El Kernel de Poisson para un dominio doblemente conexo
Resumen
En este trabajo se resuelve el problema de Dirichlet para la ecuación de Laplace en un dominio doblemente conexo. Para la solución del problema planteado se cuenta con la función armónica de Green sobre este dominio, y a partir de esta función, el Kernel de Poisson es obtenido de manera explícita. Con estos elementos y las propiedades de los mapeos conformes se obtiene la fórmula de representación integral que resuelve el problema de Dirichlet para la ecuación de Laplace. Esta aplicación para obtener el Kernel de Poisson a través de la teoría de las transformaciones conformes posibilita resolver problemas de valores de frontera sobre una variedad de dominios en los que no es admisible el conocido método de parqueting-reflection y el método vía problema de Schwarz.
Citas
Abdymanapov, S., Begehr, H., y Tungatarov, A. (2005). Some schwarz problems in a quarter plane. Eurasian Math.J.3, 22-35.
Abdymanapov, S., Begehr, H., y Tungatarov, A. (2009). Four boundary value problems for the Cauchy-Riemann equation in a quarter plane. More Progress in Analysis, 1137-1147. doi: 10.1142/97898128356350109
Adler, P. (1992). Porous media. geometry and transport. Paris: Butterworth Heinemann.
Akel, M., y Hussien, S. (2012). Two basic boundary value problems for inhomogeneous Cauchy-Riemann equation in an innite sector. Advances in Pure and Applied Mathematics, 3(3), 315-328. doi: 10.1515/apam-2012-0009
Begehr, H. (1994). Complex analytic methods for partial dierential equations. World Scientic.
Begehr, H. (2017). Fundamental solutions to the laplacian in plane domains bounded by ellipses. , 4, 293–311. doi: 10.1007/978-981-10-4642-125
Begehr, H., y Gilbert, R. (1992). Transformations, transmutations, and Kernel functions. Longman Scientic Technical.
Begehr, H., y Harutyunyan, G. (2006). Complex boundary value problems in a quarter plane, complex analysis and applications, proc.13th intern.conf.on finite or innite dimensional complex analysis and appl., Shantou, China. N. J. Y.Wang. World Sci., Ed.
Begehr, H., y Vaitsiakhovich, T. (2009a). Harmonic boundary value problems in half disc and half ring. Functiones et Approximatio Commentarii Mathematici,40(2). doi: 10.7169/facm/1246454030
Begehr, H., y Vaitsiakhovich, T. (2009b). A polyharmonic Dirichlet problem of arbitrary order for complex plane domains. Further Progress in Analysis, 327-336. doi: 110.1142/97898128373320027
Begehr, H., y Vaitsiakhovich, T. (2010a). How to find harmonic Green functions in the plane. Complex Variables and Elliptic Equations, 56(12), 1169–1181. doi: 10.1080/17476933.2010.534157
Begehr, H., y Vaitsiakhovich, T. (2010b). The parqueting-reflection principle for constructing Green functions. Analytic Methods of Analysis and Differential Equations.
Begehr, H., y Vaitsiakhovich, T. (2012). Harmonic Dirichlet problem for some equilateral triangle. Complex Variables and Elliptic Equations, 57(4), 185196. doi: 10.1080/17476933.2011.598932
Dzhuraev, A. (2000). Singular partial dierential equations. Boca Raton London: Chapman Hall CRC Press.
Gakhov, F. (1966). Boundary value problems. Pergamon Press: Oxford.
Garnett, J. (1981). Bounded analytic functions. New York: Academic Press.
Muskhelishvili, N. (1953). Singular integral equations. Noordho: Groningen.
Richardson, S. (2001). Hele-shaw ows with time-dependent free boundaries involving a multiply connected region. Euro J. Appl. Math, 12(5), 571-599.
Shupeyeva, B. (2012). Harmonic boundary value problems in a quarter ring domain. Advances in Pure and Applied Mathematics, 3, 393-419. doi: 10.1515/apam-2012-0025
Shupeyeva, B. (2013). Boundary value problems for complex partial differential equations in quarter a ring and half exagon (Doctoral Thesis). Freie Universitat Berlin.
Stein, E., y Shakarchi, R. (2003). Complex analysis (princeton lectures in analysis, no. 2). Princeton University Press Illustrated edition.
Vaitsiakhovich, T. (2007). Boundary value problems to second order complex partial differential equations in a ring domain. Siauliai Mathematical Seminar, 2(10), 117-146.
Vaitsiakhovich, T. (2008). Boundary value problems for complex partial diferential equations in a ring domain (Doctoral Thesis). Freie Universitat Berlin.
Vekua, I. (1962). Generalized analytic functions. Pergamon Press: Oxford.
Vergara, J., y Vanegas, C. (2021). A fundamental solution for the Laplace operator in a doubly connected domain. Bull CompAMa: Bulletin of Computational Applied Mathematics.
Ying, W., y Xuefang, Z. (2016). Schwarz boundary value problem for the CauchyRiemann equation in a rectangle. Boundary Value Problems, 1. doi: 10.1186/s13661-016-0520-z
Abdymanapov, S., Begehr, H., y Tungatarov, A. (2009). Four boundary value problems for the Cauchy-Riemann equation in a quarter plane. More Progress in Analysis, 1137-1147. doi: 10.1142/97898128356350109
Adler, P. (1992). Porous media. geometry and transport. Paris: Butterworth Heinemann.
Akel, M., y Hussien, S. (2012). Two basic boundary value problems for inhomogeneous Cauchy-Riemann equation in an innite sector. Advances in Pure and Applied Mathematics, 3(3), 315-328. doi: 10.1515/apam-2012-0009
Begehr, H. (1994). Complex analytic methods for partial dierential equations. World Scientic.
Begehr, H. (2017). Fundamental solutions to the laplacian in plane domains bounded by ellipses. , 4, 293–311. doi: 10.1007/978-981-10-4642-125
Begehr, H., y Gilbert, R. (1992). Transformations, transmutations, and Kernel functions. Longman Scientic Technical.
Begehr, H., y Harutyunyan, G. (2006). Complex boundary value problems in a quarter plane, complex analysis and applications, proc.13th intern.conf.on finite or innite dimensional complex analysis and appl., Shantou, China. N. J. Y.Wang. World Sci., Ed.
Begehr, H., y Vaitsiakhovich, T. (2009a). Harmonic boundary value problems in half disc and half ring. Functiones et Approximatio Commentarii Mathematici,40(2). doi: 10.7169/facm/1246454030
Begehr, H., y Vaitsiakhovich, T. (2009b). A polyharmonic Dirichlet problem of arbitrary order for complex plane domains. Further Progress in Analysis, 327-336. doi: 110.1142/97898128373320027
Begehr, H., y Vaitsiakhovich, T. (2010a). How to find harmonic Green functions in the plane. Complex Variables and Elliptic Equations, 56(12), 1169–1181. doi: 10.1080/17476933.2010.534157
Begehr, H., y Vaitsiakhovich, T. (2010b). The parqueting-reflection principle for constructing Green functions. Analytic Methods of Analysis and Differential Equations.
Begehr, H., y Vaitsiakhovich, T. (2012). Harmonic Dirichlet problem for some equilateral triangle. Complex Variables and Elliptic Equations, 57(4), 185196. doi: 10.1080/17476933.2011.598932
Dzhuraev, A. (2000). Singular partial dierential equations. Boca Raton London: Chapman Hall CRC Press.
Gakhov, F. (1966). Boundary value problems. Pergamon Press: Oxford.
Garnett, J. (1981). Bounded analytic functions. New York: Academic Press.
Muskhelishvili, N. (1953). Singular integral equations. Noordho: Groningen.
Richardson, S. (2001). Hele-shaw ows with time-dependent free boundaries involving a multiply connected region. Euro J. Appl. Math, 12(5), 571-599.
Shupeyeva, B. (2012). Harmonic boundary value problems in a quarter ring domain. Advances in Pure and Applied Mathematics, 3, 393-419. doi: 10.1515/apam-2012-0025
Shupeyeva, B. (2013). Boundary value problems for complex partial differential equations in quarter a ring and half exagon (Doctoral Thesis). Freie Universitat Berlin.
Stein, E., y Shakarchi, R. (2003). Complex analysis (princeton lectures in analysis, no. 2). Princeton University Press Illustrated edition.
Vaitsiakhovich, T. (2007). Boundary value problems to second order complex partial differential equations in a ring domain. Siauliai Mathematical Seminar, 2(10), 117-146.
Vaitsiakhovich, T. (2008). Boundary value problems for complex partial diferential equations in a ring domain (Doctoral Thesis). Freie Universitat Berlin.
Vekua, I. (1962). Generalized analytic functions. Pergamon Press: Oxford.
Vergara, J., y Vanegas, C. (2021). A fundamental solution for the Laplace operator in a doubly connected domain. Bull CompAMa: Bulletin of Computational Applied Mathematics.
Ying, W., y Xuefang, Z. (2016). Schwarz boundary value problem for the CauchyRiemann equation in a rectangle. Boundary Value Problems, 1. doi: 10.1186/s13661-016-0520-z
