I am no specialist in PDE theory but I am interested where I could find an answer for the following question.

Consider two sets $D_1 \subset D_2 \in \mathbb{R}^d$ (actually only for $d \in \{2,3\}$) and the PDEs $$\Delta u_i = f \text{ on } D_i\\u_i=0 \text{ on } \partial D_i $$ with $f:D_2 \rightarrow \mathbb{R}$. Assume that all the usual conditions on regularity for the occurring functions and boundaries are fulfilled.

Are there some kind of error estimates between the solutions $u_1, u_2$? I am sure there are, but where can I find them?

For the case I haven't been rigorous enough let me put it in words: Are there error estimates for the solution of the Dirichlet problem with homogeneous boundary conditions in case of geometric "uncertainty"?



You can use the maximum principle to get error estimates. Since $w:=u_1-u_2$ is harmonic in $D_1$ we have

$$\max_{D_1} |u_1-u_2| \leq \max_{\partial D_1}|u_1-u_2| = \max_{\partial D_1} |u_2|.$$

Now, to estimate the boundary term, you need some notion of closeness of $D_1$ and $D_2$. For example, let us set

$$\varepsilon = \max\{\text{dist}(x,\partial D_2) \, : \, x \in \partial D_1 \}.$$

Given $D_1,D_2$ are open bounded with sufficiently smooth boundaries, the solution $u_2$ is Lipschitz continuous, and so $|u_2(x)| \leq C\text{dist}(x,\partial D_2)$. Therefore $\max_{\partial D_1} |u_2| \leq C\varepsilon$ and so

$$\max_{D_1} |u_1-u_2| \leq C\varepsilon.$$

There are other conditions you can place on the closeness of $D_1$ and $D_2$. If you want to measure the difference in terms of the measure of $D_2\setminus D_1$, I would try energy methods, though this may be harder. In general the solutions can be much different if the domains are not similar.


You can translate the uncertainty from the geometry to the right hand side.

Let $\phi:D^2\to D^1$ be invertible and sufficiently smooth.

Let $v:=u_1\circ \phi$. Then $$ \Delta v= \Delta u+\epsilon = f+\epsilon \\ v(\partial D_2)=0 $$ Hence $$ \|u_2-v\|\leq C\|\epsilon\| $$ and $$ \|u_1-u_2\|\leq C \|\epsilon\| +K $$ with $\epsilon$ and $K$ depending on $\phi$ and $C$ depending on $D_2$


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