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Person# Bernard Dacorogna

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Partial differential equation

In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function.
The function is often thought of as

Differential form

In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pione

Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one (or more) f

Related publications (106)

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Let Omega subset of R-n be an open set, A is an element of R-nxn and G : Omega -> R-nxn be given. We look for a solution u : Omega -> R-n of the equation A del u + (del u)(t) A = G We also study the associated Dirichlet problem. (C) 2020 Elsevier Ltd. All rights reserved.

This manuscript extends the relaxation theory from nonlinear elasticity to electromagnetism and to actions defined on paths of differential forms. The introduction of a gauge allows for a reformulation of the notion of quasiconvexity in Bandyopadhyay et al. (J Eur Math Soc 17:1009-1039, 2015), from the static to the dynamic case. These gauges drastically simplify our analysis. Any non-negative coercive Borel cost function admits a quasiconvex envelope for which a representation formula is provided. The action induced by the envelope not only has the same infimum as the original action, but has the virtue to admit minimizers. This completes our relaxation theory program.

In this work we give optimal, i.e., necessary and sufficient, conditions for integrals of the calculus of variations to guarantee the existence of solutions-both weak and variational solutions-to the associated L-2-gradient flow. The initial values are merely L-2 functions with possibly infinite energy. In this context, the notion of integral convexity, i.e., the convexity of the variational integral and not of the integrand, plays the crucial role; surprisingly, this type of convexity is weaker than the convexity of the integrand. We demonstrate this by means of certain quasi-convex and nonconvex integrands.