# Reference request: Is if possible to estimate the local behaviour of the solution of $\nabla \cdot a(x) \nabla f=g$ via constant coefficients?

Consider the divergence form uniformly elliptic operator $$\nabla \cdot a(x) \nabla$$ where the coefficient $$a$$ are smooth and bounded and $$D$$ is a bounded and smooth domain of $$\mathbb R^d$$ $$\begin{cases} \nabla \cdot a(x) \nabla f (x)=g \text{ in } D \\ f(x) = 0 \text{ in } \partial D, \end{cases}$$ where $$g$$ for some $$g$$. Consider now $$x_0\in D$$ and $$\delta < d(x,\partial D)$$ and the function $$f_{x_0}$$ which solves $$\begin{cases} \nabla \cdot a(x_0) \nabla f^\delta_{x_0} (x)=g \text{ in } B(x_0,\delta)\\ f^\delta_{x_0}(x) = f(x) \text{ in } \partial B(x_0,\delta). \end{cases}$$

I was wondering whether it is possible to bound quantities such as $$M(x_0,\delta,r,p):=r^{-d}\|f-f^\delta_{x_0}\|_{L^p(B(x_0,r))}$$ for $$r < \delta$$ and for some $$p \in [1,\infty]$$. In particular, I was wondering about the case asymptotic behaviour for $$r \to 0^+$$. That is, can I show that $$M(p, \gamma):= \sup_{x_0 \in D} \sup_{\delta < d(x,\partial D)\wedge c_a} \sup_{r \le \delta} \frac{M(x_0,\delta,r,p)}{r^\gamma},$$ is finite for some $$\gamma>0$$ and some constant $$c_a >0$$? If so, does that bound depends on the smoothness of $$g$$?

The idea being that if $$\delta$$ is sufficiently small, $$a(x)\approx a(x_0)$$ in the ball $$B(x_0,\delta)$$ and therefore the two equations should behave similarly. I am not sure if this is indeed enough or if I would need to ask $$\delta$$ to vanish as well.

I would appreciate any references or even what are the keywords to find such type of estimates.

• If I am parsing things right, for $\gamma > 0$, if $M(p,\gamma)$ is finite this would require that $f(x_0) = f^{\delta}_{x_0}(x_0)$ for every $x_0$ and every compatible $\delta$. (This is just using the inner most $\sup$.) I don't think this can be expected. May 10, 2021 at 17:47
• On the probability side, such "freezing coefficients" approach is known as the parametrix method. But I do not know if it is really related to your question. May 10, 2021 at 18:42
• In regularity theory of PDEs and for minimization problems in calcvar similar ideas also occur. A related concept might also be that of a "blow up-limit", i.e. if you take the solution to the variable $a$ problem and rescale it around $x_0$, then in the limit will usually be a solution to the constant $a$-problem. (though on $\mathbb{R}^d$, instead of with the boundary data you requested).
– mlk
May 10, 2021 at 19:07
• @mlk Do you have a reference for this? May 11, 2021 at 13:37
• @Kernel I haven't read the book myself, but I think "Elliptic regulary theory" by Lisa Beck might be worth a read. At least from a quick glance there is a section on blow up and the book uses Morrey and Campanato spaces, which are extremely related to your definition of $M$.
– mlk
May 11, 2021 at 14:14

As Willie Wong pointed out, there are issues with the quantity you are looking at. But you can fine-tune your question in the one dimensional case, as you cannot expect better regularity than in $$1$$-d case. Not trying to insult anyone's intelligence, but the solution is in that case an explicit function of $$a$$ and $$g$$.
Take $$\Omega=(-\ell,\ell)\subset\mathbb{R},$$ and $$g=1$$. The solution in $$H_{0}^{1}\left(\Omega\right)$$ of $$\textrm{div}\left(a\nabla f\right)=1\text{ in }\left(-\ell,\ell\right)$$ is $$f=\int_{-\ell}^{x}\frac{t-z_{\ell}}{a(t)}dt,$$ and $$f^{\prime}\left(x\right)=\frac{1}{a(x)}\left(x-z_{\ell}\right),$$ with $$z_{\ell}=\frac{\int_{-\ell}^{\ell}\frac{t}{a(t)}dt}{\int_{-\ell}^{\ell}\frac{1}{a(t)}dt}$$ The solution $$f-f_{x_{0}}^{\delta}$$ satisfy \begin{align*} \textrm{div}\left(a\left(x_{0}\right)\nabla\left(f-f_{x_{0}}^{\delta}\right)\right) & =\textrm{div}\left(\left(a\left(x_{0}\right)-a\right)\nabla f\right)\text{when }\left|x-x_{0}\right|<\delta.\\ f-f_{x_{0}} & =0\text{ when }\left|x-x_{0}\right|=\delta \end{align*} in other words $$f-f_{x_{0}}^{\delta}\in H_{0}^{1}\left(x_{0}-\delta,x_{0}+\delta\right)$$ satisfies $$\left(f-f_{x_{0}}^{\delta}\right)^{\prime\prime}=\left(\left(\frac{1}{a\left(x\right)}-\frac{1}{a\left(x_{0}\right)}\right)\left(x-z_{\ell}\right)\right)^{\prime},$$ that is, $$f\left(x\right)-f_{x_{0}}^{\delta}\left(x\right)=\int_{x_0-\delta}^{x}\left(\left(\frac{1}{a\left(t\right)}-\frac{1}{a\left(x_{0}\right)}\right)\left(t-z_{\ell}\right)-\kappa\right)dt,$$ with $$\kappa=\frac{1}{2\delta}\int_{x_0-\delta}^{x_0+\delta}\left(\frac{1}{a\left(t\right)}-\frac{1}{a\left(x_{0}\right)}\right)\left(t-z_{\ell}\right)dt.$$
If for example $$a=\frac{1}{2\ell+x}$$, $$z_\ell=\frac16 \ell$$, $$\kappa=\frac13 \delta^2$$, $$f\left(x\right)-f_{x_{0}}^{\delta}\left(x\right)=\frac{(2x_0-\ell+4x)(x_0+\delta-x)(x_0-\delta-x)}{24\delta}.$$ At $$x=x_0$$, $$f\left(x_0\right)-f_{x_{0}}^{\delta}\left(x_0\right)=\frac{\delta}{24}(\ell-6x_0)$$, which is not null anywhere on the interval.