PDE

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…

Direchlet principle: $ \Omega \subset R^n$ is a compact set with $C^1$ boundary. then there exists unique solution $f$ satisfied $\Delta f=0$ in $\Omega$, $f=g$ on $ \partial \Omega$. Perron lifting and barrier function We know the standard approach of the Dirichlet principle is perron lifting and construction of barrier function on the boundary. The key point is if we define the variation energy $ E(u)=\int_{\Omega}|\nabla u|^2$, then it is easy to see for $ u_1,u_2$ is in perron set, $ E(sup (…

Gromov’s idea applicate to parabolic equation. Key point: 1.rescaling+renormalization. 2.analysis it on every scale.

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…

There is a main obstacle to improve the kakeya conjecture,remain in dimension 3,and the  result established by Tomas Wolff in 1995 is almost the best result in $ R^3$ even until now.the result establish by Katz and Tao can be view as a corollary of Wolff’s X-ray estimate. For $ f\in L_{loc}^1(R^d)$,for $ 0<\delta<1$: $ f_{\delta}^*:P^{d-1}\longrightarrow R$.$ f_{\delta}^*(e)=\sup_{T}\frac{1}{|T|}\int_{T}|f|$. $ T$ is varise in all cylinders with length 1.radius $ \delta$.axis in…

In this note we discuss the Schauder theory for uniformly elliptic linear equations and Sobelov inequality. the three main topics ars a priori estimate in Holder norms,regularity of arbitrary solutions and the solvability of the Dirichlet problem.Among these topics,a priori estimates are the most fundamental and the basis of the follows two.we will discuss both the interior Schauder estimate and global Schauder estimate. -Schauder Theory- 1. Interior Schauder Theory $ {\Omega} $ be a domain in $…