Contribution to the Theory of Upper and Lower Bounds in Boundary Value Problems

概要

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In the present paper certain approximate methods concerning self-adjoint operators H of Hilbert spaces which can be expressed in the form H=T^*T are developed. Some theorems are formulated with remarks and examples. They give a method of obtaining upper and lower bounds of the value of the solution, at each preassigned point, of a boundary value problem. As a numerical example we consider the following Poisson's equation, -Δu=1 in D, u=0 on C, where D denotes the square domain in the xy-plane with vertices at (1,-1),(1,1),(-1,1),(-1,-1) and C is the boundary of this square. Upper and lower bounds of the value of the solution u at the origin are calculated. The result of the second approximation is 0.2939≦u(0, 0)≦0.2953. In order to show that the results are applicable to many problems, several differential operators familiar in mathematical physics, for instance, Laplace's differential operators with different boundary conditions are reduced to the form T^*T.
社団法人日本物理学会の論文
1955-01-05