Description (Answer1).
In particular, we consider
Problem 1.
Let $a>0$ and $c,d\in R$ be given constants and let $u$ be a real valued continuous on $I:=[0,a]$ (more generally bounded on $J:=(0,a)$) and $C^2$ function on $J:=(0,a)$ such that (i): $|u''(x)|\leq c |u'(x)^2| +d$, $x\in (0,a)$. Check whether $|u'(x)|$ is bounded on $J$.
To get a feeling consider example $L(x)=x^s$, $s>0$, on $(0,1)$.
For several dimensional version see
[H] E. Heinz, On certain nonlinear elliptic
differential equations and univalent mappings, J. d' Anal. 5,
1956/57, 197-272.
Problem 2. Consder Laplacian-gradient inequality (known also as Poison inequality) with gradient power $d$:
$|\Delta u| \le a |\nabla u|^d + b\,.$ For example let $u$ be a real valued function defined and continuous on the closed unit disk, C^2 -function in open unit disk. Find under which condition we can conclude that partial derivatives of $u$ are bounded on the open unit disk.
In [H] the case d=2 is consder.
Theorem H. Let $u$ be a real valued function defined and continuous on the closed unit disk, C^2 -function in open unit disk and suppose that (i): the restriction of $u$ on the unit circle C^2 -function.
In addition if we suppose that (ii): $u$ satisfies the Laplacian-gradient inequality with gradient power $2$, then
partial derivatives of $u$ are bounded on the open unit disk.
Please look at these works
Article On the local properties of solutions of the nonlinear Beltra...
https://www.dopovidi-nanu.org.ua/sites/default/files/2019/2/19-02-03.pdf
Answer 2. Using Theorem H (see Answer 1) we can prove:
Let $u: \overline{\mathbb{B}^n} \rightarrow \Omega$ be a twice
differentiable quasiconformal mapping of the unit ball onto the
bounded domain $\Omega$ with $C^2$ boundary, satisfying the Laplacian-gradient inequality with gradient power $2$ (for def see Answer 1, ie.
Poisson differential inequality). Then $\nabla u$ is bounded and
$u$ is Lipschitz continuous.
We can consider the corresponding results if we suppose that $u$ is mapping of finite distortion
with integrable inner distortion instead of $u$ is quasiconformal mapping.
PS. Dear Prof Salimov, thanks for your paper. We will consider your paper
https://www.dopovidi-nanu.org.ua/sites/default/files/2019/2/19-02-03.pdf and try to find connection with Theorem H.
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