## Differential and Integral Equations

### Regularity of weak solutions of $n$-dimensional H-systems

Sławomir Kolasiński

#### Abstract

We use the method of Strzelecki [Calc. Var. 1 (2003)] to generalize the Bethuel theorem [C. R. Acad. Sci. Paris 314 (1992)] to $n$-dimensional H-systems. We prove that if $u$ is a parameterization of an $n$-dimensional hypersurface in $\mathbb R^{n+1}$, weakly satisfies the system $\Delta_n u = H(u) u_{x_1} \times \cdots \times u_{x_n}$ and additionally has $n-1$ weak derivatives in $L^{n/(n-1)}$, then $u$ is Hölder continuous. Furthermore it is continuous up to the boundary, whenever it has continuous trace. We also give an example showing that the structure of the H-system is relevant and that the assumption that $u$ has $n-1$ weak derivatives does not trivialize the problem.

#### Article information

Source
Differential Integral Equations, Volume 23, Number 11/12 (2010), 1073-1090.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356019073

Mathematical Reviews number (MathSciNet)
MR2742478

Zentralblatt MATH identifier
1240.35145

#### Citation

Kolasiński, Sławomir. Regularity of weak solutions of $n$-dimensional H-systems. Differential Integral Equations 23 (2010), no. 11/12, 1073--1090. https://projecteuclid.org/euclid.die/1356019073