Struggles with the regularity of $n$-harmonic maps

Abstract: Let us consider the Dirichlet $n$-energy $\int_{\mathcal{M}} |\nabla u|^n$ for maps $u \colon \mathcal{M}^n \to \mathcal{N}^m$ between two Riemannian manifolds. Its Euler-Lagrange system – known as the $n$-harmonic equation – is an example of a conformally invariant system with critical nonlinearity. In general such systems can have discontinuous solutions, but the regularity of $n$-harmonic maps is an open problem for $n > 2$.

Partial results in this direction rely on the jacobian structure of the $n$-harmonic equation, together with the theory of Hardy and BMO spaces. After a brief review of these methods, I will describe a new application, which leads to yet another partial result – regularity for $W^{n/2,2}$-solutions – but also gives some hope for further progress.

This is joint work with Paweł Strzelecki and Bogdan Petraszczuk.

analysis of PDEs

Audience: researchers in the discipline

Online Seminar "Geometric Analysis"

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