We develop the method of Fischer-Riesz equations for general boundary value problems elliptic in the sense of Douglis-Nirenberg. To this end we reduce them to a boundary problem for a (possibly overdetermined) first-order system whose classical symbol has a left inverse. For such a problem there is a uniquely determined boundary value problem which is adjoint to the given one with respect to the Green formula. On using a well-elaborated theory of approximation by solutions of the adjoint problem, we find the Cauchy data of solutions of our problem. 1. Introduction The aim of this paper is to bring together two areas in which integral formulas like Green’s formula for harmonic functions are of great importance. The first area is complex analysis where the method of integral representations was a central tool over the second half of the 20th century, see [1]. And the second area is the theory of elliptic boundary problems where the parametrix method led to the most refined results, such as local principle, -algebras of pseudodifferential boundary value problems [2], and so forth. The method of Fischer-Riesz equations can be specified within a larger approach which is usually referred to as the boundary element method. By this latter is meant a numerical method of solving boundary value problems which have been formulated as boundary integral equations. It can be applied in many areas of engineering and science including fluid mechanics, acoustics, electromagnetics, and fracture mechanics, see [3–6]. The boundary elements method attempts to use the given boundary conditions and other data of the problem to fit boundary values into the integral equation, rather than values throughout the space defined by a system of partial differential equations. Once this is done, the boundary integral equation can be used again to calculate numerically the solution directly at any desired point in the solution domain. More precisely, from the Cauchy data of the solution on the whole boundary one calculates readily the solution in the domain provided a left fundamental solution of the system is available in an explicit form, see for instance Lemma？？10.2.3 in [7]. The idea of the method of Fischer-Riesz equations goes back at least as far as [8]. The paper [9] was given by Picone as an invited address before the Second Austrian Mathematical Congress in Insbruck in 1949. He states in the introduction that he asked Fichera to write a certain part of the report. It is a crystallisation in the form of an abstract theory of some of the methods used by the authors and their
References
[1]
L. A. Aizenberg and A. P. Yuzhakov, Integral Representations and Residues in Many-Dimensional Complex Analysis, Nauka, Novosibirsk, Russia, 1979.
[2]
L. Boutet de Monvel, “Boundary problems for pseudo-differential operators,” Acta Mathematica, vol. 126, no. 1-2, pp. 11–51, 1971.
[3]
L. S. Wrobel and M. H. Aliabadi, The Boundary Element Method, John Wiley & Sons, New York, NY, USA, 2002.
[4]
J. T. Katsikadelis, Boundary Elements Theory and Applications, Elsevier, Oxford, UK, 2002.
[5]
W. C. Gibson, The Method of Moments in Electromagnetics, Chapman & Hall/CRC, London, UK, 2008.
[6]
G. Beer, I. Smith, and C. Duenser, The Boundary Element Method and Programming: For Engineers and Scientists, Springer, Berlin, Germany, 2008.
[7]
N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations, vol. 7 of Mathematical Topics, Akademie, Berlin, Germany, 1995.
[8]
M. Picone, “Nuovi metodi per il calcolo delle soluzioni delle equazioni a derivate parziali della fisica matematica,” Annales Scientiques de l'Universite Jassy, vol. 26, pp. 183–232, 1940.
[9]
M. Picone and G. Fichera, “Neue funktional-analytische Grundlagen für die Existenzprobleme und L？sungsmethoden von Systemen linearer partieller Differentialgleichungen,” Monatshefte für Mathematik, vol. 54, no. 3, pp. 188–209, 1950.
[10]
V. D. Kupradze, “Approximate solution of problems of mathematical physics,” Uspekhi Matematicheskikh Nauk, vol. 22, no. 2, pp. 59–107, 1967.
[11]
O. V. Karepov and N. N. Tarkhanov, “The Fischer-Riesz equations method in the ill-posed Cauchy problem for systems with injective symbols,” Journal of Inverse and Ill-Posed Problems, vol. 1, no. 4, pp. 307–330, 1993.
[12]
A. Douglis and L. Nirenberg, “Interior estimates for elliptic systems of partial differential equations,” Communications on Pure and Applied Mathematics, vol. 8, pp. 503–538, 1955.
[13]
M. H. Protter, “Overdetermined first order elliptic systems,” in Maximum Principles and Eigenvalue Problems in Partial Differential Equations (Knoxville, TN, 1987), P. W. Schaefer, Ed., vol. 175 of Research Notes in Mathematics, pp. 68–81, Pitman, New York, NY, USA, 1988.
[14]
L. R. Volevi？, “Solubility of boundary value problems for general elliptic systems,” Matematicheski？ Sbornik, vol. 68, no. 110, pp. 373–416, 1965.
[15]
C. Cosner, “On the definition of ellipticity for systems of partial differential equations,” Journal of Mathematical Analysis and Applications, vol. 158, no. 1, pp. 80–93, 1991.
[16]
C. B. Morrey, “Second-order elliptic systems of differential equations,” in Contributions to the Theory of Partial Differential Equations, vol. 33 of Annals of Mathematics Studies, pp. 101–159, Princeton University Press, Princeton, NJ, USA, 1954.
[17]
K. Krupchyk, W. M. Seiler, and J. Tuomela, “Overdetermined elliptic systems,” Foundations of Computational Mathematics, vol. 6, no. 3, pp. 309–351, 2006.
[18]
M. S. Agranovi？, “Boundary value problems for systems of first order pseudodifferential operators,” Uspekhi Matematicheskikh Nauk, vol. 24, no. 1, pp. 61–125, 1969.
[19]
Y. Roitberg, Elliptic Boundary Value Problems in the Spaces of Distributions, vol. 384 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1996.
[20]
A. P. Calderón, “Boundary value problems for elliptic equations,” in Outlines Joint Sympos. Partial Differential Equations (Novosibirsk, 1963), pp. 303–304, Academy of Sciences of the USSR Siberian Branch, Moscow, Russia, 1963.
[21]
L. R. Volevich and A. R. Shirikyan, “Stable and unstable manifolds for nonlinear elliptic equations with a parameter,” Transactions of the Moscow Mathematical Society, vol. 61, pp. 97–138, 2000.
[22]
I. V. Shestakov and A. A. Shlapunov, “On the Cauchy problem for operators with injective symbols in the spaces of distributions,” Journal of Inverse and Ill-Posed Problems, vol. 19, no. 1, pp. 127–150, 2011.
[23]
I. Stern, “Boundary value problems for generalized Cauchy-Riemann systems in the space,” in Boundary Value and Initial Value Problems in Complex Analysis: Studies in Complex Analysis and Its Applications to Partial Differential Equations, I (Halle, 1988), vol. 256 of Pitman Research Notes in Mathematics Series, pp. 159–183, Longman Scientific and Technical, Harlow, UK, 1991.
[24]
I. Stern, “On the existence of Fredholm boundary value problems for generalized Cauchy-Riemann systems in the space,” Complex Variables: Theory and Application, vol. 21, no. 1-2, pp. 19–38, 1993.
[25]
I. Stern, “Direct methods for generalized Cauchy-Riemann systems in the space,” Complex Variables: Theory and Application, vol. 23, no. 1-2, pp. 73–100, 1993.
[26]
K. O. Friedrichs and P. D. Lax, “Boundary value problems for first order operators,” Communications on Pure and Applied Mathematics, vol. 18, pp. 355–388, 1965.
