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

• Michael Struwe
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 4)

## 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.

## Keywords

Homotopy Class Partial Regularity Global Weak Solution Monotonicity Formula Monotonicity Estimate
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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