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Our concern is to estimate the solutions with explicit constants, for domains in <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:msup><mml:mrow><mml:mi>\u211d<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:math> (<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>n<\/mml:mi><mml:mo>\u2265<\/mml:mo><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:math>) of class <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:msup><mml:mrow><mml:mi>C<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">0,1<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:math>. The existence of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:msup><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u221e<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:msup><mml:mrow><mml:mi>W<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><mml:mo>,<\/mml:mo><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:math> estimates is assured for <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mi>q<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:math> and any <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mi>q<\/mml:mi><mml:mo>&lt;<\/mml:mo><mml:mi>n<\/mml:mi><mml:mo>\/<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>n<\/mml:mi><mml:mo>-<\/mml:mo><mml:mn mathvariant=\"normal\">1<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> (depending on the data), whenever the coefficient is only measurable and bounded. The proof method of the quantitative <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:msup><mml:mrow><mml:mi>L<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u221e<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:math> estimates is based on the De Giorgi technique developed by Stampacchia. By using the potential theory, we derive <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mrow><mml:msup><mml:mrow><mml:mi>W<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><mml:mo>,<\/mml:mo><mml:mi>p<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:math> estimates for different ranges of the exponent <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mrow><mml:mi>p<\/mml:mi><\/mml:mrow><\/mml:math> depending on the fact that the coefficient is either Dini-continuous or only measurable and bounded. In this process, we establish new existences of Green functions on such domains. The last but not least concern is to unify (whenever possible) the proofs of the estimates to the extreme Dirichlet and Neumann cases of the mixed problem.<\/jats:p>","DOI":"10.1155\/2014\/845760","type":"journal-article","created":{"date-parts":[[2014,3,31]],"date-time":"2014-03-31T21:20:21Z","timestamp":1396300821000},"page":"1-16","source":"Crossref","is-referenced-by-count":3,"title":["Explicit Estimates for Solutions of Mixed Elliptic Problems"],"prefix":"10.1155","volume":"2014","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1002-3711","authenticated-orcid":true,"given":"Luisa","family":"Consiglieri","sequence":"first","affiliation":[{"name":"N\u00facleo de Investigadores Cient\u00edficos, Universidad Central del Ecuador, Ciudadela Universitaria Avenida Am\u00e9rica, 290-4799 Quito, 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