On the exterior Dirichlet problem for special Lagrangian equations

Abstract

In this paper, we establish the existence and uniqueness theorem of the exterior Dirichlet problem for special Lagrangian equations with prescribed asymptotic behavior at infinity.

Symmetry of solutions of minimal gradient graph equations on punctured space

“…We will write the vector (a 1 , • • • , a n ) simply as a when no confusion can arise. Similar to the strategy as in [2,24,27], we seek for generalized symmetric subsolution of form…”

“…Remark 1.9. The study on exterior Dirichlet problem by Li-Li [24] for τ = π 4 and Li [27] for τ ∈ ( π 4 , π 2 ] also works for punctured space after minor modification. From their proof, the results in Theorem 1.7 still holds with δ 0 changed into…”

Asymptotic behavior of solutions of fully nonlinear equations over exterior domains

“…Essentially, [5] provided a incisive observation to analyse the asymptotic behavior for fully nonlinear equations. Recently, many researchers studied the Liouville theorem and asymptotic behavior for various types of fully nonlinear equations such as k-Hessian equations [2,7], parabolic k-Hessian equations [19], Hessian quotient equations [13], special Lagrangian equations [14,18], Lagrangian mean curvature equations [1], parabolic Monge-Ampère equations [21][22][23], some fully nonlinear degenerate equations [15][16][17], and the references therein. Especially, Li et al [14] investigated the asymptotic behavior at infinity for general fully nonlinear elliptic equations…”

On the exterior Dirichlet problem for Hessian type fully nonlinear elliptic equations

“…Their results were later improved by Li and Lu [30], who gave the sharp conditions for the solvability of the problems considered in [4,7]. In the same spirit, the exterior Dirichlet problem for k-Hessian equations and Hessian quotient equations with the constant right-hand side as well as for the special Lagrangian equations also has been studied in [3,25,31] in the viscosity sense, under a prescribed quadratic condition at infinity. Moreover, the extension of these studies to the corresponding equations with a general right-hand side g satisfying (1.12) were recently treated in [10,22].…”

“…Moreover, due to the abstract form of f and the variance of g, it is a delicate issue to seek appropriate subsolutions and supersolutions of (1.1) for carrying out the Perron process. Especially, we need to present a new technique for the construction of supersolutions in the more general setting (1.1), since we could neither directly pick quadratic polynomials as the desired supersolutions as adopted in [3,7,21,25,27,31] for the case g ≡ 1, nor merely try to obtain such ones parallel to seeking subsolutions as handled in [4,10,22] for those special f from (1.3) and (1.4); see Remark 4.4 for a detailed explanation.…”