Regularity for Almost Convex Viscosity Solutions of the Sigma-2 Equation

Ravi Shankar and Yu Yuan

Department of Mathematics,Box 354350,University of Washington,Seattle,WA 98195,USA.

Abstract.We establish interior regularity for almost convex viscosity solutions of the sigma-2 equation.

Key words:Regularity,viscosity solution,fully nonlinear,elliptic partial differential equation.

1 Introduction

In this paper,weestablish regularity for almost convex viscosity solutionsof theσ2equation

One quick consequence is that every entire almost convex(such as in Theorem 1.1)viscosity solution of(1.1)is a quadratic function;the smooth case was donein[5].Recall the classic rigidity results for the equations△u=1 and detD2u=1:every entire convex viscosity solution is quadratic.Our result shows that if a singular viscosity solution of(1.1)exists,then it is not convex,or even almost convex.

The interior regularity for(1.1)in general dimensions is a longstanding problem.Sixty years ago,Heinze[8]achieved a priori estimates and regularity for two dimensional Monge-Amp`ere type equations including(1.1)withn=2 by two dimensional techniques.More than ten years ago,a priori estimates and regularity for(1.1)withn=3 were obtained via the minimal surface structure of equation(1.1)in the joint work with Warren[17].Along this“integral”way,Qiu[12]has proved a priori Hessian estimates–then regularity follows–for threedimensional(1.1)withC1,1variable right hand side,and even with left hand sideλreplaced by the principal curvaturesκ.Hessian estimates for convex smooth solutions of general quadratic Hessian equations in general dimensions have been obtained via a pointwise approach by Guan and Qiu[7].Hessian estimates for almost convex smooth solutions of(1.1)in general dimensions have been derived by a compactness argument in[10],and recently for semiconvex smooth solutions in[13]using new mean value and Jacobiinequalities.

In contrast,there are Pogorelov-like singular convex viscosity solutions of the symmetric Hessian equationsσk(λ)=1 withk≥3 in dimensionn≥3.Under a strict kconvexity assumption on weak/viscosity solutions ofσk(λ)=1,a priori Hessian estimates and then regularity were obtained by Pogorelov[11]and Chou-Wang[6],fork=nand 2≤k

Extending the above a priori estimates to regularity statements about viscosity solutions of(1.1)is more subtle.In dimensions two and three,one can smoothly solve the Dirichlet problem on interior balls with smoothly approximated boundary data;a limiting procedure combined with the a priori estimates then yields the desired interior regularity for the viscosity solution.However,for dimensionn≥4,a prioriestimates are not known for general solutions of(1.1).Because the smooth approximations may not satisfy the convexity constraints,we cannot invoke the available a prioriestimates while taking the limit and deduce interior regularity.

2 Preliminaries

2.1 Smooth functions and solutions

so up to a constant,

which is,in fact,negativethe Legendretransform of striclty convex functionf(x)=?u(x),formulated in extremal form as

The Hessians are related by

2.2 Convex functions and viscosity solutions

Acknowledgments

Ravi Shankar is partially supported by NSFGraduate Research Fellowship Program under Grant No.DGE-1762114,and Yu Yuan by NSFGrant No.DMS-1800495.