As many people guessed, this article just gives a badly explained description of the spectrum of the Laplace operator with Dirichlet and Neumann boundary conditions.
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The Neumann boundary conditions for Laplace's equation specify not the function ? itself on the boundary of " D ", but its normal derivative.
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The fact that T-duality interchanges the usual Neumann boundary conditions with Dirichlet boundary conditions was discovered independently by Horava and by Dai, Leigh, and Shenker.
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Where \ nabla ^ 2 denotes the Laplacian, the Neumann boundary conditions on a domain \ Omega \ subset \ mathbb { R } ^ n take the form:
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Once the strong properties are established in terms of and the Neumann boundary conditions, the " bootstrap " regularity results can be proved exactly as for the Dirichlet problem.
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Different kinds of boundary conditions for the fields may be imposed on the fundamental fields; for example, Neumann boundary condition or Dirichlet boundary condition is acceptable for free bosonic fields.
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Parametrising the boundary conditions by b empowers us to cover the insulating Neumann boundary condition case b = 0, the Dirichlet boundary condition case b = 1, and all cases between.
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The first eigenvalue is zero, if the domain has a boundary and the Neumann boundary condition is used, or if the shape contains no boundary ( e . g . the sphere ).
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The name originates from the replacement of certain elements in the original layout with imaginary charges, which replicates the boundary conditions of the problem ( see Dirichlet boundary conditions or Neumann boundary conditions ).
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As a third case, exploiting also to the arbitrariness of S _ R, we can choose a Neumann boundary condition of F _ { h } tangent to S _ R in any point.
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