat. Mech. Anal. 198 (2010), no. 3, 869-925.

[doi:10.1007/s00205-010-0293-y]

Preprint: [arXiv:0909.1166]

Symmetry of solutions of elliptic problems (2005– )

Given a solution \(u\) to the Dirichlet problem \[ \left\{ \begin{aligned} -\Delta u &= f(u), & & \text{in \(\Omega\)},\\ u&=0 & & \text{on \(\partial \Omega\)}, \end{aligned} \right. \] one asks the question under which assumptions the solutions \(u\) does inherit the symmetries of the domain \(\Omega\). Gidas, Ni and Nirenberg have shown that if \(f\) is Lipschitz continuous, \(u\) is positive and \(\Omega\) is a ball, then \(u\) is radial.

With Michel Willem, me have extended a method due to Thomas Bartsch, Michel Willem and Tobias Weth in order to study tge symmetry of least energy nodal solutions of \[ \left\{ \begin{aligned} -\Delta u(x)+a(x)u(x) &= f(x, u(x)), & & \text{in \(x \in \Omega\)},\\ u&=0 & & \text{on \(\partial \Omega\)}, \end{aligned} \right. \] when \(f\) is not Hölder continuous.

With Denis Bonheure, Vincent Bouchez and Christopher Grumiau, nwe have studier the problem of symmetry of least energy nodal solutions of \[ \left\{ \begin{aligned} -\Delta u &= \lambda u^p, & & \text{in \(\Omega\)},\\ u&=0 & & \text{on \(\partial \Omega\)}. \end{aligned} \right. \] where \(p > 1\). We have thudies the asymptotics of the solutions when \(p \to 1\). This lead us to symmetry results when the second eigenvalue of the Laplacian is nondegenerate, some symmetry breaking and a conjecture that when \(\Omega\) is a square, least energy nodal solutions are symmetric with respect to the diagonals.

Marco Squassina and Jean Van Schaftingen, Finding critical points whose polarization is also a critical point, submitted.

Preprint: [arXiv:1108.6217]

Denis Bonheure, Vincent Bouchez, Christopher Grumiau and Jean Van Schaftingen, Asymptotics and symmetries of least energy nodal solutions of Lane-Emden problems with slow growth, Commun. Contemp. Math. 10 (2008), no. 4, 609–631.

[doi:10.1142/S0219199708002910][MR:2444849]

Jean Van Schaftingen and Michel Willem, Symmetry of solutions of semilinear elliptic problems, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 2, 439–456.

Preprint: [dvi] [ps][pdf]

Stationnary states for the nonlinear Schrödinger equation (2005– )

Stationnary states for the nonlinear Schrödinger equation are solutions of the elliptic equation \[ -\varepsilon^2 \Delta u + V u = u^p, \] in \(\mathbf{R}^n\) where \(V : \mathbf{R}^n \to \mathbf{R}\) is a given potential and \(\varepsilon\) is the adimensionalized Planck constant. In the semi-classical limit where \(\varepsilon \to 0\), one expects solutions to concentrate around critical points of \(V\).

When \(\inf V > 0\), the existence of solutions for small \(\varepsilon\) concentrating around critical point of \(V\) as \(\varepsilon \to 0\) has been shown by many authors. I have been interested in the critical-frequency case where \(V\) is positive but \(\inf V =0\). With Denis Bonheure, we have adapted the penalization method of Manuel del Pino et Patricio Felmer and we have shown the existence of solutions for potentials that do not decay too fast at infinity. With Vitaly Moroz, we have obtained some optimal results for fast decaying potentials, including compactly supported potentials. With Denis Bonheure and Jonathan Di Cosmo, we have applied these methode to obtain solutions concentrating on spheres of dimension \(k \in \{1, \dotsc, n-1\}\).

Avec Denis Bonheure, nous avons aussi étudié l'équation \[ -\Delta u + V u = Ku^p, \] dans \(\mathbf{R}^n\) où \(V : \mathbf{R}^n \to \mathbf{R}\) et \(K : \mathbf{R}^n \to \mathbf{R}\) sont des potentiels donnés. Nous avons donné des conditions sous lesquelles ce problème a une solution de moindre énergie et étudié la décroissance à l'infini de ces solutions.

Jonathan Di Cosmo and Jean Van Schaftingen, Stationary solutions of the nonlinear Schrödinger equation with fast-decay potentials concentrating around local maxima, accepted in Calc. Var. Partial Differential Equations.

[doi:10.1007/s00526-012-0518-z]

Preprint: [arXiv:1109.6773]

Denis Bonheure, Jonathan Di Cosmo and Jean Van Schaftingen, Nonlinear Schrödinger equation with unbounded or vanishing potentials: solutions concentrating on lower dimensional spheres, J. Differential Equations 252 (2012), no. 1, 941–968.

[doi:10.1016/j.jde.2011.10.004]

Preprint: [arXiv:1009.2600]

Denis Bonheure and Jean Van Schaftingen, Groundstates for the nonlinear Schrödinger equation with potential vanishing at infinity, Ann. Mat. Pura Appl. (4) 189 (2010), 273-301.

[doi:10.1007/s10231-009-0109-6]

Vitaly Moroz and Jean Van Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials, Calc. Var. Partial Differential Equations 37 (2010), no. 1, 1—27.

[doi:10.1007/s00526-009-0249-y]

Preprint: [arXiv:0902.0722]

Vitaly Moroz and Jean Van Schaftingen, Existence and concentration for nonlinear Schrödinger equations with fast decaying potentials, C. R. Math. Acad. Sci. Paris 347 (2009), no. 15-16, 921-926.

[doi:10.1016/j.crma.2009.05.009]

Denis Bonheure and Jean Van Schaftingen, Bound state solutions for a class of nonlinear Schrödinger equations, Rev. Mat. Iberoam. 24 (2008), no. 1, 297–351.

Preprint: [dvi] [ps][pdf]

Denis Bonheure and Jean Van Schaftingen, Nonlinear Schrödinger equations with potentials vanishing at infinity, C. R. Math. Acad. Sci. Paris 342 (2006), no. 12, 903–908.

[doi:10.1016/j.crma.2006.04.011][MR:2235608]

Preprint: [dvi] [ps][pdf]

Symmetrization by rearrangement (2001–2009

Symmetrizations are a tool used to prove that solutions of variational problems are symmetrical. To every function \(u\), a more symmetrical fonction \(u^*\) is associated. This nonlinear transformation preserves the measure of sublevel sets, so that many integral functionals are preserved or decrease when the function they contain is symmetrized.

I have worked on

the relationships between the properties of the symmetrization of sets and the symmetrization of functions,
the approximation of symmetrizations by simpler symmetrizations, by simpler symmetrizations, in particular by polarizations, and random approximation,
the symmetry of critical points obtained by minimax methods (Mountain Pass Theorem, Linking Theorem, Krasnsoselskii genus),
anisotropic symmetrizations, i.e. symmetrization with respect to a noneuclidean norm.

Jean Van Schaftingen, Explicit approximation of the symmetric rearrangement by polarizations, Archiv der Mathematik 93 (2009), no. 2, 181-190.

[doi:10.1007/s00013-009-0018-3]

Preprint: [arXiv:0902.0637]

Jean Van Schaftingen, Approximation of symmetrizations and symmetry of critical points, Topol. Methods Nonlinear Anal. 28 (2006), no. 1, 61–85.

Preprint: [dvi] [ps][pdf]

Jean Van Schaftingen, Anisotropic symmetrization, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 4, 539–565.

[doi:10.1016/j.anihpc.2005.06.001][MR:2245755]

Preprint: [dvi] [ps][pdf]

Jean Schaftingen, Universal approximation of symmetrizations by polarizations, Proc. Amer. Math. Soc. 134 (2006), no. 1, 177–186 (electronic).

[doi:10.1090/S0002-9939-05-08325-5][MR:2170557]

Preprint: [dvi] [ps][pdf]

Jean Van Schaftingen, Symmetrization and minimax principles, Commun. Contemp. Math. 7 (2005), no. 4, 463–481.

[doi:10.1142/S0219199705001817][MR:2166661]

Preprint: [dvi] [ps][pdf]

J. Van Schaftingen and M. Willem, Set transformations, symmetrizations and isoperimetric inequalities, in Nonlinear analysis and applications to physical sciences, Springer Italia, Milan, 2004, 135–152.

Homogeneization (2004–2006)

L’homogeneization is the mathematical study of problems with parameters oscillating on a small scale with respect to the size of the problem, in order e.g. to obtain effective models for composite materials. In the case of periodic homogeneization, Cioranescu, Damlamian and Griso have developed the powerful unfolding method.

Nicolas Meunier and I have applied this method to nonlinear elliptic problem. In collaboration with Alain Damlamian we have extended the results to the case of maximal monotone graphs and subdifferentials of convex functions.

Alain Damlamian, Nicolas Meunier and Jean Van Schaftingen, Periodic homogenization for convex functionals using Mosco convergence, Ricerche Mat. 57 (2008), no. 2, 209–249.

[doi:10.1007/s11587-008-0038-5]

Alain Damlamian, Nicolas Meunier and Jean Van Schaftingen, Periodic homogenization of monotone multivalued operators, Nonlinear Anal. 67 (2007), no. 12, 3217–3239.

[doi:10.1016/j.na.2006.10.007]

Preprint: [dvi] [ps][pdf]

Nicolas Meunier and Jean Van Schaftingen, Periodic reiterated homogenization for elliptic functions, J. Math. Pures Appl. (9) 84 (2005), no. 12, 1716–1743.

[doi:10.1016/j.matpur.2005.08.003][MR:2180388]

Nicolas Meunier and Jean Van Schaftingen, Reiterated homogenization for elliptic operators, C. R. Math. Acad. Sci. Paris 340 (2005), no. 3, 209–214.

[doi:10.1016/j.crma.2004.10.026][MR:2123030]

Preprint: [dvi] [ps][pdf]

Miscellaneous

Jean Van Schaftingen, Proving the existence of eigenvalues and eigenvectors by Weierstrass's theorem, submitted.

Preprint: [arXiv:1109.6821]

Vitaly Moroz and Jean Van Schaftingen, Existence, stability and oscillation properties of slow decay positive solutions of supercritical elliptic equations with Hardy potential, submitted.

Preprint: [arXiv:1108.4668]

Vincent Bouchez and Jean Van Schaftingen, Extremal functions in Poincaré-Sobolev inequalities for functions of bounded variation, in Nonlinear Elliptic Partial Differential Equations, Amer. Math. Soc., Contemporary Mathematics, No. 540, 2011, 47−58.

Preprint: [arXiv:1001.4651]

Tianling Jin, Vladimir Maz'ya and Jean Van Schaftingen, Pathological solutions to elliptic problems in divergence form with continuous coefficients, C. R. Math. Acad. Sci. Paris 347 (2009), no. 13-14, 773-778.

[doi:10.1016/j.crma.2009.05.008]

Preprint: [arXiv:0904.1674]

Augusto C. Ponce and Jean Van Schaftingen, The continuity of functions with N-th derivative measure, Houston J. Math. 33 (2007), no. 3, 927–939.

[web][MR:2335744]

Preprint: [dvi] [ps][pdf]

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