Global existence and partial regularity results for the evolution of harmonic maps

Abstract

Consider a compact, m-dimensional Riemannian manifold M with metric $\gamma = {({\gamma _{\alpha \beta }})_{1 \leqslant \alpha ,\beta \leqslant m}}$ and with $\partial M = O,\;or\;M = {R^m},m \geqslant 2$. For a compact, ℓ-dimensional manifold N, ∂N = Ø, with metric $g = {({g_{ij}})_{{1_{ \leq i,j \leqslant l}}}}$ and C¹-maps u: M → N let

$$ E\left( u \right) = \int\limits_M {e\left( u \right)}dM $$

be the energy of u, with density

$$ \begin{array}{*{20}{c}}{e\left( u \right) = \frac{1}{2}{\gamma ^{\alpha \beta }}{g_{ij}}\left( u \right)}&{\frac{\partial }{{\partial {x_\beta }}}} \end{array}{u^j} $$

and volume element

$$ dM = \sqrt {\left| \gamma \right|} dx,\left| \gamma \right| = \det \left( {{\gamma _{\alpha \beta }}} \right), $$

in local coordinates.