Estimates of solutions of initial-irregular oblique derivative problems for linear parabolic equations of second order with measurable coefficients

The initial-irregular oblique derivative boundary value problems for linear and nondivergence parabolic complex equations of second order in multiply connected domains are dealt with, where the coefficients of equations are measurable. Firstly the uniqueness of solutions for the above problems is introduced, and then somea priori estimates of solutions for the problems are given. By using the above estimates and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems can be proved. The results are generalizations of corresponding theorems in literature. Project supported by the National Natural Science Foundation of China (Grant No. 19671006).     

INITIAL-OBLIQUE DERIVATIVE PROBLEMS FOR NONLINEAR PARABOLIC SYSTEMS WITH MEASURABLE COEFFICIENTS

This paper deals with some initial-oblique derivative boundary value problems for nonlinear nondivergent parabolic systems of several second order equations with measurable coefficients in multiply connected domains. Firstly, a priori estimates of solutions for the initial-boundary value problems are given, and then by using the above estimates of solutions and the Leray-Schauder theorem, the existence and uniqueness of solutions for the problems are proved.     

AN IRREGULAR OBLIQUE DERIVATIVE PROBLEM FOR SOME NONLINEAR ELLIPTIC EQUATIONS OF SECOND ORDER

1FormulationofirregularObliqueDerivativeProblemLetDbean(N 1)-connecteddomainwithboundaryr=UU=,r,6Cj(0相似文献   

Oblique derivative problems for nonlinear elliptic systems with measurable coefficients

1 Problem formulation Let Q be a bounded multiply connected domain in RN and the boundary OQ E C2. W6consider the nonlinear elliptic system of second order equationsUnder certain conditions, system (1) can be reduced to the formwhere u = (ul,' t u.), Da = (u..), DZu = (u:..,), andSuppose that (1) (or (2)) satisfiesCondition C For arbitrary functions u'(x), u'(z) E Cd(~) n W::(Q), Fk(x,u,Du,DZu)(k = 1,' I m) satisfy the conditionswhere 0 < g < 1, u == al - u2, and al:), bit), of*),…     

A priori estimates, existence and non-existence of positive solutions of generalized mean curvature equations

This paper concerns a priori estimates and existence of solutions of generalized mean curvature equations with Dirichlet boundary value conditions in smooth domains. Using the blow-up method with the Liouville-type theorem of the p laplacian equation, we obtain a priori bounds and the estimates of interior gradient for all solutions. The existence of positive solutions is derived by the topological method. We also consider the non-existence of solutions by Pohozaev identities.     

Polynomial modules and a generalized cauchy problem for systems of holomorphic partial differential operators

In this paper, we study holomorphic solutions of overdetermined systems of partial differential equatoins with constant coefficients. We parametrize the solution space, using an ideal of holomorphic functions generated by certain polynomials associated to the principal symbols of the operators, by proving a Cauchy-Kowalevsky type theorem for these systems. In this theorem, we introduce a non-characteristicity condition for systems which is a direct generalization of a condition for single equations introduced by the author and H.S. Shapiro in a previous paper.

As a tool for studying the systems mentioned above, certain estimates, which may be of independent interest, for elements of analytic modules generated by polynomials are obtained.     

带有可测系数的二阶非线性抛物型方程组的初—混合边值问题

在多连通区域上研究带有可测系数的二阶非线性抛物型方程组的初－混合边值问题，首先我们将其化为复形式的方程组，并给出在一定条件下的上述初－这值问题解的先验估计，然后利用解的估计和列紧性原理，证明了这种初－边值问题解的存在性。在论证过程中，我们始终用复分析方法讨论文中所提出的问题，没有看到国外有人使用这种方法处理此类问题。     

Estimates of solutions of initial-irregular oblique derivative problems for fully nonlinear parabolic equations

In this paper the initial-irregular oblique derivative problems for fully nonlinear parabolic equations of second order are proposed, and then some a priori estimates of solutions for the above problems are given.     

二阶非线性椭圆型方程带位移的边值问题

In this paper, we consider the boundary value problem with the shift for nonlinear uniformly elliptic equations of second order in a multiply connected domain. For this sake, we propose a modified boundary value problem for nonlinear elliptic systems of first order equations, and give a priori estimates of solutions for the modified boundary value problem. Afterwards we prove by using the Schauder fixedpoint theorem that this boundary value problem with some conditions has a solution. The result obtained is the generlization of the corresponding theorem on the Poincare boundary value problem.     

拟线性正对称方程组的边值问题及其对混合型方程的应用

<正> 1.引言 本文有两个目的.第一个目的是讨论拟线性正对称方程组的边值问题.如所知,线性的正对称方程组是一类相当广泛的偏微分方程组,许多常见的偏微分方程都可以化为正对称方程组去.对于这个方程组,可以利用能量不等式来证明许多边值问题的适定性.本文充分利用线性正对称方程组的结果,经过适当的估计,用压缩映照原理证明了拟线性正对称方程组的边值问题的解的存在性.对求解的区域而言,问题是大范围的,但     

An Existence Theorem of Smooth Nonlocal Center Manifolds for Systems Close to a System with a Homoclinic Loop

Summary. In this paper we give a proof of the existence of smooth nonlocal center manifolds for systems close to a system with a homoclinic orbit to a saddle-type equilibrium point. Our proof is based on a consideration of some class of the boundary value problems (see Section 3). We obtain estimates for solutions of the boundary value problems that allow us to prove the theorem on the center manifolds at the C ¹ -assumptions for the smoothness of systems. Received June 4, 1997; final revision received April 24, 1998     

SOME BOUNDARY VALUE PROBLEMS FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS OF SECOND ORDER

The present article deals with some boundary value problems for nonlinear elliptic equations with degenerate rank 0 including the oblique derivative problem.Firstly the formulation and estimates of solutions of the oblique derivative problem are given, and then by the above estimates and the method of parameter extension,the existence of solutions of the above problem is proved.In this article,the complex analytic method is used,namely the corresponding problem for degenerate elliptic complex equations of first order is firstly discussed,afterwards the above problem for the degenerate elliptic equations of second order is solved.     

Two-sided estimates for solutions to the Cauchy problem for Wazewski linear differential systems with delay

The Cauchy problem is considered for Wazewski linear differential systems with finite delay. The right-hand sides of systems contain nonnegative matrices and diagonal matrices with negative diagonal entries. The initial data are nonnegative functions. The matrices in equations are such that the zero solution is asymptotically stable. Two-sided estimates for solutions to the Cauchy problem are constructed with the use of the method of monotone operators and the properties of nonsingular M-matrices. The estimates from below and above are zero and exponential functions with parameters determined by solutions to some auxiliary inequalities and equations. Some estimates for solutions to several particular problems are constructed.     

On extending existence theory from scalar ordinary differential equations to infinite quasimonotone systems of functional equations

In this paper we use Tarski's fixed point theorem to extend in a systematic way the existence of extremal solutions from scalar initial value problems to boundary value problems for infinite quasimonotone functional systems of differential equations.

     

Boundary value problems for first-order mixed complex equations with degenerate curve

This article deals with some boundary value problems for quasilinear mixed (elliptic-hyperbolic) complex equations of first order with degenerate hyperbolic curve in a simply connected domain. First, we give the representation theorem and prove the uniqueness of solutions for the above boundary value problems. Then using the method of successive iteration, the existence of solutions for the above problems is proved.     

Regular Degenerate Separable Differential Operators and Applications

The boundary value problems for differential-operator equations with variable coefficients, degenerated on all boundary are studied. Several conditions for the separability, fredholmness and resolvent estimates in L _p -spaces are given. In applications degenerate Cauchy problem for parabolic equation, boundary value problems for degenerate partial differential equations and systems of degenerate elliptic equations on cylindrical domain are studied.     

Uniform and optimal estimates for solutions to singularly perturbed parabolic equations

The Cauchy problem for singularly perturbed parabolic equations is considered, and weighted L²-estimates as well as certain decay properties of bounded classical solutions to it are established. These do not depend on the value of the small perturbation parameter, and allow to prove global in time existence of strong solutions to certain boundary-value problems for ultraparabolic equations with unbounded coefficients. Optimal decay estimates are proved for such solutions. All results concerning ultraparabolic equations apply, in particular, to the Kolmogorov equation for diffusion with inertia, to the (linear) Fokker-Planck equation, to the linearized Boltzmann equation, and to some nonlinear integro-differential ultraparabolic equations of the Fokker-Planck type, arising from biophysics. Optimal decay estimates are derived for global in time strong solutions to such equations.     

Some Riemann boundary value problems in Clifford analysis

In this paper, we mainly study the R_m (m>0) Riemann boundary value problems for functions with values in a Clifford algebra C?(V_{3, 3}). We prove a generalized Liouville‐type theorem for harmonic functions and biharmonic functions by combining the growth behaviour estimates with the series expansions for k‐monogenic functions. We obtain the result under only one growth condition at infinity by using the integral representation formulas for harmonic functions and biharmonic functions. By using the Plemelj formula and the integral representation formulas, a more generalized Liouville theorem for harmonic functions and biharmonic functions are presented. Combining the Plemelj formula and the integral representation formulas with the above generalized Liouville theorem, we prove that the R_m (m>0) Riemann boundary value problems for monogenic functions, harmonic functions and biharmonic functions are solvable. Explicit representation formulas of the solutions are given. Copyright © 2009 John Wiley & Sons, Ltd.     

Three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations

In this paper, we consider the existence of at least three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations. The associated Green’s function for the boundary value problems is first given. The proofs of our main results are based upon the Leggett–Williams fixed point theorem. Finally, we give an example to demonstrate our result.     

奇异耦合分数阶微分方程组Dirichlet型边值问题的正解

研究了一非线性奇异非自治耦合半正分数阶微分方程组Dirichlet型边值问题.利用Schauder不动点定理,获得了该非自治耦合分数阶微分方程组Dirichlet型边值问题的正解.