Question

Let for every $i=1,\dots,n$ and $j=1,\dots,n$ be given functions $a^{ij}\colon\mathbb{R}^n\to \mathbb{R}$, $b^{i}\colon\mathbb{R}^n\to \mathbb{R}$, $c\colon\mathbb{R}^n\to \mathbb{R}$ from space $C^\infty_b(\mathbb{R}^n,\mathbb{R})$ of all bounded real functions on $\mathbb{R}^n$, having bounded derivatives of all orders. Let me define differential operator $L$ by equality

$$(Lu)(x)=\sum_{i=1}^n\sum_{j=1}^na^{ij}(x) \frac{\partial^2}{\partial x_i\partial x_j}u(x) + \sum_{i=1}^nb^i(x)\frac{\partial}{\partial x_i}u(x)+c(x)u(x).$$

Let's assume that exists constant $\varkappa>0$, such that for $\xi=(\xi_1,\dots,\xi_n)\in\mathbb{R}^n$ and all $x\in\mathbb{R}^n$ ellipticity condition holds: $\sum_{i=1}^n\sum_{j=1}^na^{ij}(x)\xi_i\xi_j\geq\varkappa\|\xi\|^2.$ Let's also assume that $c(x)\leq 0$ for all $x\in\mathbb{R}^n$.

Let constant $\lambda>0$ be fixed.

Let function $\psi\in C^\infty_b(\mathbb{R}^n,\mathbb{R})$ be also fixed, and $f$ be a solution (it exists and is unique in a class $C^\infty_b(\mathbb{R}^n,\mathbb{R})$) of an equation $$Lf-\lambda f=\psi.$$

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QUESTION. What can we say about uniform upper estimate for derivatives of $f$, i.e. for $\sup_{x\in\mathbb{R}^n}|\nabla f(x)|$ и $\sup_{x\in\mathbb{R}^n}\left|\frac{\partial^2}{\partial x_i\partial x_j} f(x)\right|?$ We can use in estimates $\lambda, \varkappa$ and all derivatives of $\psi$. Also in estimating we can use maximum of absolute values of coefficients of operator $L$, but not their derivatives.

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I know, for instance, that $\|Lf-\lambda f\|\geq \lambda\|f\|$, and thus on the strength of equation $\|f\|\leq \frac{1}{\lambda}\|\psi\|$. But what can we say about derivatives of $f$?

It will be ideal if you post link for an article or a book in which the estimate is proven.

I have already looked in books by Krylov, by Ladyzhenskaya and Uraltseva, and by Gilbarg and Trudinger, but found only estimates in Holder spaces involving Holder norm of coefficients of $L$, which is not helpful because I can control only maximum of absolute values of coefficients of $L$.

I also heard something about de Giorgi estimates, but I don't know what are they.

Thank you in advance!

Answer 1

In 2 dimensions, Taylor's PDE III (Theorem 16.1) gives an interior estimate of the type you are seeking on $C^{1,\mu}$-norm of the solution. The result is due to Morrey. Taylor mentions that this is stronger than DeGiorgi-Nash-Moser estimates, which work for higher dimensions and give only a $C^{0,\mu}$-bound. I don't know if DeGiorgi-Nash-Moser can somehow be massaged to produce the desired estimates, at least in some special circumstances.