BCAM COURSE | Limiting Sobolev Estimates for Vector Fields and Cancelling Differential Operators
Fecha: Lun, Mar 3 - Vie, Mar 7 2025
Hora: 11:00-13:00
Ubicación: Maryam Mirzakhani Seminar Room at BCAM
Ponentes: Jean Van Schaftingen
Registro: Course Website and Registration Link
Jean Van Schaftingen will present Sobolev–Gagliardo–Nirenberg endpoint estimates for classes of homogeneous vector differential operators.
I will present Sobolev–Gagliardo–Nirenberg endpoint estimates for classes of homogeneous vector differential operators. Away of the endpoint cases, the classical Calderón–Zygmund estimates show that the ellipticity is necessary and sufficient to control all the derivatives of the vector field. In the endpoint case, Ornstein has showed that there is no nontrivial estimate on same-order derivatives and the ellipticity is necessary for endpoint Sobolev estimates. Such endpoint estimates were proved first for the deformation operator (Korn–Sobolev inequality by M.J. Strauss) and for the Hodge complex (Bourgain and Brezis). The class of operators for which estimates holds can be characterized by a cancelling condition. The estimates rely on a duality estimate for L1 vector fields satisfying some conditions on the derivatives, combined with classical algebraic and harmonic analysis techniques. This characterisation unifies classes of known inequalities and extends to the case of Hardy inequalities. The estimates obtained in such a way are stronger than those obtained thanks to real Hardy spaces and Sobolev embeddings.
More information about the program at the Course Website (Link).
Organizadores:
BCAM
Ponentes confirmados:
Jean Van Schaftingen (Université Catholique de Louvain)
Jean Van Schaftingen is a Belgian mathematical analyst whose research focuses on functional spaces and inequalities, the calculus of variations, and partial differential equations. His contributions span a wide range of topics, including Bourgain-Brezis endpoint Sobolev estimates, characterizations of Sobolev spaces, approximation, extension of traces, and lifting for Sobolev mappings between manifolds, Ginzburg-Landau relaxations, desingularization of vortices for the Euler and lake equations, symmetrization by rearrangement, semilinear elliptic equations of Schrödinger and Choquard type and homogenization.
After obtaining his doctoral degree in 2005 at UCLouvain under the supervision of Michel Willem, and completing a post-doctoral stay in Paris with Haïm Brezis, he joined UCLouvain as a professor in 2007. Since 2021, he has served as an ordinary professor and chair of the School of Mathematics. He has been awarded the Prix Adolphe Wetrems in 2011 and the Prix Jacques Deruyts in 2020 by the Académie royale des Sciences, des Lettres et des Beaux-Arts de Belgique.
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