The Dirichlet problem for Hessian equations on Riemannian manifolds

Isometric Embedding of Negatively Curved Disks in the Minkowski Space

“…Since ψ u ≥ 0, taking A = max x∈D ψ(x,φ),φ = max ∂D ϕ in Lemma 3.4 we obtain an admissible subsolution for the Dirichlet problem (3.9), (3.11). Theorem 3.3 now follows from Theorem 1.3 in [6]. The proof of Theorem 3.3 is complete and therefore so is that of Theorem 1.2.…”

“…By Theorem 1.3 in [6], in order to prove Theorem 3.3 we only need to construct an admissible subsolution attaining the same boundary data. …”

“…We shall modify the constructions in [2] and [6]. For convenience we assume σ k is normalized so that…”

Locally convex hypersurfaces of constant curvature with boundary

“…In [11], page 611, and subsequent papers by the first author (see [7], [8]) we used T (u−u) instead of T (u−ϕ) in similar calculations. As a result, the constants in the corresponding inequalities to (2.30) depend on the third derivatives of u.…”

Existence of viscosity solutions with asymptotic behavior of exterior problems for Hessian equations

“…There exist many excellent results in the case of bounded domains, see for examples [2,3,7,13,16] and the references therein. Caffarelli, Nirenberg and Spruck [2,3] and Trudinger [15] established the classical solvability of the Dirichlet problems under various hypothesis.…”