Sobolev and Trace Spaces

The natural setting for the Conductivity Equation is the Sobolev space

$$H^1(\Omega )=\{u\in L^2(\Omega ):\nabla u\in L^2(\Omega {)}^n\},\Vert u{\Vert }_{H^1}^2=\Vert u{\Vert }_{L^2}^2+\Vert \nabla u{\Vert }_{L^2}^2.$$

A function in $H^1(\Omega )$ has no pointwise boundary values. It does, however, have a well-defined trace. On a Lipschitz domain the trace operator

$$\mathrm{tr}:H^1(\Omega )\to H^{1/2}(\partial \Omega ),u\mapsto u{|}_{\partial \Omega },$$

is bounded and surjective. Its image $H^{1/2}(\partial \Omega )$ is the space of admissible boundary voltages. Boundary currents $\gamma {\partial }_{\nu }u$ live in the dual space $H^{-1/2}(\partial \Omega )$. The pairing $⟨g,f⟩$ extends ${\int }_{\partial \Omega }gf\mathrm{d}s$.

Useful subspaces:

• $H_0^1(\Omega )$: functions with zero trace (test space of the Dirichlet Problem);
• $H^1(\Omega )/\mathbb{R}$: functions modulo constants (solution space of the Neumann Problem);
• $H_{\diamond }^{\pm 1/2}(\partial \Omega )$: boundary functions with zero mean, the domain and range of the Neumann-to-Dirichlet Map.

On a smooth closed curve, $H^s(\partial \Omega )$ can be described by Fourier coefficients: $\Vert f{\Vert }_{H^s}^2\simeq {\sum }_k(1+k^2{)}^s|{\hat{f}}_k{|}^2$. This is the idea behind the Discrete Fractional Sobolev Norms.

Poincaré–Wirtinger inequality. $\Vert u-\bar{u}{\Vert }_{L^2(\Omega )}\le C\Vert \nabla u{\Vert }_{L^2(\Omega )}$, where $\bar{u}$ is the mean of $u$. So $\Vert \nabla u{\Vert }_{L^2}$ is a norm on $H^1(\Omega )/\mathbb{R}$.

References

1. L. C. Evans (2010). Partial Differential Equations, 2nd ed. AMS GSM 19. doi:10.1090/gsm/019
2. W. McLean (2000). Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press. ISBN 978-0-521-66375-5