Existence and regularity results for harmonic maps with potential (Q5946903)

Harmonic maps with potential are the critical points of the functional: NEWLINE\[NEWLINE E_{G}(u,\Omega) = \int_{\Omega} [e(u) - G(u)] v_{g} ,NEWLINE\]NEWLINE for a compact domain \(\Omega\) of a Riemannian manifold \((M,g)\), a smooth function \(G\) defined on a Riemannian manifold \((N,h)\) and where \(e(u)\) is the energy density of \(u: (M,g) \to (N,h)\). The associated Euler-Lagrange equation is: \(\tau(u) + \nabla G =0,\) \(\tau(u)\) being the tension field. NEWLINENEWLINENEWLINEWhen \((N,h)\) is the canonical 2-sphere and \(G\) the inner product with a fixed vector, this is the static Landau-Lifshitz equation. NEWLINENEWLINENEWLINEThis article addresses the problem of the existence of minimisers of \(E_{G}\) in a class \(X_{f,R}\) of Sobolev maps with \(L^{\infty}\)-norm bounded by \(R\), under a Dirichlet boundary condition \(f\in H^{1,2}\cap L^{\infty}(\Omega, {\mathbb R}^{n})\). NEWLINENEWLINENEWLINEThe existence is obtained by the direct method of calculus of variations but under the blanket assumption of sub-polynomial growth for the potential and the availability of an upper bound of the maximal radial curvature, by a multiple of the distance function (this ensures normal coordinates and estimates of the nonlinear terms of the tension field). NEWLINENEWLINENEWLINEA finer analysis shows that, if the dimension of the domain is between 2 and 4 and the tangential derivative of the potential also has sub-polynomial growth, then such a minimiser is also a weak harmonic map with potential. NEWLINENEWLINENEWLINEMoreover, the solution is \(C^{2,\alpha}\) on the interior and Hölder continuous, when the boundary data is regular enough. NEWLINENEWLINENEWLINEThe subsequent correction [ibid. 25, No. 1, 205--207 (2002)] modifies the proof of Theorem 2.3 but not its statement.