Integral Equations and Inequalities with Refinements of Some Classical Results
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Date
2024-02-21
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Addis Ababa University
Abstract
In this PhD thesis, we study two closely related mathematical subjects: Integral equations and integral inequalities with refinements. The first part of the thesis
deals with various generalizations and refinements of some classical inequalities.
We prove and discuss some new Hardy-type inequalities in Banach function space settings. In particular, such a result is proved and applied for a new general Hardy operator, which generalizes the usual Hardy kernel operator. Next, we prove some new refined Hardy-type inequalities again in Banach function space settings. We apply superquadraticity technique to find some refinements of the Jensen, Minkowski and Beckenbach-Dresher inequalities. These results both generalize and unify several results of this type. For the case 0 < p ≤ q < ∞, some new Cochran-Lee inequalities in higher dimensions are proved and good two-sided estimates of the sharp constants are obtained. Using these results a new multidimensional weighted Cochran-Lee inequality with sharp constant is also proved. Further, these results are extended to P´olya-Knopp type inequalities on homogeneous groups using a direct method. In the second part of the thesis, the Dirichlet and Neumann boundary value problems (BVPs) for the linear second-order scalar elliptic differential equation with variable coefficients in a bounded two-dimensional domain are considered. The right-hand side the PDE belongs to H−1(Ω) or eH−1(Ω), when neither classical nor canonical conormal derivatives of solutions are well defined. The two-operator approach and appropriate parametrix (Levi function) are used to reduce each of the problem to two different systems of two-operator boundary-domain integral equations (BDIEs). Although the theory of BDIEs in 3D is well developed, the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the associated Sobolev spaces or choose appropriate scaling parameter in the parametrix form, to insure the invertibility of the corresponding parametrix-based integral layer potentials and hence the unique solvability of BDIEs. The equivalence of the two-operator BDIE systems to the original problems, BDIE system solvability, solution uniqueness/nonuniqueness and invertibility BDIE system are analyzed in the appropriate Sobolev spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible, and appropri ate finite-dimensional perturbations are constructed leading to invertibility of the perturbed operators.
This PhD thesis is written as a monograph. Some of the results, with the candidate as author or coauthor, are already published in international journals (see [15], [19], [94], [132], [133] and the new book [91]). Finally, we give motivation and examples of our future plans to study the application of Hardy type inequalities to solve some interesting problems concerning differential and integral equations.
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Integral Equations, Inequalities, Refinements, Some Classical Results