Elliptic Systems with Almost Regular Coefficients: Singular Weight Integral Operators

Stanislav Antontsev ()
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Stanislav Antontsev: Universidade da Beira Interior, Departamento de Matemática

A chapter in Factorization, Singular Operators and Related Problems, 2003, pp 25-41 from Springer

Abstract: Abstract We consider the linear elliptic system of two first-order equations % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfKttLearuavP1wzZbItLDhis9wBH5garm % Wu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9 % q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir % -Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGa % aeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbiabgkGi2+aadaWgaaWcba % WdbiqbdQha6zaaraaapaqabaGcpeGaeqyYdCNaey4kaSIaeqiVd02d % amaaBaaaleaapeGaeGymaedapaqabaGcpeWaaeWaa8aabaWdbiabdQ % ha6bGaayjkaiaawMcaaiabgkGi2+aadaWgaaWcbaWdbiabdQha6bWd % aeqaaOWdbiabeM8a3jabgUcaRiabeY7aT9aadaWgaaWcbaWdbiabik % daYaWdaeqaaOWdbmaabmaapaqaa8qacqWG6bGEaiaawIcacaGLPaaa % daqdaaWdaeaapeGaeyOaIy7damaaBaaaleaapeGaemOEaOhapaqaba % GcpeGaeqyYdChaaiabg2da9iabdgeabnaabmaapaqaa8qacqWG6bGE % aiaawIcacaGLPaaacqaHjpWDcqGHRaWkcqWGcbGqdaqadaWdaeaape % GaemOEaOhacaGLOaGaayzkaaGafqyYdCNbaebacqGHRaWkcqWGgbGr % daqadaWdaeaapeGaemOEaOhacaGLOaGaayzkaaaaaa!609F! $$ {{\partial }_{{\bar{z}}}}\omega + {{\mu }_{1}}\left( z \right){{\partial }_{z}}\omega + {{\mu }_{2}}\left( z \right)\overline {{{\partial }_{z}}\omega } = A\left( z \right)\omega + B\left( z \right)\bar{\omega } + F\left( z \right) $$ , where % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfKttLearuavP1wzZbItLDhis9wBH5garm % Wu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9 % q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir % -Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGa % aeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbiabdEha3naabmaapaqaa8 % qacqWG6bGEcqGGSaalcuWG6bGEgaqeaaGaayjkaiaawMcaaiabg2da % 9iabdwha1jabgUcaRiabdMgaPjabdAha2baa!3CCB! $$ w\left( {z,\bar{z}} \right) = u + iv $$ is an unknown complex-valued function, and the related integral operators and boundary value problems. We assume that % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfKttLearuavP1wzZbItLDhis9wBH5garm % Wu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9 % q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir % -Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGa % aeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbiabdgeabjabcYcaSiabdk % eacjabcYcaSiabdAeagjabgIGiolabdYeam9aadaWgaaWcbaWdbiab % dchaWbWdaeqaaOWdbmaabmaapaqaa8qacqqHPoWvaiaawIcacaGLPa % aacqGGSaalcqWGWbaCcqGHKjYOcqaIYaGmaaa!4124! $$ A,B,F \in {{L}_{p}}\left( \Omega \right),p \leqslant 2 $$ , in contrast to the regular Vekua’s theory where p > 2. We prove that in this case the solutions of the system still preserve the properties, which correspond to the regular case with respect to: the structure of zeros, Liouville’s theorem, solvability of Riemann-Hilbert boundary value problems etc.

Keywords: Elliptic system; generalized analytic functions; Riemann-Hilbert problems; integral operators. (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-017-0227-0_3

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DOI: 10.1007/978-94-017-0227-0_3

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