Chaabi, Slah; Rigat, Stéphane Decomposition theorem and Riesz basis for axisymmetric potentials in the right half-plane. (English) Zbl 1327.30055 Eur. J. Math. 1, No. 3, 582-640 (2015). Summary: We consider the Weinstein equation, also known as the equation governing generalized axisymmetric potentials (GASP), with complex coefficients \(L_mu={\Delta } u+(m/x)\partial _x u =0\), \(m\in {\mathbb {C}}\). We generalize results known for \(m\in {\mathbb {R}}\) to the case \(m\in {\mathbb {C}}\). In particular, explicit expressions of fundamental solutions for Weinstein operators and their estimates near singularities are presented, a Green’s formula for GASP in the right half-plane \({\mathbb {H}}^+\) for \(\mathrm{Re}\,m<1\) is established. We prove a new decomposition theorem for the GASP in annular domains for \(m\in {\mathbb {C}}\), which is in fact a generalization of the Bôcher’s decomposition theorem. In particular, using bipolar coordinates, it is proved for annuli that a family of solutions for the GASP equation in terms of associated Legendre functions of first and second kind is complete. This family is shown to be a Riesz basis in some non-concentric circular annuli. Cited in 1 Document MSC: 30G20 Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.) 35J15 Second-order elliptic equations Keywords:Weinstein differential equation; fundamental solutions PDFBibTeX XMLCite \textit{S. Chaabi} and \textit{S. Rigat}, Eur. J. Math. 1, No. 3, 582--640 (2015; Zbl 1327.30055) Full Text: DOI arXiv References: [1] Ablowitz, M.J., Fokas, A.S.: Complex Variables. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1997) [2] Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. Dover, New York (1992) · Zbl 0171.38503 [3] Alladio, F., Crisanti, F.: Analysis of MHD equilibria by toroidal multipolar expansions. Nuclear Fusion 26(9), 1143-1164 (1986) [4] Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory. Graduate Texts in Mathematics, vol. 137, 2nd edn. Springer, New York (2001) · Zbl 0959.31001 [5] Baratchart, L., Borichev, A., Chaabi, S.: Pseudo-holomorphic functions at the critical exponent. J. Eur. Math. Soc. (2013, to appear). arXiv:1309.3079 · Zbl 1353.30044 [6] Baratchart, L., Fischer, Y., Leblond, J.: Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation. Complex Var. Elliptic Equ. 59(4), 504-538 (2014) · Zbl 1293.30086 [7] Baratchart, L., Leblond, J., Rigat, S., Russ, E.: Hardy spaces of the conjugate Beltrami equation. J. Funct. Anal. 259(2), 384-427 (2010) · Zbl 1196.42022 [8] Blum, J.: Numerical Simulation and Optimal Control in Plasma Physics. Wiley/Gauthier-Villars Series in Modern Applied Mathematics. John Wiley & Sons, Chichester (1989) · Zbl 0717.76009 [9] Blum, J., Boulbe, C., Faugeras, B.: Reconstruction of the equilibrium of the plasma in a Tokamak and identification of the current density profile in real time. J. Comput. Phys. 231(3), 960-980 (2012) · Zbl 1382.76295 [10] Brelot, M.: Équation de Weinstein et potentiels de Marcel Riesz. In: Hirsch, E., Mokobodzki, G. (eds.) Séminaire de Theorie du Potentiel 3. Lecture Notes in Mathematics, vol. 681, pp. 18-38. Springer, Berlin (1978) · Zbl 0386.31007 [11] Brelot-Collin, B., Brelot, M.: Représentation intégrale des solutions positives de l’équation \[{L}_k(u)=\sum_1^n \partial^2 u/\partial x_i^2+(k/x_n) \partial u/{\partial } x_n=0\, (k\] Lk(u)=∑1n∂2u/∂xi2+(k/xn)∂u/∂xn=0(k constante réelle) dans le demi-espace \[E\, (x_n>0)\] E(xn>0), de \[{\mathbb{R}}^nRn\]. Acad. Roy. Belg. Bull. Cl. Sci. (5) 58, 317-326 (1972) · Zbl 0248.31004 [12] Brelot-Collin, B., Brelot, M.: Allure à la frontière des solutions positives de l’équation de Weinstein \[{L}_k(u)={{\Delta }} u+(k/x_n)\partial u/\partial x_n=0\] Lk(u)=Δu+(k/xn)∂u/∂xn=0 dans le demi-espace \[E\] E \[(x_n>0)\](xn>0) de \[{\mathbb{R}}^n \[Rn\](n \ge 2)\](n≥2). Acad. Roy. Belg. Bull. Cl. Sci. (5) 59, 1100-1117 (1973) · Zbl 0282.31001 [13] Brelot-Collin, B., Brelot, M.: Étude à la frontière des solutions locales positives de l’équation \[(1) {L}_k(u)={{\Delta }} u+(k/x_n)\partial u/\partial x_n=0\] Lk(u)=Δu+(k/xn)∂u/∂xn=0 dans le demi-espace E \[(x_n>0)\](xn>0) de \[\mathbb{R}^n \[Rn\](n\ge 2)\](n≥2). Acad. Roy. Belg. Bull. Cl. Sci. (5) 62, 322-340 (1976) · Zbl 0346.35082 [14] Chalendar, I., Partington, J.R.: Phragmén-Lindelöf principles for generalized analytic functions on unbounded domains. Complex Anal. Oper. Theory. doi:10.1007/s11785-015-0453-z · Zbl 1338.30042 [15] Cohl, H.S., Tohline, J.E., Rau, A.R.P., Srivastava, H.M.: Developments in determining the gravitational potential using toroidal functions. Astron. Nachr. 321(5-6), 363-372 (2000) · Zbl 0990.70012 [16] Copson, E.T.: On sound waves of finite amplitude. Proc. Roy. Soc. London. Ser. A. 216(1127), 539-547 (1953) · Zbl 0050.19106 [17] Copson, E.T.: On Hadamard’s elementary solution. Proc. Roy. Soc. Edinburgh. Sect. A 69(1), 19-27 (1970) · Zbl 0203.10501 [18] Copson, E.T.: Partial Differential Equations. Cambridge University Press, Cambridge (1975) · Zbl 0308.35001 [19] Dautray, R., Lions, J.-L.: Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, vol. II. Collection Enseignement, Masson, Paris (1987) [20] Diaz, J.B., Weinstein, A.: On the fundamental solutions of a singular Beltrami operator. In: Studies in Mathematics and Mechanics Presented to Richard von Mises, pp. 97-102. Academic Press, New York (1954) · Zbl 0057.08003 [21] Erdélyi, A. (ed.): Higher Transcendental Functions, vol. I. McGraw-Hill, New York (1953) [22] Erdélyi, A.: Singularities of generalized axially symmetric potentials. Commun. Pure Appl. Math. 9(3), 403-414 (1956) · Zbl 0071.32002 [23] Erdélyi, A.: An application of fractional integrals. J. Anal. Math. 14, 113-126 (1965) · Zbl 0135.33801 [24] Erdélyi, A.: Axially symmetric potentials and fractional integration. J. Soc. Indust. Appl. Math. 13(1), 216-228 (1965) · Zbl 0158.12504 [25] Fischer, Y., Leblond, J.: Solutions to conjugate Beltrami equations and approximation in generalized Hardy spaces. Adv. Pure Appl. Math. 2(1), 47-63 (2011) · Zbl 1208.35169 [26] Fischer, Y., Leblond, J., Partington, J.R., Sincich, E.: Bounded extremal problems in Hardy spaces for the conjugate Beltrami equation in simply-connected domains. Appl. Comput. Harmon. Anal. 31(2), 264-285 (2011) · Zbl 1253.30080 [27] Fischer, Y., Marteau, B., Privat, Y.: Some inverse problems around the tokamak Tore Supra. Commun. Pure Appl. Anal. 11(6), 2327-2349 (2012) · Zbl 1267.30107 [28] Garabedian, P.R.: Partial Differential Equations. AMS Chelsea, Providence (1998) · Zbl 0913.35001 [29] Gilbert, R.P.: Some properties of generalized axially symmetric potentials. Amer. J. Math. 84(3), 475-484 (1962) · Zbl 0121.32501 [30] Gilbert, R.P.: On generalized axially symmetric potentials. J. Reine Angew. Math. 212, 158-168 (1963) · Zbl 0123.29203 [31] Gilbert, R.P.: Poisson’s equation and generalized axially symmetric potential theory. Ann. Mat. Pura Appl. 61, 337-348 (1963) · Zbl 0121.32502 [32] Gilbert, R.P.: Bergman’s integral operator method in generalized axially symmetric potential theory. J. Math. Phys. 5, 983-997 (1964) · Zbl 0152.23304 [33] Hall, N.S., Quinn, D.W., Weinacht, R.J.: Poisson integral formulas in generalized bi-axially symmetric potential theory. SIAM J. Math. Anal. 5(1), 111-118 (1974) · Zbl 0242.31005 [34] Henrici, P.: On the domain of regularity of generalized axially symmetric potentials. Proc. Amer. Math. Soc. 8(1), 29-31 (1957) · Zbl 0077.30301 [35] Hörmander, L.: The Analysis of Linear Partial Differential Operators, Vol. I. Grundlehren der Mathematischen Wissenschaften, vol. 256. Springer, Berlin (1990) · Zbl 0687.35002 [36] Huber, A.: A theorem of Phragmén-Lindelöf type. Proc. Amer. Math. Soc. 4(6), 852-857 (1953) · Zbl 0053.39202 [37] Huber, A.: On the uniqueness of generalized axially symmetric potentials. Ann. Math. 60(2), 351-358 (1954) · Zbl 0057.08802 [38] Huber, A.: Some results on generalized axially symmetric potentials. In: Proceedings of the Conference on Differential Equations, pp. 147-155. University of Maryland Book Store, College Park (1956) · Zbl 0072.31406 [39] Lebedev, N.N.: Special Functions and their Applications. Prentice-Hall, Englewood Cliffs (1965) · Zbl 0131.07002 [40] Lenells, J., Fokas, A.S.: Boundary-value problems for the stationary axisymmetric Einstein equations: a rotating disc. Nonlinearity 24(1), 177-206 (2011) · Zbl 1221.35416 [41] Liu, H.: The Cauchy problem for an axially symmetric equation and the Schwarz potential conjecture for the torus. J. Math. Anal. Appl. 250(2), 387-405 (2000) · Zbl 0966.32023 [42] Love, J.D.: The dielectric ring in a uniform, axial, electrostatic field. J. Math. Phys. 13, 1297-1304 (1972) [43] Nikolski, N.K.: Operators, Functions, and Systems, Vol. II. Mathematical Surveys and Monographs, vol. 93. American Mathematical Society, Providence (2002) [44] Savina, T.V.: On splitting up singularities of fundamental solutions to elliptic equations in \[{\mathbb{C}}^2\] C2. Cent. Eur. J. Math. 5(4), 733-740 (2007) · Zbl 1145.35308 [45] Segura, J., Gil, A.: Evaluation of toroidal harmonics. Comput. Phys. Commun. 124(1), 104-122 (2000) · Zbl 0949.65016 [46] Shafranov, V.D.: On magnetohydrodynamical equilibrium configurations. Soviet Phys. JETP 6(3), 545-554 (1958) · Zbl 0081.21801 [47] Shushkevich, G.Ch.: Electrostatic problem for a torus and a disk. Technical Phys. 42(4), 436-438 (1997) [48] van Milligen, B.Ph., Lopez Fraguas, A.: Expansion of vacuum magnetic fields in toiroidal harmonics. Comput. Phys. Commun. 81(1-2), 74-90 (1994) [49] Vekua, I.N.: New Methods for Solving Elliptic Equations. North-Holland Series in Applied Mathematics and Mechanics, vol. 1. North-Holland, Amsterdam (1967) · Zbl 0146.34301 [50] Virchenko, N., Fedotova, I.: Generalized Associated Legendre Functions and their Applications. World Scientific, River Edge (2001) · Zbl 0974.33001 [51] Weinacht, R.J.: Fundamental solutions for a class of singular equations. Contributions to Differential Equations 3, 43-55 (1964) [52] Weinacht, R.J.: A mean value theorem in generalized axially symmetric potential theory. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 8(38), 610-613 (1965) · Zbl 0141.09702 [53] Weinacht, R.J.: Fundamental solutions for a class of equations with several singular coefficients. J. Austral. Math. Soc. 8(3), 575-583 (1968) · Zbl 0157.18301 [54] Weinstein, A.: Discontinuous integrals and generalized potential theory. Trans. Amer. Math. Soc. 63(2), 342-354 (1948) · Zbl 0038.26204 [55] Weinstein, A.: On generalized potential theory and on the torsion of shafts. In: Studies and Essays Presented to R. Courant on his 60th Birthday, pp. 451-460. Interscience, New York (1948) · Zbl 0032.36602 [56] Weinstein, A.: On the torsion of shafts of revolution. In: Proceedings of the Seventh International Congress on Applied Mechanics, vol. 1, pp. 108-119 (1948) [57] Weinstein, A.: Transonic flow and generalized axially symmetric potential theory. In: Symposium on Theoretical Compressible Flow, pp. 73-82. Naval Ordnance Laboratory, White Oak (1950) [58] Weinstein, A.: On Tricomi’s equation and generalized axially symmetric potential theory. Acad. Roy. Belgique. Bull. Cl. Sci. 5(37), 348-358 (1951) · Zbl 0043.10003 [59] Weinstein, A.: Generalized axially symmetric potential theory. Bull. Amer. Math. Soc. 59(1), 20-38 (1953) · Zbl 0053.25303 [60] Weinstein, A.: The singular solutions and the Cauchy problem for generalized Tricomi equations. Commun. Pure Appl. Math. 7(1), 105-116 (1954) · Zbl 0056.09301 [61] Weinstein, A.: On a class of partial differential equations of even order. Ann. Mat. Pura Appl. 39, 245-254 (1955) · Zbl 0065.33102 [62] Weinstein, A.: The generalized radiation problem and the Euler-Poisson-Darboux equation. Summa Brasil. Math. 3, 125-147 (1955) [63] Weinstein, A.: The method of axial symmetry in partial differential equations. In: Convegno Internazionale sulle Equationi Lineari alle Derivate Parziali, pp. 86-96. Edizioni Cremonese, Roma (1955) · Zbl 0121.32501 [64] Weinstein, A.: Elliptic and hyperbolic axially symmetric problems. In: Proceedings of the International Congress of Mathematicians, vol. 3, pp. 264-269. North-Holland, Amsterdam (1956) [65] Weinstein, A.: Sur une classe d’équations aux dérivées partielles singulières. In: Colloques Internationaux du Centre National de la Recherche Scientifique, vol. 71, pp. 179-186. Centre National de la recherche Scientifique, Paris (1956) · Zbl 0075.08903 [66] Weinstein, A.: On a singular differential operator. Ann. Mat. Pura Appl. 49, 359-365 (1960) · Zbl 0094.06101 [67] Weinstein, A.: Singular partial differential equations and their applications. In: Fluid Dynamics and Applied Mathematics, pp. 29-49. Gordon and Breach, New York (1962) [68] Weinstein, A.: Some applications of generalized axially symmetric potential theory to continuum mechanics. Applications of the Theory of Functions in Continuum Mechanics, vol. 2, pp. 440-453. 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