论文标题
与梯度无关的完全非线性方程椭圆形分支的粘度解的比较原理
Comparison principles for viscosity solutions of elliptic branches of fully nonlinear equations independent of the gradient
论文作者
论文摘要
研究了可变系数完全非线性梯度无势能理论中比较原理的有效性，然后用于证明确定合适潜在理论的完全非线性部分偏微分方程的比较原理。该方法结合了受Krylov（Trans。Amer。Math。Soc。1995）启发的适当椭圆分支的概念与Harvey and Lawson（Comm。PureAppl。Math。2009）发起的单调性二元方法。在可变系数的非线性潜在理论中，定义潜在理论的适当椭圆形图$θ$的Hausdorff连续性发挥了特殊作用。在操作员$ f $定义的非线性方程式的应用中，将确定$ f $的结构条件，在$θ$ -subharmonics/superharmonics/superharmonics和可接收的粘度子和非线性方程式的可接受的粘度子和等式的相比，该方程与相关的兼容潜在理论相比。将检查一般结果和差异几何学的明确模型。
The validity of the comparison principle in variable coefficient fully nonlinear gradient free potential theory is examined and then used to prove the comparison principle for fully nonlinear partial differential equations which determine a suitable potential theory. The approach combines the notions of proper elliptic branches inspired by Krylov (Trans. Amer. Math. Soc. 1995) with the monotonicity-duality method initiated by Harvey and Lawson (Comm. Pure Appl. Math. 2009). In the variable coefficient nonlinear potential theory, a special role is played by the Hausdorff continuity of the proper elliptic map $Θ$ which defines the potential theory. In the applications to nonlinear equations defined by an operator $F$, structural conditions on $F$ will be determined for which there is a correspondence principle between $Θ$-subharmonics/superharmonics and admissible viscosity sub and supersolutions of the nonlinear equation and for which comparison for the equation follows from the associated compatible potential theory. General results and explicit models of interest from differential geometry will be examined.
