On first and second order planar elliptic equations with degeneracies

In [1, 2, 3, 4, 6], the authors posed and discussed some boundary‐value problems of second order elliptic equations with parabolic degeneracy. In [5], the authors posed and discussed some boundary‐value problems of second order mixed equations with degenerate curve on the sides of an angle, but the coefficients of the equations possess strong restrictions. The present paper deals with the mixed problem for elliptic equations with degenerate curve on the sides of an angle, where the coefficients satisfy general conditions. We first give the formulation of the problem and estimates of solutions of the problem for the equations, and then prove the existence of solutions for the above problem by the Leray–Schauder theorem. In this paper, we use the complex analytic method, namely we first introduce the new notation (2.1) below and reduce the degenerate elliptic equations of second order to the corresponding problem for degenerate elliptic complex equations of first order, afterwards the above problem of second order degenerate elliptic equations can be solved. The results in this paper will be used in a subsequent paper to handle the Tricomi problem of second order equations of mixed type with degenerate curve on the sides of an angle. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A method for the numerical solution of the time dependent two-dimensional magneto-plasma dynamic coupled problem in the low magnetic Reynolds number case, is presented. The method is based on a time dependent two-dimensional description of the fluid dynamics iteratively coupled to a steady state two-dimensional description of the electrodynamics. This is allowed as the characteristic times of variation of the electrodynamic quantities are much lower than those of the fluid dynamic quantities. The fluid is assumed to be compressible and viscous, and governed by the Navier-Stokes equations. A semi-implicit finite difference pressure scheme has been utilised for the solution of the fluid dynamic problem. The electrodynamics is described by a second order elliptic partial differential equation obtained from the Maxwell equations and generalised Ohm's law. The elliptic equation is solved by means of an exponential fitting finite difference method. In order to test the algorithm described above, the transient of a plasma flow in a channel, caused by the variation of the externally applied magnetic field, has been studied.

The aim of this chapter is to discuss some full (that is everywhere) regularity results for scalar-valued weak solutions to second order elliptic equations in divergence forms. Weak solutions only belong to some Sobolev space (consequently, neither are they necessarily continuous nor do derivatives a priori exist in the classical sense), and hence, their regularity needs to be investigated. We here prove local Holder regularity of weak solution and present two different (and classical) strategies of proof dating back to the late 1950s. First, we explain De Giorgi’s level set technique, in a unified approach that applies both to weak solutions of elliptic equations and to minimizers of variational integrals, via the study of Q-minimizers of suitable functionals. We then address, for the specific case of linear elliptic equations, an alternative proof of the everywhere regularity result of weak solutions via Moser’s iteration method.

We compare a block successive overrelaxation method (BSOR) with a direct method for solving difference equations derived from fourth order elliptic partial differential equations. The direct method appears to be competitive in time and accuracy for a general problem.

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

Necessary and sufficient conditions are established for the existence of limits in the sense, 1$ SRC=http://ej.iop.org/images/0025-5734/48/2/A19/toc_tex_sm_2693_img2.gif/>, on the boundary of a domain, of solutions of second order elliptic equations in domains with Lyapunov boundaries.Bibliography: 8 titles.

This paper is devoted to study of an auxiliary spaces preconditioner for H(div) systems and its application in the mixed formulation of second order elliptic equations. Extensive numerical results show the efficiency and robustness of the algorithms, even in the presence of large coefficient variations. For the mixed formulation of elliptic equations, we use the augmented Lagrange technique to convert the solution of the saddle point problem into the solution of a nearly singular H(div) system. Numerical experiments also justify the robustness and efficiency of this scheme.

The positive solutions of a semilinear second order elliptic equation with critical non linear exponent are studied and existence and nonexistence theorems of a solution are given.

In this paper we show how a second order scalar uniformly elliptic equation in divergence form with measurable coefficients and Dirichlet boundary conditions can be transformed into a first order elliptic system with half-Dirichlet boundary condition. This first order system involves Hodge-Dirac operators and can be seen as a natural generalization of the Beltrami equation in the plane and we develop a theory for this equation, extending results from the plane to higher dimension. The reduction to a first order system applies both to linear as well as quasilinear second order equations and we believe this to be of independent interest. Using the first order system, we give a new representation formula of the solution of the Dirichlet problem both on simply and finitely connected domains. This representation formula involves only singular integral operators of convolution type and Neumann series there of, for which classical Calderón-Zygmund theory is applicable. Moreover, no use is made of any fundamental solution or Green's function beside fundamental solutions of constant coefficient operators. Remarkably, this representation formula applies also for solutions of the fully non-linear first order system. We hope that the representation formula could be used for numerically solving the equations. Using these tools we give a new short proof of Meyers' higher integrability theorem. Furthermore, we show that the solutions of the first order system are Hölder continuous with the same Hölder coefficient as the solutions of the second order equations. Finally, factorization identities and representation formulas for the higher dimensional Beurling-Ahlfors operator are proven using Clifford algebras, and certain integral estimates for the Cauchy transform is extended to higher dimensions.

In this paper we study the uniform approximation of continuous functions on compact subsets of R n by solutions of second order elliptic equations. Also we discuss the problem of existence of a continuous best approximation and the connection of the approximation problem to some isoperimetric inequalities.KeywordsMaximum PrincipleElliptic EquationDirichlet ProblemUniform ApproximationIsoperimetric InequalityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

For an elliptic equation with a partially Muckenhoupt weight we find a sufficient condition for the regularity of a boundary point for the solution to the Dirichlet problem with a continuous boundary function and obtain an estimate for the modulus of continuity for the solution at this point.

This paper contains new uniqueness results of the boundary blow-up viscosity solutions of second order elliptic equations, generalizing a well known result of Marcus-Veron for the Laplace operator.

In this paper we discuss some problems about critical point theory. In the first part of the paper we study existence and multiplicity results of semilinear second order elliptic equation: $$ \begin{cases} -\Delta u=f(x,u) &\text{for } x\in \Omega, \\ u=0 &\text{for } x\in \partial \Omega, \end{cases} $$ In [Z. Li. Liu and S. J. Li, Contractibility of level sets of functionals associated with some elliptic boundary value problems and applications , NoDEA 10 (2003), 133–170], the authors study the contractibility of level sets of functionals associated with some elliptic boundary value problems. In this paper by using Morse theory and minimax method we give a more precise description of topological construction of level set of critical value of energy functional for mountain pass type critical point. It is well known that nondegenerate critical point is isolated, so if a critical point is not isolated, it must be a degenerate critical point. In the second part we will give an example that all the critical points of functional of a class of oscillating equation with Neumann boundary condition are isolated and the equation has only constant solutions. Moreover, critical groups of each critical point of the functional are trivial. The elliptic sine-Gordon equation originates from the static case of the hyperbolic sine-Gordon equation modelling the Josephson junction in superconductivity, which is of contemporary interest to physicists. The problem is similar to the elliptic sine-Gordon equation so we believe that it derives from profound physical backdrop.

The Legendre spectral Galerkin method of self-adjoint second order elliptic equations usually results in a linear system with a dense and ill-conditioned coefficient matrix. In this paper, the linear system is solved by a preconditioned conjugate gradient (PCG) method where the preconditioner M is constructed by approximating the variable coefficients with a (T+1)-term Legendre series in each direction to a desired accuracy. A feature of the proposed PCG method is that the iteration step increases slightly with the size of the resulting matrix when reaching a certain approximation accuracy. The efficiency of the method lies in that the system with the preconditioner M is approximately solved by a one-step method based on the incomplete LU factorization technique with no fill-in, denoted by ILU(0). The ILU(0) factorization of \(M\in {\mathbb {R}}^{(N-1)^d\times (N-1)^d}\) can be computed using \({\mathcal {O}}(T^{2d} N^d)\) operations, and the number of nonzeros in the factorization factors is of \({\mathcal {O}}(T^{d} N^d)\), \(d=1,2,3\). A conclusion of the algorithm is to fast solve the resulting system from the Legendre Galerkin spectral method for Poisson equations with Dirichlet boundary conditions, which has a complexity of \({\mathcal {O}}(N^d)\). To further speed up the PCG method, an algorithm is developed for fast matrix-vector multiplications by the resulting matrix of Legendre-Galerkin spectral discretization, without the need to explicitly form it. The complexity of the fast matrix-vector multiplications is of \({\mathcal {O}}(N^d (\log _2 N)^{2})\). In view that T is independent of N in one dimension and is set to be of order \({\mathcal {O}}(\log _2 N)\) in two and three dimensions, the PCG method has a \({\mathcal {O}}(N^d (\log _2 N)^{2d})\) quasi-optimal complexity for a d dimensional domain with \((N-1)^d\) unknows, \(d=1,2,3\). In addition, a fast direct solver for the three-dimensional Poisson equation is developed, which is of \({\mathcal {O}}(N^{3} (\log _2 N)^{2})\) and improves the existing results on the computational complexity. Numerical examples are given to demonstrate the efficiency of proposed preconditioners and the algorithm for fast matrix-vector multiplications.

The purpose of this paper is to estimate the rate of convergence for some natural difference analogues of Dirichlet's problem for uniformly elliptic differential equations, $$\begin{gathered} \sum\limits_{j,k = 1}^N {\frac{\partial }{{\partial x_j }}} \left( {a_{jk} \frac{{\partial u}}{{\partial x_k }}} \right) = F in R, \hfill \\ u = f on B, \hfill \\ \end{gathered}$$ in aN-dimensional domainR with boundaryB. These schemes will in general not be of positive type, and the analysis will therefore be carried out in discreteL2-norms rather than in the maximum norm. Since our approximation of the boundary condition is rather crude, we will only arrive at a rate of convergence of first order for smoothF andf. Special emphasis will be put on appraising the dependence of the rate of convergence on the regularity ofF andf.