$\vec{w}h\alpha\mathfrak{t}\;\; i\mathbb{S}\ldots$

the direct method in the calculus of variations?

Beatrice Bugert (WIAS)
Urania Berlin, at the BMS Loft (3rd floor)
About what?

One of the fundamental problems in the calculus of variations consists of finding a function $u$ minimizing the integral functional \[ I(u) = \int_\Omega f(x, u(x), Du(x)) \ dx \] over all the functions $u$ satisfying $u = u_0$ on the boundary $\partial \Omega$ of $\Omega$, where $u_0$ is a given function. Euler--often referred to as the founder of the calculus of variations--treated this problem by deducing the so-called Euler-Lagrange equation from the integral functional. He proved that in the case of convex functionals solutions of this equation are already minimizers of $I(u)$. As this method is hard to implement for higher dimensional integrals (i.e., not one-dimensional ones), there was a great need to find an alternative method avoiding the Euler-Lagrange equations. It was Riemann who finally succeeded at this task and introduced the so-called direct method in the calculus of variations, which provides the existence of minimizing functions $u$ directly from the properties of the functional $I$. This talk will give an overview of Riemann's method for convex functionals and show how it has further developed over almost two centuries under the influence of the Italian mathematicians Tonelli and De Giorgi.