= Trace operator
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{title2=$T:H^1(\Omega)\to L^2(\partial\Omega)$}
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= Sobolev trace operator
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= Sobolev trace
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On a bounded <Lipschitz domain>, the <Sobolev trace operator> is the continuous linear extension to $H^1(\Omega)$ of the boundary restriction of smooth functions. Continuity and density make this extension unique; its existence is the <Sobolev trace theorem>. The kernel prescribes a homogeneous <Dirichlet boundary condition>. Vanishing <Sobolev trace> on only part of the boundary instead gives the form domain for a <mixed boundary condition>, with the natural <Neumann boundary condition> on the remainder. On an assembly of congruent tiles, taking a <Sobolev trace> commutes with a constant matrix acting on the tile restrictions. This lets a <transplantation theorem> be proved on <quadratic form> domains even when corner eigenfunctions have singular classical derivatives.
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