Singular Solutions of Hessian Fully Nonlinear Elliptic Equations
Singular Solutions of Hessian Fully Nonlinear Elliptic Equations
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Hessian全非线性椭圆方程的奇异解
DOI：
10.1016/j.aim.2011.06.030
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发表时间：
2008
期刊：
影响因子：
--
通讯作者：
S. Vladuts
中科院分区：
文献类型：
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作者：
N. Nadirashvili;S. Vladuts
我们研究了Hessian完全非线性一致椭圆型方程，并证明了这些方程（在12维或更多维）的粘性解的二阶导数可以在域的内点爆破。我们证明了这类解的最优内部正则性不超过C1+ π，从而证明了已知内部正则性结果的最优性.同样的道理也适用于Isaacs方程。证明了11维完全非线性Hessian一致椭圆型方程非光滑解的存在性。我们还研究了定义在点邻域内的Hessian方程解的可能奇异性，证明了穿孔球上的Hessian一致椭圆方程的齐次0<α<1阶解应该是径向的.
We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+ϵ, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.