Hardy space methods for nonlinear partial differential equations
Unexpected solutions of first and second order partial differential equations.
Higher integrability of determinants and weak convergence in L1.
On the singular support of the distributional determinant
Compactifications of C3 with reducible boundary divisor.
On the non-triviality of the Griffith group.
Syzygies and the Abel-Jacobi map for cyclic coverings.
Chow-Künneth decomposition for some moduli spaces.
Convex integration with constraints and applications to phase transitions and partial differential equations
We study solutions of first order partial differential relations , where is a Lipschitz map and is a bounded set in matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of and second we replace Gromov’s −convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our work was originally...
Algebraic cycles on a general complete intersection of high multi-degree of a smooth projective variety
Algebraic cycles on certain Calabi-Yau threefolds.
The p-harmonic system with measure-valued right hand side
A nonlinear model for inextensible rods as a low energy Γ-limit of three-dimensional nonlinear elasticity
Weak compactness of wave maps and harmonic maps
Harmonic maps on planar lattices
A-quasiconvexity : weak-star convergence and the gap
Differential Equations associated to Families of Algebraic Cycles
We develop a theory of differential equations associated to families of algebraic cycles in higher Chow groups (, motivic cohomology groups). This formalism is related to inhomogenous Picard–Fuchs type differential equations. For a families of K3 surfaces the corresponding non–linear ODE turns out to be similar to Chazy’s equation.
Sufficient conditions for the validity of the Cauchy-Born rule close to
The Cauchy–Born rule provides a crucial link between continuum theories of elasticity and the atomistic nature of matter. In its strongest form it says that application of affine displacement boundary conditions to a monatomic crystal will lead to an affine deformation of the whole crystal lattice. We give a general condition in arbitrary dimensions which ensures the validity of the Cauchy–Born rule for boundary deformations which are close to rigid motions. This generalizes results of Friesecke...
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