Boundary value problems for analytic and harmonic functions in nonstandard Banach function spaces (Q2914661)

The monograph may be considered as a systematic and detailed analysis of various boundary value problems (BVPs) for analytic and harmonic functions of two variables in domains with complicated geometrical structure (``bad domains''). The authors explore new aspects of the following problems:NEWLINENEWLINE(i) The Riemann BVP: find a function \(\Phi(z)\) from a given class of functions, analytic in the plane, cut along a rectifiable curve \(\Gamma\), whose boundary values satisfy the conjugacy condition NEWLINE\[NEWLINE\Phi^+(t)=G(t)\Phi^-(t)+g(t),NEWLINE\]NEWLINE where \(G(t)\) and \(g(t)\) are functions prescribed on \(\Gamma\), and \(\Phi^\pm(t)\) are bo\-un\-da\-ry values of \(\Phi(z)\) on \(\Gamma\);NEWLINENEWLINE(ii) The Riemann-Hilbert BVP: find a function \(\Phi(z)\) such that its boundary values \(\Phi^+(t)\) satisfy the one-sided boundary condition NEWLINE\[NEWLINE\text{Re}[(a(t)+ib(t))\Phi^+(t)]=c(t),\quad t\in\Gamma,NEWLINE\]NEWLINE where \(a(t)\), \(b(t)\) and \(c(t)\) are functions given on \(\Gamma\), and also, more generally, the Riemann-Hilbert-Poincaré BVP, for which the one-sided boundary condition contains derivatives of an unknown function.NEWLINENEWLINE(iii) The Dirichlet and Neumann BVPs. Though the Dirichlet problem for the real parts of Smirnov classes of analytic functions (or Cauchy type integrals) is a particular case of the Riemann-Hilbert problem, nevertheless the results related to this problem take up a considerable amount of space of this book as its solution is obtained not only in simply-connected but also in doubly-connected domains with piecewise smooth boundary. Along with this, the results in this direction are more informative and transparent.NEWLINENEWLINEChapter 1, ``The Smirnov classes of harmonic functions and the Dirichlet problem'', deals with criteria for a domain in which the Dirichlet problem for harmonic functions from the Smirnov class is uniquely solvable. The authors introduce the Smirnov type 2 classes of harmonic functions in simply connected domains and prove their non-coincidence. A difference between solvability pictures of the Dirichlet problem in the above-mentioned classes is determined.NEWLINENEWLINEIn Chapter 2, ``Singular integrals in nonstandard Banach function spaces'', the authors establish a boundedness criterion of the Cauchy singular integral operator in weighted variable exponent Lebesgue spaces and in generalized grand Lebesgue spaces. These results play a crucial role in the solution of BVPs functions with data from the above-mentioned new function spaces.NEWLINENEWLINEIn the third chapter titled ``Variable exponent Hardy and Smirnov classes of analytic functions'' various types of weighted variable exponent Hardy and Smirnov classes of analytic functions in simple and doubly-connected domains are introduced and studied. In particular, a wide class of those domains is revealed in which the functions from the above mentioned classes are representable by Cauchy type integrals with densities of weighted variable exponent Lebesgue spaces.NEWLINENEWLINEChapter 4, ``Boundary value problems for analytic functions'', deals with the solution of the Riemann BVP with boundary conditions from various nonstandard Banach function spaces. The authors mainly focus on the case of piecewise continuous conjugacy coefficients. This chapter consists three parts. In Part I, the Riemann BVP is investigated when the boundary function belongs to the variable exponent Lebesgue space. In Part II, the problem is studied in a more general setting. Namely, the authors treat the so-called Riemann BVP value problem with shift which in literature is known as the Haseman problem. Part III deals with the problem of linear conjugation with data from grand Lebesgue spaces. In all above mentioned cases, the classes of sought functions are sets of Cauchy type integrals with densities from the same classes as the data.NEWLINENEWLINEIn Chapter 5 titled ``The Riemann-Hilbert problem in weighted classes of Cauchy type integrals'', the Riemann-Hilbert problem is considered in the class of functions representable in the form \(\Phi(z)=\omega^{-1}(z)K_\Gamma\varphi(z)\), where \(K_\Gamma\varphi(z)\) is a Cauchy type integral with density from the class \(L^{p(\cdot)}(\Gamma)\) and \(\omega(z)\) is given by NEWLINE\[NEWLINE\omega(z)=\prod_{k=1}^\nu(z-t_k)^{\alpha_k},\quad t_k\in\Gamma,\quad \alpha_k\in\mathbb R.\tag{1}NEWLINE\]NEWLINE The set of all such functions \(\Phi(z)\) is called the weighted class of Cauchy type integrals with density from \(L^{p(\cdot)}(\Gamma;\omega)\) and is denoted by \(K^{p(\cdot)}(D;\omega)\). When a weight function \(\omega\) belongs to \(W^{p(\cdot)}(\Gamma)\), the authors show that \(K^{p(\cdot)}(D;\omega)\) and \(K^{p(\cdot)}(\Gamma;\omega)\) coincide for the wide class of curves \(\Gamma\) and variable exponents \(p(t)\).NEWLINENEWLINE This Chapter consists of two parts. In Part I, the problem is considered in the class \(K^{p(\cdot)}(D;\omega)\), where \(D\) is a simply connected domain bounded by a simply piecewise Lyapunov curve with angular points \(A_k\), at which the angle sizes with respect to \(D\) are equal to \(\pi\nu_k\), \(0<\nu_k\leq2\). The weight \(\omega\) is assumed to be an arbitrary power function of the form (1), while the coefficients \(a(t)\) and \(b(t)\) are piecewise Hölder with the condition inf\(\,(a^2(t)+b^2(t))>0\) and \(c(t)\in L^{p(\cdot)}(\Gamma;\omega)\). For the function \(p(t)\) it is required that it belongs to \({\mathcal P}(\Gamma)\). Under these assumptions, the authors obtain a complete picture of the solvability: conditions for the problem to be solvable are derived and solutions are constructed. These conditions and solutions largely depend both on the values of \(p(t)\) at the angular points of \(\Gamma\) and on the angle sizes at these points. An important contribution to the picture of solvability is made by the exponents \(\alpha_k\) of the weight function and the jump values at the discontinuity points of \(a(t)\) and \(b(t)\). In Part II, the problem is studied when the boundary \(\Gamma\) is a piecewise smooth curve, the coefficients \(a(t)\) and \(b(t)\) are piecewise continuous functions but the function \(p(t)\) belongs to the class \(\widetilde{\mathcal P}(\Gamma)\) which is rather narrow compared to \({\mathcal P}(\Gamma)\). The presented results are new even for the constant exponents \(p>1\).NEWLINENEWLINEChapter 6, ``The Dirichlet problem for variable exponent Smirnov classes harmonic functions in domains with arbitrary piecewise smooth boundaries'', contains four parts. In Part I, the authors consider the Dirichlet problem in a simply connected domain Laplace equation, assuming that the unknown function is the real part of an analytic function representable by a Cauchy type integral with density in \(L^{p(\cdot)}(\Gamma)\). In that case, the considered problem reduces to the Dirichlet problem for a circle. The Muskhelishvili method plays an important role in reducing this problem to the Riemann problem in the class \(h^{p(\cdot)}(\omega)\). For \(p(t)=\text{const}>1\), the Cauchy type integral with a density from \(L^p(\Gamma)\), where \(\Gamma\) is a simple closed Carleson curve bounding the domain \(D\), belongs to the Smirnov class \(E^p(D)\), and thus we have an additional useful information on the problem solution. It turns out, in particular, that if \(\Gamma\) is a simple closed piecewise Lyapunov curve bounding the domain \(D\), then the Cauchy type integral with density from \(L^{p(\cdot)}(\Gamma)\) belongs to the Smirnov class \(E^p(D)\). The Dirichlet problem is thereby solved in the class \(e^{p(\cdot)}(D)=\text{Re}\,E^{p(\cdot)}(D)\), too. However, the important and interesting case of domains with arbitrary piecewise smooth boundary has so far been remaining unsolved. In Part II, for such domains the authors study the Dirichlet problem for harmonic functions from \(e^{p(\cdot)}(D)\). Using conformal mappings it again reduces to a problem for the circle, but this time in the weight class \(h^{p(\cdot)}(\omega)=\text{Re}\,H^{p(\cdot)}(\omega)\), where the weight \(\omega\) is more general than the power weight. After solving the problem in the class \(h^{p(\cdot)}(\omega)\), the authors proceed to the main goal, which is to study the class of ``bad domains''. In Part III, the authors consider the Dirichlet problem in doubly-connected domains with an arbitrary piecewise smooth boundary \(\Gamma\) in the class of those harmonic functions which are real parts of functions analytic in \(D\) that belong to the Smirnov class \(E^{p_1(\cdot),p_2(\cdot)}(D)\). Part IV is devoted to an investigation of the Dirichlet problem with data from grand Lebesgue spaces. Depending on the boundary geometry and the values of the exponents at angular points, the Dirichlet problem may be unsolvable, solvable uniquely or nonuniquely. For the unsolvable case, a necessary and sufficient condition is found for the boundary function, which provides the problem solvability. For all solvability cases, solutions are constructed in explicit form.NEWLINENEWLINEIn many BVPs of function theory, the boundary conditions contain not only the sought function, but also its derivatives up to certain order. Therefore it is useful to have formulas giving an integral representation of such analytic functions. One form of such representations, quite convenient for applications, was proposed by I.~Vekua, and used for investigating quite a general BVP, namely, the Riemann-Hilbert-Poincaré problem. In Chapter 7, ``The Riemann-Hilbert-Poincaré problem'', the authors consider the question whether the Vekua integral representations can be used to present analytic functions whose \(m\)-th derivatives are representable by a Cauchy type integral with a density from \(L^{p(\cdot)}(\Gamma;\omega)\), where \(\Gamma\) is a piecewise smooth curve. Relations are found between the values of the function \(p(t)\) at the angular points of the curve \(\Gamma\), the values of these angles and the power exponents of weight functions \(\omega\), which provide the required representation. This is achieved thanks to the well studied Riemann-Hilbert problem in the considered classes. For this the authors extend the results obtained in Chapter 5 for bounded domains to the case of unbounded domains. The representations obtained are used for an investigation of the Riemann-Hilbert-Poincaré problem under quite general assumptions.NEWLINENEWLINEThe book is suitable for mathematicians working in function theory and mathematical physics as well as experts in the theory of elasticity. The text of the book is self-contained. It can be read without looking frequently into other sources; however, some knowledge on function spaces, operator theory and complex analysis is needed. The book can also be used for teaching purposes.