Recently I've conisdered a functional derivative estimate on the first spatial derivative of bounded classical solutions $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ to $$ u_t - u_{xx} - f(u) = 0 \ \ \ (x,t)\in \mathbb{R}\times (0,T], $$ $$ u=u_0 \ \ \ (x,t)\in \mathbb{R}\times \{ 0 \} .$$ Here we suppose that $u_0:\mathbb{R}\to\mathbb{R}$ satisfies $u_0\in C^1(\mathbb{R})$ and $f:\mathbb{R}\to\mathbb{R}$ is continuous. We denote the set of bounded, continuous, sufficiently smooth functions $u$ which solve the above Cauchy problem as $I_{T,f,u_0}$. It follows (via a standard Schauder estimate) that $$ ||u_x(\cdot , t)||_\infty \leq \mathcal{F}_t(f,u_0,u)\ \ \ (*) $$ where the funtional $\mathcal{F}_t:C(\mathbb{R})\times C^1(\mathbb{R})\times I_{T,f,u_0}\to\mathbb{R} $ is given by $$ \mathcal{F}_t(f,u_0,u) = ||u_0'||_\infty + \frac{1}{\sqrt{\pi}}\int_0^t \frac{||f(u(\cdot , \tau ))||_\infty}{(t-\tau )^{1/2}} d\tau .$$ In $(*)$ equality holds trivially if we chose $(0,0,0)\in C(\mathbb{R})\times C^1(\mathbb{R})\times I_{T,f,u_0}$. However, we were interested in whether or not equality in $(*)$ could hold non trivially, i.e., if for any $\epsilon >0$, there exists a sequence $$ (f_n,u_{0,n},u_n)\in C(\mathbb{R})\times C^1(\mathbb{R})\times I_{T,f,u_0} $$ for which $||u_{nx}(\cdot , T)||\geq \epsilon $ for all $n\geq N$ and $$ \lim_{n\to\infty} \left( \inf_{t\in (0,T]} \left( ||(u_{nx}(\cdot ,t )||_\infty -\mathcal{F}_t(f_n,u_{0,n},u_n)\right) \right) =0 .$$ Long story short, it does hold non-trivially, so finally, does anyone know of any references to similar results for functional derivative estimates for solutions to elliptic or parabolic boundary value problems?
