K-curvature equation

I wish to gain some understanding of the MVP of nonlinear elliptic equation by geometric intuition. Linear elliptic equation case First of all, I have a very good geometric explain of the MVP of Laplace equation, i.e. MVP of laplace equation $\Delta u=0$ in $\Omega$ , $\forall B(x_0,r)\subset \Omega$ is a Ball, we have following identity: $\frac{1}{\mu(\partial(B))}\int_{\partial B}u(x)dx=u(x_0)$ I need to point out first, this property is not difficult to proof by standard integral by part meth…

Understanding the k-Hessian Equation and k-Curvature Equation k-Hessian Equation The k-Hessian equation is given by: $H_k(u)=\sigma_k(D^2(u))=f$ (*) where u is admissible, i.e. $\forall 1\leq i\leq k$, $\sigma_i(D^2(u))\geq 0$. This condition ensures that (*) is an elliptic equation. The most important results are as follows: This result could be used to establish some results on singularity points that can be removable. k-Curvature Equation My Ideas: Looking at it as the “average” o…

this is a note after reading the article”” of Cafferalli. in his article,a large type of fully nonlinear elliptic equation has been established.in particular,including the k-curvature equation.and use the continue method,we just need to establish a ingredient estimate,$latex C^2$ estimate in the interior and $latex C^2$ estimate near the boundary.we establish these estimate step by step,base on construct special flexible function and use the maximum principle to establish the first a…