For my master thesis, I solved a PDE under the assumption of the domain being smooth and small. I wanted to patch these domains and solutions somehow together, hoping that I can get a global result.
With that in mind, I asked myself, whether one can tile the plane using bounded domains with smooth boundary.
By tiling, I mean a family of closed sets covering the entire plane, such that the interiours are pairwise disjoint. My reasonable assumptions on the sets are, that they should be closed domains, with a uniform bound on the diameter, having a smooth boundary and the family needs to be locally finite.
It seems impossible to me, if one restricts to locally finite tilings, since when two adjacent boundaries separate, they always leave a cusp, which can’t be filled by a smooth and locally finite tiling.
Does anyone have a clue if such a smooth tilings can exist or if they have a name? Maybe I missed something, but I think the cusps are inevitable.
Edit 1: here are the reasons for my assumptions: If one removes one of the assumptions (smooth boundary, uniformly bounded and locally finite)
without smooth boundary, one can take for example a square grid tiling the plane
without boundedness, one could take two half planes. Without a uniform bound on the diameter, one could take concentric rings which tile the plane.
without the local finiteness, one could use a Vitali-like covering using balls.
Edit 2: My sketch of the proof, that it is impossible to tile the plane with these sets goes as follows. Assume such a tiling exists. If two adjacent sets touch, then the touching boundaries either seperate at some point, or the sets share a connected component of their boundaries. If they seperate, one sees by local graph description of the sets, that he have a cusp, which can’t be filled by a single smoothly bounded set (maybe by a sequence of domains,but that is against the local finiteness). Therefore, sets that share a point at their boundary must actually share the entire connected component of the boundary. For each set $A$, there is some set $B$ that touches the most outer boundary enclosing the set $A$ and must therefore share this outermost connected component. This makes it, so that $B$ has strictly larger diameter, since $A$ is surrounded by $B$. This process continues for the outermost boundary of $B$ and if we use the local finiteness, one can continue arbitrary often to find that the sets have unbounded diameter. This is a contradiction to the hypothesis, that such a tiling exists.