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Register a new userAsymptotic behavior at infinity of solutions of Lagrangian mean curvature equations
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We studied the asymptotic behavior of solutions with quadratic growth condition of a class of Lagrangian mean curvature equations $F_{tau}(lambda(D^2u))=f(x)$ in exterior domain, where $f$ satisfies a given asymptotic behavior at infinity. When f(x) is a constant near infinity, it is not necessary to demand the quadratic growth condition anymore. These results are a kind of exterior Liouville theorem, and can also be regarded as an extension of theorems of Pogorelov, Flanders and Yuan.
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We obtain a quantitative high order expansion at infinity of solutions for a family of fully nonlinear elliptic equations on exterior domain, refine the study of the asymptotic behavior of the Monge-Amp`ere equation, the special Lagrangian equation and other elliptic equations, and give the precise gap between exterior maximal (or minimal) gradient graph and the entire case.
We show that convex viscosity solutions of the Lagrangian mean curvature equation are regular if the Lagrangian phase has Holder continuous second derivatives.
We classify regularity for a class of Lagrangian mean curvature type equations, which includes the potential equation for prescribed Lagrangian mean curvature and those for Lagrangian mean curvature flow self-shrinkers and expanders, translating solitons, and rotating solitons. We first show that convex viscosity solutions are regular provided the Lagrangian angle or phase is $C^2$ and convex in the gradient variable. We next show that for merely Holder continuous phases, convex solutions are regular if they are $C^{1,beta}$ for sufficiently large $beta$. Singular solutions are given to show that each condition is optimal and that the Holder exponent is sharp. Along the way, we generalize the constant rank theorem of Bian and Guan to include arbitrary dependence on the Legendre transform.
We obtain asymptotic mean value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. We study both when the set of coefficients is bounded and unbounded (each case requires different techniques). The families of equations that we consider include well-known operators such as Pucci, Issacs, and $k-$Hessian operators.
We prove some non-existence theorems for translating solutions to Lagrangian mean curvature flow. More precisely, we show that translating solutions with an $L^2$ bound on the mean curvature are planes and that almost-calibrated translating solutions which are static are also planes. Recent work of D. Joyce, Y.-I. Lee, and M.-P. Tsui, shows that these conditions are optimal.
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