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 $latex \Delta u=0$ in $latex \Omega$ , $latex \forall B(x_0,r)\subset \Omega$ is a Ball, we have following identity: $latex \frac{1}{\mu(\partial(B))}\int_{\partial B}u(x)dx=u(x_0)$   I need to point out first, this property is not difficult to p…

here is the problem, how to understand k-hessian equation and k-curvature equation. k-hessian equation k-hessian equation is: $latex H_k(u)=\sigma_k(D^2(u))=f$ (*) where u is admissible, i.e. $latex \forall 1\leq i\leq k$, $latex \sigma_i(D^2(u))\geq 0$. this is just the condition to make (*) be a elliptic equation. The most important result is the following three: 1.sovable (*) with direchlet boundary condition. This is mainly the contribution of Caffaralli in 90’s. According flexible fun…

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…