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Results tagged with laplacian
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user 85906
The Laplacian matrix is the representation of a graph in matrix form.
3
votes
Interpolation spaces
mathrm{dom}_{L^2(\Omega)}(-\Delta_D)^{(1-\theta)\alpha}$$ and the elliptic regularity result $$\mathrm{dom}_{L^2(\Omega)}(-\Delta_D) = H^2(\Omega) \cap H^1_0(\Omega),$$
which are valid for the Dirichlet Laplacian … also follows from the first formula for $\alpha = \frac12$ and $$\mathrm{dom}_{L^2(\Omega)}(-\Delta_D)^{\frac12} = H^1_0(\Omega),$$ which is the rather famous Kato square root property for the Dirichlet Laplacian …
- 2,670
3
votes
Embedding of domain of fractional power of Laplacian into Sobolev space for cylindrical domains
This Kato square root property is indeed true from abstract principles and without any regularity on the domain since $-\Delta_D$ is selfadjoint. (For $H^1_0(\Omega)$ being the closure of test functio …
- 2,670