# Aleksandrov maximum principle for semiconvex function

Let $$u:\Omega \rightarrow \mathbb{R}$$.

A function $$u$$ is called semiconvex if $$u=v+w$$ for some $$v\in C^{1,1}(\Omega)$$ and convex function $$w$$ (it's equivalent saying that $$u$$ is semiconvex if exists a $$\lambda$$ such that the function $$z(x)=u(x)+\dfrac{|x|^2}{2\lambda}$$ is convex).

Consider the elliptic operator of the form $$Lu=a^{ij}D_{ij}u+b^iD_iu$$ and let $$L$$ be uniformly elliptic.

I want to show the following statement:

Theorem(Aleksandrov maximum principle): Let $$u$$ be semiconvex in $$\Omega$$ and suppose $$Lu+f\geq0$$ almost everywhere in $$\Omega$$ for some $$f\in L^{n}(\Omega)$$. We then have the following estimates: $$\sup_{\Omega}u \leq \sup_{\partial\Omega}u+ C ||f||_{L^n(\Gamma^+)}$$

where $$\Gamma^+$$ is upper contact set of $$u$$ ( a sub domain of $$\Omega$$ where the Hessian of $$u$$ is negative define).

I know this result for subsolution $$u\in W^{2,n}(\Omega)$$, an extension through molltification of the case $$u\in C^2(\Omega)$$. So I thought that I can deduced the validity of my Aleksandrov maximum principle from its validity for classical subsolution, by mollification or something like this.