Cauchy Problem and Analytic Continuation for Systems of First Order Elliptic Equations with Analytic Coefficients

Let

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)

Euler type partial differential operators on real analytic functions

In this paper we study the topological properties of the level sets,$ \S_{t}(u)=\left\{x:~u(x)= t \right\}$, of solutions $u$of second order elliptic equations with vanishing zeroth order terms. Weshow that the total Betti number of level sets $\S_{t}$ is a uniformlybounded function of the parameter $t$. The uniform bound can be estimatedin terms of the analytic coefficients as well as the generalized degreesof the corresponding solutions. Such an estimate is also valid for thenodal sets of solutions of the same type equations with zerothorder terms. In general, it is possible to derive from our analysis anestimate for the total Betti numbers of level sets, for large measure setof $t's$, when coefficients are sufficiently smooth, and therefore a $L^{p}$bound on Betti numbers as a function of $t$. These estimatesare obtained by a quantitative Stability Lemma and a quantitative MorseLemma.

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.

Several partial answers are given to the question: Suppose U is a solution of the Douglis-Nirenberg elliptic system L U = F LU = F where F is analytic and L has analytic coefficients. If U = 0 U = 0 in some appropriate sense on a hyperplane (or any analytic hypersurface) must U vanish identically? One answer follows from introducing a so-called formal Cauchy problem for Douglis-Nirenberg elliptic systems and establishing existence and uniqueness theorems. A second Cauchy problem, in some sense a more natural one, is discussed for an important subclass of the Douglis-Nirenberg elliptic systems. The results in this case give a second partial answer to the original question. The methods of proof employed are largely algebraic. The systems are reduced to systems to which the Cauchy-Kowalewski theorem applies.

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.

This paper deals with elliptic equations in the plane with degen- eracies. The equations are generated by a complex vector field that is elliptic everywhere except along a simple closed curve. Kernels for these equations are constructed. Properties of solutions, in a neighborhood of the degen- eracy curve, are obtained through integral and series representations. An application to a second order elliptic equation with a punctual singularity is given.

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.

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.

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, a system of nonlinear Kaup equations with a loaded additional term in the class of periodic functions with respect to the spatial variable is considered. The invariance of the spectrum is proved and an analog of the Dubrovin system is derived for the evolution of the spectral parameters of a quadratic pencil of Sturm-Liouville operators on the entire line, the periodic coefficients of which are the solution of the Cauchy problem posed for the loaded system of nonlinear Kaup equations. Using trace formulas and an analog of the Dubrovin system, it is shown that the loaded system of nonlinear Kaup equations can be integrated by the inverse spectral problem method. An algorithm for solving the Cauchy problem for the loaded nonlinear Kaup’s system in the class of periodic functions with respect to the spatial variable is obtained. It is shown that if the initial functions are real analytic functions, then the solution will also be an analytic function with respect to the spatial variable. The 𝜋/2 -periodicity of the solution with respect to the spatial variable is revealed for the 𝜋/2 -periodicity of the initial functions.

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

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.

Introduction. In his study of the structure of distribution kernels (centering around his celebrated kernel theorem), L. Schwartz [11; 13] has studied the subclasses of regular and very regular distribution kernels, which may be characterized roughly by the fact that they carry infinitely differentiable functions into infinitely differentiable functions. It is our object in the present paper to extend this study to kernels which are analytically regular, i.e., which, again roughly, carry analytic functions into analytic functions. The motivation for such a study is provided by the fundamental solutions of elliptic equations with analytic coefficients. One consequence of our results is that for such equations the standard theorems on the analyticity of regular solutions imply the analyticity of the fundamental solution. The principal tool in our investigation is the general theory of topological vector spaces (particularly in certain forms given to it by A. Grothendieck) and especially the theory of topological tensor products. In ?1.1, we summarize the definitions from the theory of topological vector spaces which we shall use. In ?1.2, we give a brief summary of the definitions and principal results of Grothendieck'stheory of topological tensor products. In ?I.3, we summarize the results of Schwartz's study of distribution kernels, at least insofar as they relate to the generalizations to be given to analytic kernels. ?II is devoted to an intensive consideration of the space Gt(K) of real analytic functions on a compact subset of Rn. ?III states and proves our principal results on analytic distribution kernels. The writer should like to thank Professors L. Nachbin and F. E. Browder for discussion, suggestions, and criticism. The results of the present paper constitute a portion of the writer's Doctoral Dissertation [1] at the University of Sao Paulo, prepared while in residence at the Institute for Pure and Applied Mathematics, Rio de Janeiro.