Clifford Analysis and Elliptic Boundary Value Problems in Unbounded Domains

The studying of overdetermined boundary value problems for elliptic partial differential equations was initiated by J. Serrin in 1971. In his work, he established a property of radial symmetry for solutions of some overdetermined Poisson problem. Apart of a significant independent interest, the problems of such kind have important applications in the potential theory, integral geometry, hydrodynamics and capillarity theory. Usually, the resolving of these problems is based on Hopf lemma on an angular boundary point and the method of hyperplanes motion introduced by A.A. Alexandrov for studying some geometric problems related with characterizing the spheres. Among other more modern methods not involving the maximum principle for the considered problems we mention the duality method, the method of volume derivative as well as an integral method.

This paper provides an existence theorem for the Dirichlet problem for a quasilinear partial differential equation of elliptic type in an unbounded domain. The principal new feature of this work is that the results are obtained under weaker monotonicity conditions on the coefficients of the equation than those employed in earlier work in this area.

The purpose of this study is to develop a numerical method for bifurcation analysis of nonlinear partial differential equations, based on the reduction of partial differential equations to ordinary ones, using the Kolmogorov-Arnold theorem. Methods. This paper describes a method for reducing partial differential equations to ordinary ones using the Kolmogorov-Arnold theorem, as well as methods for the bifurcation analysis of nonlinear boundary value problems for ordinary differential equations. Results. The paper presents a new method for solving and bifurcation analysis of nonlinear boundary value problems for partial differential equations, which allow variational formulation. The method was applied to a nonlinear two-dimensional Bratu problem with Dirichlettype boundary conditions. Conclusion. A new method of bifurcation analysis for nonlinear partial differential equations has been developed. Specifically, a method has been proposed for reducing partial different equations to ordinary equations, which allows the use of the developed apparatus of bifurcation analysis for boundary value problems of ordinary differential equations. This method allows conducting bifurcation analysis for arbitrary nonlinear partial differential equations.

This work presents a review and analysis of modern asymptotic methods for analysis of singularly perturbed problems with interior and boundary layers. The central part of the work is a review of the work of the author and his colleagues and disciples. It highlights boundary and initial-boundary value problems for nonlinear elliptic and parabolic partial differential equations, as well as periodic parabolic problems, which are widely used in applications and are called reaction–diffusion and reaction–diffusion–advection equations. These problems can be interpreted as models in chemical kinetics, synergetics, astrophysics, biology, and other fields. The solutions of these problems often have both narrow boundary regions of rapid change and inner layers of various types (contrasting structures, moving interior layers: fronts), which leads to the need to develop new asymptotic methods in order to study them both formally and rigorously. A general scheme for a rigorous study of contrast structures in singularly perturbed problems for partial differential equations, based on the use of the asymptotic method of differential inequalities, is presented and illustrated on relevant problems. The main achievements of this line of studies of partial differential equations are reflected, and the key directions of its development are indicated.

In this paper, we introduce the operators related to mixed boundary value problems for general linear elliptic partial complex differential equations in the unit disc of the complex plane. The solvability of the relevant boundary value problems will be studied by transforming them into singular integral equations.

SynopsisIn this paper, we prove various existence and non-existence results for semilinear elliptic problems in unbounded domains. In particular we prove for general classes of unbounded domains that there exists no solution distinct from 0 offor any smooth f satisfying f(0) = 0. This result is obtained by the use of new identities that solutions of semilinear elliptic equations satisfy.

On the variational iteration method and other iterative techniques for nonlinear differential equations

This study focuses on nonlocal boundary value problems for elliptic ordinary and partial differential-operator equations of arbitrary order, defined in Banach-valued function spaces. The region considered here has a varying bound and depends on a certain parameter. Several conditions are obtained that guarantee the maximal regularity and Fredholmness, estimates for the resolvent, and the completeness of the root elements of differential operators generated by the corresponding boundary value problems in Banach-valued weighted spaces. These results are applied to nonlocal boundary value problems for regular elliptic partial differential equations and systems of anisotropic partial differential equations on cylindrical domain to obtain the algebraic conditions that guarantee the same properties.

On existence and uniqueness of solutions of boundary value problems of fourth order elliptic partial differential equations with variable coefficients

In this study, we investigate the application of variational methods to establish the existence of weak solutions for boundary value problems (BVPs) associated with nonlinear elliptic partial differential equations (PDEs). By employing the direct method in the calculus of variations, we reformulate the given PDEs as minimisation problems over suitable Sobolev spaces, where weak solutions correspond to critical points of appropriately defined energy functionals. The analytical framework incorporates foundational tools such as Sobolev embeddings, compactness theorems, and properties of coercivity and weak lower semi-continuity to ensure the well-posedness of the variational formulation. We present and prove an existence theorem under well-defined conditions on the nonlinear coefficients of the equations, specifically those involving the Carathéodory property, polynomial growth bounds, monotonicity, and symmetry. These conditions are shown to satisfy the hypotheses of the Fucik-Kufner theorem, which guarantees the existence - and, under convexity assumptions, uniqueness - of weak solutions. To demonstrate the utility of the theoretical results, we apply the theorem to selectednonlinear elliptic problems and verify the fulfilment of all required criteria for the existence of solutions. Through this approach, we provide a rigorous and general framework for analysing nonlinear elliptic BVPs without resorting to classical linearization techniques. The findings confirm that variational methods offer powerful, flexible tools for addressing existence questions in complex PDE systems, with relevance to mathematical modelling in physics, engineering, and applied sciences.

Efficient space-time Legendre rational spectral method for parabolic problems in unbounded domains

This chapter presents the application of the boundary element method to high-frequency Helmholtz problems in unbounded domains. Based on a standard combined integral equation approach for sound-hard scattering problems we discuss the discretization, preconditioning and fast evaluation of the involved operators. As engineering problem, the propagation of high-intensity focused ultrasound fields into the human rib cage will be considered. Throughout this chapter we present code snippets using the open-source Python boundary element software BEM++ to demonstrate the implementation.

The paper deals with the solutions of ordinary differential semi-linear equa- tion, the coefficients of which depend on several real parameters. If the coefficient is chosen so that the equation does not contain the first-order derivative of the unknown function, it will be the case of Emden - Fowler equation. Asymp- totic behavior of Emden - Fowler equation solutions at infinity is described in the book of Richard Bellman. The equations with the first-order derivative, considered in this work, erase in some problems for elliptic partial differential equations in unbounded domains. The sign of the coefficient in first-order deriva- tive term essentially influences on the description of solutions. Partly the result of this paper can be obtained from the works of I.T. Kiguradze. In present work we use lemmas about the behavior of solutions of the linear equations with a strongly (weakly) increasing potential. The paper deals with the solutions of ordinary differential semi-linear equa- tion, the coefficients of which depend on several real parameters. If the coefficient is chosen so that the equation does not contain the first-order derivative of the unknown function, it will be the case of Emden - Fowler equation. Asymp- totic behavior of Emden - Fowler equation solutions at infinity is described in the book of Richard Bellman. The equations with the first-order derivative, considered in this work, erase in some problems for elliptic partial differential equations in unbounded domains. The sign of the coefficient in first-order deriva- tive term essentially influences on the description of solutions. Partly the result of this paper can be obtained from the works of I.T. Kiguradze. In present work we use lemmas about the behavior of solutions of the linear equations with a strongly (weakly) increasing potential.

The finite element method with nonuniform mesh sizes is employed to approximately solve elliptic boundary value problems in unbounded domains. Consider the following model problem: \[ − Δ u = f in Ω C , u = g on ∂ Ω , ∂ u ∂ r + 1 r u = o ( 1 r ) as r = | x | → ∞ , - \Delta u = f\quad {\text {in}}\;{\Omega ^C},\quad u = g\quad {\text {on}}\;\partial \Omega ,\quad \frac {{\partial u}}{{\partial r}} + \frac {1}{r}u = o\left ( {\frac {1}{r}} \right )\quad {\text {as}}\;r = |x| \to \infty , \] where Ω C {\Omega ^C} is the complement in R 3 {R^3} (three-dimensional Euclidean space) of a bounded set Ω \Omega with smooth boundary ∂ Ω \partial \Omega , f and g are smooth functions, and f has bounded support. This problem is approximately solved by introducing an artificial boundary Γ R {\Gamma _R} near infinity, e.g. a sphere of sufficiently large radius R . The intersection of this sphere with Ω C {\Omega ^C} is denoted by Ω R C \Omega _R^C and the given problem is replaced by \[ − Δ u R = f in Ω R C , u R = g on ∂ Ω , ∂ u R ∂ r + 1 r u R = 0 on Γ R . - \Delta {u_R} = f\quad {\text {in}}\;\Omega _R^C,\quad {u_R} = g\quad {\text {on}}\;\partial \Omega ,\quad \frac {{\partial {u_R}}}{{\partial r}} + \frac {1}{r}{u_R} = 0\quad {\text {on}}\;{\Gamma _R}. \] This problem is then solved approximately by the finite element method, resulting in an approximate solution u R h u_R^h for each h > 0 h > 0 . In order to obtain a reasonably small error for u − u R h = ( u − u R ) + ( u R − u R h ) u - u_R^h = (u - {u_R}) + ({u_R} - u_R^h) , it is necessary to make R large. This necessitates the solution of a large number of linear equations, so that this method is often not very good when a uniform mesh size h is employed. It is shown that a nonuniform mesh may be introduced in such a way that optimal error estimates hold and the number of equations is bounded by C h − 3 C{h^{ - 3}} with C independent of h and R .

During the last 50 years several methods for the solution of fundamental problems of equations of mathematical physics (difference methods, finite element methods, Galerkin method boundary element methods) have been developed and successfully applied. In this paper, special techniques in approximation theory and in numerical collocation are studied. Using the theory of quaternionic-valued functions and the operational calculus we construct an effective numerical algorithm. Numerical experiments for 2-dimensional boundary value problems of elliptic partial differential equations are presented. This algorithm can successfully be used for domains with complicated boundary.