Open Access Research

The structure of fractional spaces generated by a two-dimensional elliptic differential operator and its applications

Allaberen Ashyralyev12, Sema Akturk1 and Yasar Sozen13*

Author Affiliations

1 Department of Mathematics, Fatih University, 34500, Buyukcekmece, Istanbul, Turkey

2 Department of Mathematics, ITTU, Ashgabat, Turkmenistan

3 Present address: Fen Fakultesi, Matematik Bolumu, Hacettepe Universitesi, 06800, Beytepe, Ankara, Turkey

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Boundary Value Problems 2014, 2014:3  doi:10.1186/1687-2770-2014-3

Published: 2 January 2014

Abstract

We consider the two-dimensional differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M1">View MathML</a> defined on functions on the half-plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M2">View MathML</a> with the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M4">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M6">View MathML</a>, are continuously differentiable and satisfy the uniform ellipticity condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M8">View MathML</a>. The structure of the fractional spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/3/mathml/M9">View MathML</a> generated by the operator A is investigated. The positivity of A in Hölder spaces is established. In applications, theorems on well-posedness in a Hölder space of elliptic problems are obtained.

MSC: 35J25, 47E05, 34B27.

Keywords:
positive operator; fractional spaces; Green’s function; Hölder spaces