Mathematically, for a function u harmonic in a domain \ Omega \ subset R ^ n, the Dirichlet-to-Neumann operator maps the values of u on the boundary of \ Omega to the normal derivative \ partial u / \ partial n on the boundary of \ Omega.
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The most well known boundary value problems for elliptic PDEs are either the " Dirichlet problem " where u is prescribed on the boundary, or the " Neumann problem " where the normal derivative, denoted by u _ { \ omega }, is prescribed on the boundary.
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The geometric interpretation of this condition is as follows : if data for " u " are prescribed on the surface " S ", then it may be possible to determine the normal derivative of " u " on " S " from the differential equation.
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The Dirichlet problem for " L " is to find a function " u ", given a function " f " and some appropriate boundary values, such that " Lu = f " and such that " u " has the appropriate boundary values and normal derivatives.
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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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As " r " tends to zero, the first term on the right hand side tends to ? ( 0 ) and the second to 0, since " r " log " r " tends to 0 and the normal derivatives of ? are uniformly bounded . ( That both sides are equal even before taking limits follows from the fact that the average of a harmonic function on the boundary of a disk equals it value at the centre, while the integral of its normal derivative vanishes .)
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As " r " tends to zero, the first term on the right hand side tends to ? ( 0 ) and the second to 0, since " r " log " r " tends to 0 and the normal derivatives of ? are uniformly bounded . ( That both sides are equal even before taking limits follows from the fact that the average of a harmonic function on the boundary of a disk equals it value at the centre, while the integral of its normal derivative vanishes .)
