There are no constraints here on the values of the function " f " ( " x ", " y " ) on the boundary of the lattice grid, thus this is the case of the homogeneous Neumann boundary condition, i . e ., free boundary.
22.
The equations of motion of string theory require that the endpoints of an open string ( a string with endpoints ) satisfy one of two types of boundary conditions : The Neumann boundary condition, corresponding to free endpoints moving through spacetime at the speed of light, or the Dirichlet boundary conditions, which pin the string endpoint.
23.
Mixed Dirichlet / Neumann boundary conditions were first considered by Warren Siegel in 1976 as a means of lowering the critical dimension of open string theory from 26 or 10 to 4 ( Siegel also cites unpublished work by Halpern, and a 1974 paper by Chodos and Thorn, but a reading of the latter paper shows that it is actually concerned with linear dilation backgrounds, not Dirichlet boundary conditions ).
24.
In other words, we can solve for " ? ( x ) " everywhere inside a volume where either ( 1 ) the value of " ? ( x ) " is specified on the bounding surface of the volume ( Dirichlet boundary conditions ), or ( 2 ) the normal derivative of " ? ( x ) " is specified on the bounding surface ( Neumann boundary conditions ).
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