On the asymptotic Dirichlet problem for the minimal hypersurface equation in a Hadamard manifold (Q1681864)

Let \((M,g)\) be an \(n\)-dimensional complete and simply connected Riemannian manifold endowed with negative sectional curvature and let \(\mathcal{Q}[u]\) be the operator defined as the divergence of \(\mathcal{A}(|\nabla u|^2)\nabla u\) such that \(\mathcal{A}\) is a real positive (smooth) function on \(\mathbb R_+=[0,\infty)\). By definition the asymptotic Dirichlet problem on \((M,g)\) for \(\mathcal{Q}\) (ADP for short) states that for a given continuous function \(h\) on \(M(\infty)\), the sphere at infinity, does there exist a unique continuous function \(u\) on \(M\cup M(\infty)\) satisfying \(\mathcal{Q}[u]_{\restriction_{M}}=0\) and \(u_{\restriction_{M(\infty)}}=h\)? In the introduction, the authors recall some known results on the solvability of the ADP for the particular cases, e.g., when \(\mathcal{Q}\) is the Laplace-Beltrami operator, that is, when \(\mathcal{A}\) is the unit map (Theorems 1.1 and 1.2). The main result of the present article states that the ADP is solvable, whenever \(\text{Sect}_x(P)\in(-(b\circ\rho)^2(x),-(a\circ\rho)^2(x))\) such that \(\text{Sect}_x(P)\) represents the the sectional curvature of a plane \(P\), a subset of the tangent space of \(M\) at the point \(x\in M\), \(\rho(\cdot)=d(o,\cdot)\) for a fixed point \(o\in M\) where \(d(\cdot,\cdot)\) is the Riemannian distance in \(M\), \(a\) and \(b\) are smooth positive functions on \(\mathbb R_+\) and satisfying additional assumptions (Theorem 1.6). Then, the authors deduce the solvability of the ADP for the minimal graph equation, i.e., when \(\mathcal{A}(t)=(1+t)^{-1/2}\) (Theorem 1.5).