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Louis Nirenberg 1. Introduction In recent years the interest in extending the theory of boundary value problems of second order elliptic equations to equations of higher order and to systems has grovm considerably. The Dirichlet problem for such equations has been investigated by M. I. Vishik , p. E. Browder  and L. Garding , see also K. 0....
Louis Nirenberg 1. Introduction In recent years the interest in extending the theory of boundary value problems of second order elliptic equations to equations of higher order and to systems has grovm considerably. The Dirichlet problem for such equations has been investigated by M. I. Vishik , p. E. Browder  and L. Garding , see also K. 0. Priedrichs  and C. B. Morrey . The methods used for a single equation extend to theD irichlet problem for an elliptic system inH functions u,,...,i which is of the same order in all functions, and vjhlch is strongly elliptic in the sense defined by Vishik . Other boundary value problems have been considered by Browder  and JacquesL ouis Lions  .T he successful treatment of the general equation has been based on some form of the projection theorem inH llbert space, as employed for example by H. Veyl in [2?]. The method is, of couT se, related to the classical variational method, theD irichlet principle, which has been of extensive use in treating elliptic equations, see for instance Courant-H ilbert  The applicability of the method is based on a single inequality expressing the positive definiteness of theD irichlet integral associated vith the differential operator. This inequality furnishes boiinds on the square integrals of derivatives of solutions. For general equations with variable coefficients this inequality is due to L. Garding , and extends immediately to Vishik sstrongly elliptic systems. For second order systems with constant coefficients the inequality was derived earlier by L. Van Hove(Typographical errors above are due to OCR software and don't occur in the book.)About the Publisher Forgotten Books is a publisher of historical writings, such as: Philosophy, Classics, Science, Religion, History, Folklore and Mythology.Forgotten Books' Classic Reprint Series utilizes the latest technology to re
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