12
$\begingroup$

Let $\Gamma$ be a smooth hypersurface in $\mathbb{R}^n$. We can define the fractional Sobolev space $$X = \{ u \in L^2(\Gamma) \mid |u|_X^2 := \int_\Gamma \int_\Gamma \frac{|u(x)-u(y)|^2}{|x-y|^{n+1}} < \infty\}$$ with norm $\lVert u \rVert_X^2 := \lVert u \rVert_{L^2}^2 + |u|_X^2$.

We can also define $Y=(L^2(\Gamma), H^1(\Gamma))_{\frac 12, J}$ as the interpolation space using the J method. It is known that these spaces are equivalent with an equivalence of norms: $$C_1\lVert u \rVert_X \leq \lVert u \rVert_Y \leq C_2\lVert u \rVert_X$$

My question is, do the constants in the equivalence of norms depend on $\Gamma$ only in a nice way i.e. on the diameter of $\Gamma$, eigenvalues of the Laplacian, curvature, etc? Is there some explicit expression for these constants?

I tried Demengel, Adams and Triebel without much success.

$\endgroup$
0
$\begingroup$

Tartar's An Introduction to Sobolev Spaces and Interpolation Spaces Chapter 36 (and the chapters before) might be helpful. His computations seem quite explicit.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.