Heat flow and boundary value problem for harmonic maps (Q1263142)

The paper is concerned with the existence of global solutions \(f: [0,+\infty[ \times M \to N\) to the parabolic system \[ \begin{alignedat}{2} \partial_ t f(t,x) &= \underline\Delta f(t,x) &\qquad&\text{in }[0,+\infty[\times M,\\ f(0,x) &= \varphi(x) &\qquad&\text{on }M,\\ f(t,x) &= \chi(x) &\qquad&\text{on }[0,+\infty[ \times \partial M, \end{alignedat} \] where \(M\) and \(N\) are given Riemannian manifolds and \(\varphi : M \to N\) and \(\chi : \partial M \to N\) are assigned maps with \(\varphi|_{\partial M} = \chi\). The main result asserts the existence of a classical solution f, provided that the energy \(\frac12 \int_{M} |\nabla\varphi|^2 dV\) is sufficiently small. Moreover, along some sequence \((t_ j) \to +\infty\), \(f(t_ j,\cdot)\) is convergent to some harmonic map \(\tilde u\) with \(\tilde u |_{\partial M} = \chi\). As a consequence, some known results, concerning the existence of harmonic maps, are re-obtained. Also a result of Ljusternik-Schnirelmann type is given.