Boundary behavior of solutions to some singular elliptic boundary value problems

Let Ω be a bounded domain with smooth boundary in . For the more general weight b, some nonlinearities f and singularities g, by two kinds of nonlinear transformations, a new perturbation method, which was introduced by García Melián in [J. García Melián, Boundary behavior of large solutions to elliptic equations with singular weights, Nonlinear Anal. 67 (2007) 818–826], and comparison principles, we show that the boundary behavior of solutions to a boundary blow-up elliptic problem Δw=b(x)f(w),w>0,xΩ,w|_∂Ω=∞ and a singular Dirichlet problem −Δu=b(x)g(u),u>0,xΩ,u|_∂Ω=0 has the same form under the nonlinear transformations, which can be determined in terms of the inverses of the transformations.     

Some results for non-linear elliptic problems with mixed boundary conditions

We prove existence and uniqueness results for non-linear elliptic equations with lower order terms, L¹ data, and mixed boundary conditions that include as particular cases the Dirichlet and the Neumann problems. Mathematics Subject Classification (2000) 35J25, 35D05, 35J70, 35J60     

M ‐functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems

In this paper, we combine results on extensions of operators with recent results on the relation between the M ‐function and the spectrum, to examine the spectral behaviour of boundary value problems. M ‐functions are defined for general closed extensions, and associated with realisations of elliptic operators. In particular, we consider both ODE and PDE examples where it is possible for the operator to possess spectral points that cannot be detected by the M ‐function (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Full asymptotic expansion of the heat trace for non–self–adjoint elliptic cone operators

The operator e^–tA and its trace Tr e^–tA, for t > 0, are investigated in the case when A is an elliptic differential operator on a manifold with conical singularities. Under a certain spectral condition (parameter–ellipticity) we obtain a full asymptotic expansion in t of the heat trace as t → 0⁺. As in the smooth compact case, the problem is reduced to the investigation of the resolvent (A–λ)^–1. The main step consists in approximating this family by a parametrix of A – λ constructed within a suitable parameter–dependent calculus.     

Matrix representations of fourth-order boundary value problems with coupled or mixed boundary conditions

An equivalence between a class of regular self-adjoint fourth-order boundary value problems with coupled or mixed boundary conditions and a certain class of matrix problems is investigated. Such an equivalence was previously known only in the second-order case and fourth-order case with separated boundary conditions.     

Degenerate elliptic boundary value problems

In this paper we find conditions that guarantee that regular boundary value problems for elliptic differential-operator equations of the second order in an interval are coercive and Fredholm, and we prove the compactness of a resolvent. We apply this result to find some algebraic conditions that guarantee that regular boundary value problems for degenerate elliptic differential equations of the second order in cylindrical domains have the same properties. Note that considered boundary value conditions are nonlocal and are differential only in their principal part, and a domain is nonsmooth.     

Existence of nontrivial solutions for discrete elliptic boundary value problems

In this article, the existence of nontrivial solutions for the discrete elliptic boundary value problems is considered by using the extremum principle. Such system admits at least 2ⁿ nontrivial solutions when the nonlinear term is superlinear or sublinear. An explanation example is also given. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

The method of fundamental solutions for elliptic boundary value problems

The aim of this paper is to describe the development of the method of fundamental solutions (MFS) and related methods over the last three decades. Several applications of MFS-type methods are presented. Techniques by which such methods are extended to certain classes of non-trivial problems and adapted for the solution of inhomogeneous problems are also outlined. This revised version was published online in June 2006 with corrections to the Cover Date.     

A weak Galerkin finite element method for the second order elliptic problems with mixed boundary conditions

In this paper, a weak Galerkin finite element method is proposed and analyzed for the second-order elliptic equation with mixed boundary conditions. Optimal order error estimates are established in both discrete $H^1$ norm and the standard $L^2$ norm for the corresponding WG approximations. The numerical experiments are presented to verify the efficiency of the method.     

Localized boundary‐domain singular integral equations of Dirichlet problem for self‐adjoint second‐order strongly elliptic PDE systems

The paper deals with the three‐dimensional Dirichlet boundary value problem (BVP) for a second‐order strongly elliptic self‐adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary‐domain integral equations. The equivalence between the Dirichlet BVP and the corresponding localized boundary‐domain integral equation system is studied. We establish that the obtained localized boundary‐domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener–Hopf factorization method, we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces. Copyright © 2016 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons, Ltd.     

The coupling of hyperbolic and elliptic boundary value problems with variable coefficients

We consider a one‐dimensional coupled problem for elliptic second‐order ODEs with natural transmission conditions. In one subinterval, the coefficient ϵ>0 of the second derivative tends to zero. Then the equation becomes there hyperbolic and the natural transmission conditions are not fulfilled anymore. The solution of the degenerate coupled problem with a flux transmission condition is corrected by an internal boundary layer term taking into account the viscosity ϵ. By using singular perturbation techniques, we show that the remainders in our first‐order asymptotic expansion converge to zero uniformly. Our analysis provides an a posteriori correction procedure for the numerical treatment of exterior viscous compressible flow problems with coupled Navier–Stokes/Euler models. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical methods for fourth‐order nonlinear elliptic boundary value problems

The aim of this article is to present several computational algorithms for numerical solutions of a nonlinear finite difference system that represents a finite difference approximation of a class of fourth‐order elliptic boundary value problems. The numerical algorithms are based on the method of upper and lower solutions and its associated monotone iterations. Three linear monotone iterative schemes are given, and each iterative scheme yields two sequences, which converge monotonically from above and below, respectively, to a maximal solution and a minimal solution of the finite difference system. This monotone convergence property leads to upper and lower bounds of the solution in each iteration as well as an existence‐comparison theorem for the finite difference system. Sufficient conditions for the uniqueness of the solution and some techniques for the construction of upper and lower solutions are obtained, and numerical results for a two‐point boundary‐value problem with known analytical solution are given. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:347–368, 2001     

Multiple solutions of semilinear degenerate elliptic boundary value problems

The purpose of this paper is to study a class of semilinear elliptic boundary value problems with degenerate boundary conditions which include as particular cases the Dirichlet problem and the Robin problem. The approach here is based on the super‐sub‐solution method in the degenerate case, and is distinguished by the extensive use of an L^p Schauder theory elaborated for second‐order, elliptic differential operators with discontinuous zero‐th order term. By using Schauder's fixed point theorem, we prove that the existence of an ordered pair of sub‐ and supersolutions of our problem implies the existence of a solution of the problem. The results extend an earlier theorem due to Kazdan and Warner to the degenerate case. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions

The paper is devoted to verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model consisting of a linear elliptic (reaction-diffusion) equation with a mixed Dirichlet/Neumann/Robin boundary condition is considered in this work. On the base of this model, we present simple technologies for straightforward constructing computable upper and lower bounds for the error, which is understood as the difference between the exact solution of the model and its approximation measured in the corresponding energy norm. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions and are “flexible” in the sense that they can be, in principle, made as close to the true error as the resources of the used computer allow. This work was supported by the Academy Research Fellowship No. 208628 from the Academy of Finland.     

Transition-layer solutions of quasilinear elliptic boundary blow-up problems and dirichlet problems

We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation:

On fourth-order elliptic boundary value problems

This paper is concerned with the existence and uniqueness of a solution for a class of fourth-order elliptic boundary value problems. The existence of a solution is proven by the method of upper and lower solutions without any monotone nondecreasing or nonincreasing property of the nonlinear function. Sufficient conditions for the uniqueness of a solution and some techniques for the construction of upper and lower solutions are given. All the existence and uniqueness results are directly applicable to fourth-order two-point boundary value problems.