# Questions tagged [sobolev-spaces]

For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

3,498 questions
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### Derivation of weak formulation for surface Laplacian

For the surface Laplacian $\Delta_S= \nabla_S \cdot \nabla_s$ and a twice differentiable function $u$ on the surface $S$ we have $$\Delta_S u=\Delta u-\kappa\frac{\partial u}{\partial n}-n^TH(u)n,$$ ...
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### Density of $C^\infty(\overline{\Omega})$ in $W^{k,2}(\Omega)$

First, $\Omega\subseteq \mathbb{R}^n$ and $$\Omega=\prod_{i=1}^n\left(a_i,b_i\right).$$ I know that the space of test functions $C^\infty_c(\Omega)$ is indeed dense in $W^{k,2}(\Omega)$. Is there ...
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### Differentiability under the integral sign

Do we have some kind of Lebesgue theorem in the case of an integral defined over a boundary? $\frac{d}{dt}\int_{\partial\Omega}f(t,x)d\sigma$
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### Compact embedding of fractional space

Is the space $H^\lambda((0,T); H^1(K))$ for $0 <\lambda <1$ where $K$ is compact subset of $\mathbb{R}^n$ compactly embedded in $L^2( (0,T) \times K)$? \$H^\lambda((0,T); H^1(K))\hookrightarrow \...