Dirichlet–Neumann bracketing

Before cutting, a Sobolev function must match across an internal interface. Requiring it additionally to vanish there gives a smaller form domain, corresponding to decoupled pieces with Dirichlet cuts. At the other extreme, allow independent Sobolev functions on the pieces without matching traces. This gives a larger direct-sum form domain, whose natural interface condition is Neumann. The energy remains the sum of piece energies, so the change is a change of admissibility rather than a new integrand.

On a bounded Dirichlet domain, the original outer boundary still has zero trace in both comparisons. Only the artificial interfaces are modified. Replacing every exterior condition by Neumann would define a different comparison and should not be described as leaving the original problem intact. For a closed manifold all interfaces are artificial, making the construction especially transparent. Smooth pieces or suitable Lipschitz pieces permit the standard Sobolev trace and compactness arguments behind the form comparison.

The spectrum of the decoupled operator is the union of the piece spectra, sorted into one nondecreasing list with multiplicity. Equal eigenvalues from two different pieces remain separate entries because their eigenfunctions are independent. Decoupled Neumann pieces can also create several constant zero modes, even when the original connected object had only one. This extra low-energy freedom is consistent with the lower bound, rather than a defect to remove by hand.

For a threshold, the counting function records how many eigenvalues are at most that threshold. Lower eigenvalues produce a larger count. Thus the counting inequalities run from the Dirichlet-cut count below the original count to the Neumann-cut count above it. Threshold equality matters: a mode exactly at the threshold is included under our convention. Switching between less-than and at-most conventions midway through a finite example can falsely appear to violate an otherwise correct bracket.

Rectangular boxes and intervals have explicit separated-variable spectra. Covering or partitioning a more complicated region into small pieces therefore turns spectral asymptotics into lattice counts plus a boundary error. The two brackets squeeze the global leading term when the total volume of the troublesome boundary pieces becomes negligible. This is the central geometric idea of a bracketing proof of Weyl's law, even though a full proof must control the partition and its limit carefully.

Refining a partition does not make the finite-threshold bounds uniformly sharp without further information. Artificial cuts can create many low Neumann modes and many high Dirichlet constraints. Bracketing is effective when the scale of the partition is chosen relative to the spectral threshold and the error being sought. It supplies inequalities, not an exact assembly rule for matching eigenfunctions across interfaces; rebuilding the original eigenfunction also requires its continuity and flux compatibility.