Abstract
In many problems of mathematical physics it is desired to find the numerical value of a quadratic integral of an unknown function, where the unknown function is a solution of a linear boundary value problem consisting of a linear partial differential equation plus linear boundary condition. The quadratic integral in question is usually the quadratic form occurring in a Green's identity for the differential operator involved in the boundary value problem. The present exposition is concerned with two particular instances of this general situation. In section 2, which is based upon references [6] and [ 7] in the bibliography, upper and lower bounds for the torsional rigidity of a cylindrical beam are derived from Schwarz's inequality.
Section 3 is devoted to the estimation of the capacity, and is based upon references [l] and [13] in the bibliography.
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Diaz, J.B. (2011). Upper and Lower Bounds for the Torsional Rigidity and the Capacity, Derived from the Inequality of Schwarz. In: Fichera, G. (eds) Autovalori e autosoluzioni. C.I.M.E. Summer Schools, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10994-2_5
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