| 1. |
- Akhmetkaliyeva, Raya
(författare)
-
Maximal regularity of the solutions for some degenerate differential equations and their applications
- 2018
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This PhD thesis deals with the study of existence and uniqueness together with coercive estimates for solutions of certain differential equations.The thesis consists of six papers (papers A, B, C, D, E and F), two appendices and an introduction, which put these papers and appendices into a more general frame and which also serves as an overview of this interesting field of mathematics.In the text below the functionsr = r(x), q = q(x), m = m(x) etc. are functions on (−∞,+∞), which are different but well defined in each paper. Paper A deals with the study of separation and approximation properties for the differential operator in the Hilbert space (here is the complex conjugate of ). A coercive estimate for the solution of the second order differential equation is obtained and its applications to spectral problems for the corresponding differential operator is demonstrated. Some sufficient conditions for the existence of the solutions of a class of nonlinear second order differential equations on the real axis are obtained.In paper B necessary and sufficient conditions for the compactness of the resolvent of the second order degenerate differential operator in is obtained. We also discuss the two-sided estimates for the radius of fredholmness of this operator.In paper C we consider the minimal closed differential operator in , where are continuously differentiable functions, and is a continuous function. In this paper we show that the operator is continuously invertible when these coefficients satisfy some suitable conditions and obtain the following estimate for : ,where is the domain of .In papers D, E, and F various differential equations of the third order of the form are studied in the space .In paper D we investigate the case when and .Moreover, in paper E the equation (0.1) is studied when . Finally, in paper F the equation (0.1) is investigated under certain additional conditions on .For these equations we establish sufficient conditions for the existence and uniqueness of the solution, and also prove an estimate of the form for the solution of equation (0.1).
|
|
| 2. |
- Kikonko, Mervis, 1973-
(författare)
-
Some new results concerning general weighted regular Sturm-Liouville problems
- 2016
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this PhD thesis we study some weighted regular Sturm-Liouville problems in which the weight function takes on both positive and negative signs in an appropriate interval [a,b]. With such problems there is the possible existence of non-real eigenvalues, unlike in the definite case (i.e. left or right definite) in which only real eigenvalues exist.This PhD thesis consists of five papers (papers A-E) and an introduction to this area, which puts these papers into a more general frame.In paper A we give some precise estimates on the Richardson number for the two turning point case, thereby complementing the work of Jabon and Atkinson from 1984 in an essential way. We also give a corrected version of their result since there seems to be a typographical error in their paper.In paper B we show that the interlacing property, which holds in the one turning point case, does not hold in the two turning point case. The paper consists of a detailed presentation of numerical results of the case in which the weight function is allowed to change its sign twice in the interval (-1, 2). We also present some theoretical results which support the numerical results. Moreover, a number of new open questions are raised. We also observe that the real and imaginary parts of a non-real eigenfunction either have the same number of zeros in the interval (-1,2) or the numbers of zeros differ by two.In paper C, we obtain bounds on real and imaginary parts of non-real eigenvalues of a non-definite Sturm-Liouville problem, with Dirichlet boundary conditions, thus complementing the results obtained in a paper byBehrndt et.al. from 2013 in an essential way.In paper D we obtain a lower bound on the eigenvalue of the smallest modulus associated with a Dirichlet problem in the general case of a regular Sturm-Liouville problem.In paper E we expand upon the basic oscillation theory for general boundary problems of the form -y''+q(x)y=λw(x)y, on I = [a,b], where q(x) and w(x) are real-valued continuous functions on [a,b] and y is required to satisfy a pair of homogeneous separated boundary conditions at the end-points. Already in 1918 Richardson proved that, in the case of the Dirichlet problem, if w(x) changes its sign exactly once and the boundary problem is non-definite, then the zeros of the real and imaginary parts of any non-real eigenfunction interlace. We show that, unfortunately, this result is false in the case of two turning points, thus removing any hope for a general separation theorem for the zeros of the non-real eigenfunctions. Furthermore, we show that when a non-real eigenfunction vanishes inside I, then the absolute value of the difference between the total number of zeros of its real and imaginary parts is exactly 2.
|
|
| 3. |
- Burtseva, Evgeniya, 1988-
(författare)
-
Operators and Inequalities in various Function Spaces and their Applications
- 2016
-
Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This Licentiate thesis is devoted to the study of mapping properties of different operators (Hardy type, singular and potential) between various function spaces.The main body of the thesis consists of five papers and an introduction, which puts these papers into a more general frame.In paper A we prove the boundedness of the Riesz Fractional Integration Operator from a Generalized Morrey Space to a certain Orlicz-Morrey Space, which covers the Adams resultfor Morrey Spaces. We also give a generalization to the case of Weighted Riesz Fractional Integration Operators for a class of weights.In paper B we study the boundedness of the Cauchy Singular Integral Operator on curves in complex plane in Generalized Morrey Spaces. We also consider the weighted case with radial weights. We apply these results to the study of Fredholm properties of Singular Integral Operators in Weighted Generalized Morrey Spaces.In paper C we prove the boundedness of the Potential Operator in Weighted Generalized Morrey Spaces in terms of Matuszewska-Orlicz indices of weights and apply this result to the Hemholtz equation with a free term in such a space. We also give a short overview of some typical situations when Potential type Operators arise when solving PDEs.In paper D some new inequalities of Hardy type are proved. More exactly, the boundedness of multidimensional Weighted Hardy Operators in Hölder Spaces are proved in cases with and without compactification.In paper E the mapping properties are studied for Hardy type and Generalized Potential type Operators in Weighted Morrey type Spaces.
|
|
| 4. |
- Ragusa, Maria Alessandra
(författare)
-
Partial differential equations and systems related to Morrey spaces
- 2012
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This PhD thesis deals with the study of well posedness, existence and regularity properties of solutions of partial differential equations and systems. Preparatory to the study of partial differential equations is the action of some integral operators, that are extensively used. Such results are very useful to obtain regularity properties of solutions of elliptic, parabolic and ultraparabolic equations of second order with discontinuous coefficients, and later of systems. The thesis consists of five papers (Paper A -- Paper F), an introduction, which put these papers into a more general frame and which also serves as an overview of this interesting field of mathematics, and an appendix, where the behavior of Morrey spaces in connection with a lot of other spaces is presented and discussed. In paper A we study the local regularity in the Lebesgue spaces $L^{p},$ $1systems of arbitrary order in nondivergence form with coefficients, which can be discontinuous. In the case of continuous coefficients we review, discuss and complement the results obtained by S. Agmon, A. Douglis and L. Nirenberg in 1959, 1964 and by S. Campanato in 1977. In paper B we study the Cauchy-Dirichlet problem related to a linear parabolic equation of second order in divergence form with discontinuous coefficients. Moreover, we prove estimates in the space $\,\,H^{1- \frac1p},\,\,$ for every $\,1\,<\,p\,<\,\infty.\,$ The $H^{\frac12}$ local regularity of $u$ was also studied by A. Marino and A. Maugeri in 1989. Moreover, they obtained relevant results about $L^{p}-$local regularity for the gradient of $u$ and existence of solutions of a nonhomogeneous Cauchy-Dirichlet problem. We emphasize that the regularity we study in this paper is not of the type considered by N. Meyers in 1963 and by M. Giaquinta in 1983. These papers are concerned only with $p$ close to $2.$ Our results are in fact true for any value of $p$ in the range $]1,\,+ \infty[.$ Paper C is devoted to the study of a Cauchy-Dirichlet problem related to nondivergence form parabolic equation with discontinuous coefficients and more precisely we derive and discuss existence, uniqueness and regularity of the solution. This problem is inspired by the study made by M. Bramanti and C. Cerutti in 1993. We point out that our fundamental tools are some properties related to the products of the lower order terms with the solution $u $ and its derivatives. The technique used to derive these results is based on dividing a cylinder in sections and obtaining the requested estimates in each part of the subdivision. In paper D we derive and discuss some estimates in Morrey Spaces for the derivatives of local minimizers of variational integrals. The considered kind of functionals arise as the energy of maps between Riemannian manifolds. From this point of view, the geometric interest may occur on the above functionals. Moreover, we observe that some methods of proofs of regularity for classes of nonlinear elliptic systems can also be applied to a lot of equations in nonlinear Hodge theory. Paper E deals with the study of some qualitative properties of positive solutions of a chr\"odinger type equation of the kind $ L u+V u=0, $ having discontinuous coefficients, where $L $ is the Kolmogorov operator in $\R^{n+1} $ and $V$ belongs to a Stummel-Kato class. Moreover, we consider the Green function for the constant coefficients operator $L_0 $ and build the Green function for the operator $L. $ We also produce the proof of interior regularity and a uniqueness result for the Cauchy-Dirichlet problem associated to $L, $ making in both cases the additional assumption that the solution $u$ is bounded. Moreover, a density argument allows us to remove the extra condition of boundedness on $u$ from the uniqueness result and then we use this fact to remove the additional assumption from the regularity theorem. To motivate our interest in this kind of operators, we recall that they arise e.g. in the stochastic theory and in the theory of diffusion processes. For instance, the linear Fokker-Planck equation can be written as a particular case of the above equation.
|
|
| 5. |
|
|
