It is well known that for Lipschitz domains $\Omega\subset \Bbb{R}^2$ one can define linear extension operators $$ E: H^1(\Omega) \to H^1(\Bbb{R}^2).$$ I am interested in explicit upper bounds for the norm of the extension operator. In most proofs it is indicated that the constant depends on the dimension and the Lipschitz constant, but are there works in the literature which give explicit constants for simple cases like:

  • triangles?
  • (regular) polygons?

I am aware of the reflection, cutoff technique: reflect as many times as needed the contents of $D$ in a neighborhood. The Lipschitz constant will give an upper bound on the number of copies needed. Use a cutoff function to eliminate points far away, with a gradient related to the diameter of $\Omega$.

I am curious if one can do better than this, especially in simple cases, like the ones mentioned above.



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