Uniqueness of harmonic maps with small energy
Let \(B\) be the unit ball in \({\mathbb R}^3\) with boundary \(S^2\). Let \(N\) be a smooth compact Riemannian manifold without boundary. Let \(N\) be isometrically embedded in some Euclidean space \({\mathbb R}^n\). For a map \(g\in H^{1,2}(S^2;N)\) let \[ E(g;S^2) = {1 \over 2} \int_{S^2} |\nabla g|^2 dA \] be its Dirichlet energy. For an \(H^{1,2}\)-map \(u:B\to N\) the Dirichlet energy \(E(u;B)\) is defined similarly. For \(g\in H^{1,2}(S^2;N)\) with \(E(g;S^2)<\infty\) put \[ H^{1,2}_g(B;N) = \{ u\in H^{1,2}(B;N)\;|\;u|_{\partial B} = g \}. \] A weakly harmonic map is a critical point of \(E(\cdot;B)\) in \(H^{1,2}_g(B;N)\). Equivalently, \(u\) solves \[ -\Delta = A(u)(\nabla u,\nabla u) \] in the distributional sense where \[ A(p):T_pM \times T_pM \to (T_pM)^\perp \] denotes the second fundamental form of \(N\) in \({\mathbb R}^n\). It is shown that there exist constants \(\epsilon=\epsilon(N)>0\) and \(C=C(N)\) such that for each \(g\in H^{1,2}(S^2;N)\) with \[ E(g;S^2)<\epsilon \] there exists a unique weakly harmonic \(u_g \in H^{1,2}_g(B;N)\) satisfying \[ \sup_{x_0\in B, r>0} \left\{ r^{-1}\int_{B\cap B_r(x_0)} |\nabla u_g|^2 dx \right\} \leq C\cdot E(g;S^2). \] Uniqueness is known to fail if one drops this last condition of uniformly small scaled energy. Moreover, it is shown that \(u_g\) minimizes \(E(\cdot;B)\) in \(H^{1,2}_g(B;N)\), \(u_g\) is smooth in \(B\) and the \(L^3\)- and \(H^{1/2,2}\)-norms of \(\nabla u_g\) can be bounded in terms of \(E(g;S^2)\).
scientific article; zbMATH DE number 1208704