论文标题
外部域上的Hessian商方方程
Hessian quotient equations on exterior domains
论文作者
论文摘要
众所周知，Monge-ampère方程的著名Jörgens-Calabi-Pogorelov定理指出，任何经典的（粘度）凸（d^2u）= 1 $ in $ \ MATHBB {r}^n $都必须是Quadratic polynomial。因此，研究这种完全非线性偏微分方程在外部领域上具有适当的无限渐近条件的外部域上的dirichlet问题的存在和唯一定理是一个有趣的话题。作为caffarelli-li在Monge-ampère方程和Bao-li-li方程的延续，本文致力于hessian商的外部迪里奇特问题的解决性，hessian商人方程式$σ_k（λ（λ（d^2u）/quyq niq le for nis（λ） l <k \ leq n $在所有维度上$ n \ geq 2 $。通过引入广义对称亚树种的概念，然后使用Perron的方法，我们建立了粘度溶液的存在定理，其规定的渐近行为与无限的某些二次多项式相近。
It is well-known that a celebrated Jörgens-Calabi-Pogorelov theorem for Monge-Ampère equations states that any classical (viscosity) convex solution of $\det(D^2u)=1$ in $\mathbb{R}^n$ must be a quadratic polynomial. Therefore, it is an interesting topic to study the existence and uniqueness theorem of such fully nonlinear partial differential equations' Dirichlet problems on exterior domains with suitable asymptotic conditions at infinity. As a continuation of the works of Caffarelli-Li for Monge-Ampère equation and of Bao-Li-Li for $k$-Hessian equations, this paper is devoted to the solvability of the exterior Dirichlet problem of Hessian quotient equations $σ_k(λ(D^2u))/σ_l(λ(D^2u))=1$ for any $1\leq l<k\leq n$ in all dimensions $n\geq 2$. By introducing the concept of generalized symmetric subsolutions and then using the Perron's method, we establish the existence theorem for viscosity solutions, with prescribed asymptotic behavior which is close to some quadratic polynomial at infinity.
