an:03875850
Zbl 0549.35071
Simon, Leon
Asymptotics for a class of non-linear evolution equations, with applications to geometric problems
EN
Ann. Math. (2) 118, 525-571 (1983).
00283939
1983
j
35L10 49Q20 58J35 58J45 58E15 58E20 53C42
evolution equations; geometric problems; asymptotic behaviour; compact Riemannian manifold; small initial data
The first part of this paper is concerned with the study of the asymptotic behaviour of a smooth function u(x,t) defined on \(\Sigma\times [0,T)\) and satisfying either \((1)\quad\dot u-{\mathcal M}(u)=f\) or \((2)\quad\ddot u+\dot u-{\mathcal M}(u)-{\mathcal R}(u)=f.\) Here \(\Sigma\) is a compact Riemannian manifold, f(x,t) decays exponentially for \(t\to\infty \), and \({\mathcal M}(u)\) is a second order elliptic Euler-Lagrange operator associated with an energy functional \({\mathcal E}(u)=\int_{\Sigma}E(x,u,\nabla u)\); E depends analytically on u and \(\nabla u\) for \(\| u\|_{C^ 1(\Sigma)}\) sufficiently small and \({\mathcal M}(0)=0\). The term \({\mathcal R}(u)\) is assumed to be negligible in a certain sense. For a solution u of (1) with small initial data it is shown that either \({\mathcal E}(u(t))\) becomes negative for some t, or u(t) converges asymptotically to a solution of \({\mathcal M}(\omega)=0.\)
The main result concerning the hyperbolic equation states that a solution with small initial data and not too fast growth converges to a solution of the stationary equation provided \({\mathcal E}(u(t))-{\mathcal E}(0)\geq\delta \), \(\delta\) sufficiently small. Among the important applications to geometric problems are the following. The shown behaviour of the solution of (2) at infinity implies that an n-dimensional minimal submanifold of \({\mathbb{R}}^{n+k}(k\geq 1)\) with a singularity at 0 and which has a multiplicity 1 ''tangent cone'' must converge asymptotically to this cone. This improves results obtained by \textit{W. K. Allard} and \textit{F. J. Almgren} [ibid. 113, 215-265 (1981; Zbl 0437.53045)].
Another application shows that ''tangent maps'' for an energy minimizing harmonic map at an isolated singularity is unique, in case the target manifold is an analytic submanifold isometrically imbedded in an Euclidean space and in case at least one of the tangent maps are smooth.
R.Landes
Zbl 0437.53045