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A Green's function can also be thought of as a right inverse of L. Aside from the difficulties of finding a Green's function for a particular operator, the integral in equation 3 may be quite difficult to evaluate. However the method gives a theoretically exact result.
Green's functions can be expanded in terms of the basis elements (harmonic functions) which are determined using the separable coordinate systems for the linear partial differential equation. There are many expansions in terms of special functions for the Green's function. In the case of a boundary put at infinity with the boundary condition ...
A nonlinear operator = (,, | |) is elliptic if its linearization is; i.e. the first-order Taylor expansion with respect to u and its derivatives about any point is an elliptic operator. Example 1 The negative of the Laplacian in R d given by − Δ u = − ∑ i = 1 d ∂ i 2 u {\displaystyle -\Delta u=-\sum _{i=1}^{d}\partial _{i}^{2}u} is a ...
is the derivative of the Green's function along the inward-pointing unit normal vector ^. The integration is performed on the boundary, with measure d s {\displaystyle ds} . The function ν ( s ) {\displaystyle \nu (s)} is given by the unique solution to the Fredholm integral equation of the second kind,
The function K(x,y) is variously known as a Green's function, or the kernel of an integral. It is sometimes called the nucleus of the integral, whence the term nuclear operator arises. In the general theory, x and y may be points on any manifold; the real number line or m-dimensional Euclidean space in the simplest cases.
The name comes from the Green's functions used to solve inhomogeneous differential equations, to which they are loosely related. (Specifically, only two-point "Green's functions" in the case of a non-interacting system are Green's functions in the mathematical sense; the linear operator that they invert is the Hamiltonian operator, which in the ...
A Green's function always exists, but unless the domain Ω can be readily decomposed into one-variable problems (see below), it may not be possible to write it down explicitly. Other methods for obtaining Green's functions include the method of images, separation of variables, and Laplace transforms (Cole, 2011).
If X has been selected correctly (as it has in this example), then for μ 0 >> 0 the operator L + μ 0 is positive, and then employing elliptic estimates, one can prove that L + μ 0 : dom(L) → X is a bijection, and its inverse is a compact, everywhere-defined operator K from X to X, with image equal to dom(L).