求解具有混合边界条件的拟线性扩散方程的有限元方法的误差估计问题

在M.F.Wheeler的[5]中,对一类拟线性抛物型方程的F.E.M,进行了颇为深入的理论分析。但是,[5]所考虑的方程,其高阶项的系数,尚有某种局限性,以致不能应用于一般的各向异性问题;对于混合边界的情形,也未加讨论。此外,[5]中所涉及的条件也是较强的,如要求解函数u(x,t)∈c~2(Ω×[0,T])等等。 本文对实践中常常遇到的具有第三混合边界条件的一类拟线性扩散问题的F.E.M,在较[5]为弱的条件下,进行了讨论,把有关拟线性问题的误差估计问题归结为某一线性椭圆边值问题F.E.M的误差估计问题。本文的结果是[1]的推广。     

Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian

In this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional Laplacians and the critical exponents. This work can be seen as a nonlocal analog of the results of Han (1991) [24] and Rey (1990) [35].     

Existence and multiplicity of solutions for elliptic problems with indefinite discontinuous nonlinearities

In this paper we study the critical points for a locally Lipschitz functional that in some sense will be solutions of an elliptic problem with indefinite discontinuous nonlinearities. We should mention that our results were inspired by the work of Ambrosetti-Badiale [3], Arcoya-Calahorrano [5], Alama-Tarantello [1] and Chang [8]. For the problem studied in [3] we introduce indefinite nonlinearities as in [1] and [6]. To obtain the existence and multiplicity of solutions we use the critical points theory developed by Chang. Applications for Plasma Physics are considered with nonlinearities that change sign. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Remarks on elliptic singular perturbation problems

We show the effectiveness of viscosity-solution methods in asymptotic problems for second-order elliptic partial differential equations (PDEs) with a small parameter. Our stress here is on the point that the methods, based on stability results [3], [16], apply without hard PDE calculations. We treat two examples from [11] and [23]. Moreover, we generalize the results to those for Hamilton—Jacobi—Bellman equations with a small parameter.H. Ishii was supported in part by the AFOSR under Grant No. AFOSR 85-0315 and the Division of Applied Mathematics, Brown University.     

Nonexistence of solutions for singular elliptic equations with a quadratic gradient term

This paper concerns the nonexistence of solutions for singular elliptic equations with a quadratic gradient term. The main results complement and partly extend some works by Arcoya et al. (2009) [1]. As a by-product of the main results, we fill in a gap in one of their works.     

A concentration-compactness principle at infinity and positive solutions of some quasilinear elliptic equations in unbounded domains

In this paper, we formulate a concentration-compactness principle at infinity which extends a result introduced by J. Chabrowski [Calc. Var. Partial Differential Equations 3 (1995) 493-512]. Then we consider some quasilinear elliptic equations in some classes of unbounded domains by solving their corresponding constrained minimization problems under certain conditions. We show the existence of positive solutions of those equations via the concentration-compactness principle at infinity, which extends some results in [Differential Integral Equations 6 (1993) 1281-1298].     

On computation of solution curves for semilinear elliptic problems

We consider computation of solution curves for semilinear elliptic equations. In case solution is stable, we present an algorithm with monotone convergence, which is a considerable improvement of the corresponding schemes in [4] and [5]. For the unstable solutions, we show how to construct a fourth-order evolution equation, for which the same solution will be stable.     

Deformation Argument under PSP Condition and Applications

In this paper we introduce a new deformation argument, in which C~0-group action and a new type of Palais-Smale condition PSP play important roles. This type of deformation results are studied in [17, 21] and has many different applications [10,11, 17, 21] et al. Typically it can be applied to nonlinear scalar field equations. We give a survey in an abstract functional setting. We also present another application to nonlinear elliptic problems in strip-like domains. Under conditions related to [5,6], we show the existence of infinitely many solutions. This extends the results in [8].     

Some existence results for fully nonlinear elliptic equations of monge-ampère type

We study the existence and uniqueness of solutions of Monge-Ampère-type equations. This type of equations has been studied extensively by Caffarelli, Nirenberg, Spruck and many others. (See [5] through [8] and the references therein.) We present some existence and uniqueness results for this type of equations on compact Riemannian manifolds with non-negative sectional curvature. We have also generalized some results in [7].     

Degree Theory for Second Order Nonlinear Elliptic Operators and its Applications

We introduce, along the lines of [4], an integer valued degree for second order fully nonlinear elliptic operators which is invariant under homotopy within elliptic operators. We also give some applications to the bifurcation problem for nonlinear elliptic equations. Applications to the existence of solutions of certain fully nonlinear elliptic equations on compact manifolds can be found in [7].     

Regularity of refinable functions with exponentially decaying masks

The regularity of refinable functions is an important issue in all multiresolution analysis and has a strong impact on applications of wavelets to image processing, geometric and numerical solutions of elliptic partial differential equations. The purpose of this paper is to characterize the regularity of refinable functions with exponentially decaying masks and a dilation matrix whose eigenvalues have the same modulus. The main results of this paper are really extensions of some results in Cohen et al. (1999) [5], Jia (1999) [17] and Lorentz and Oswald (2000) [28].     

Elliptic operators, conormal derivatives and positive parts of functions (with an appendix by Haïm Brezis)

Haïm Brezis and Augusto Ponce introduced and studied several extensions of Kato's inequality, in particular Kato's inequalities up to the boundary involving the Laplacian and the normal derivative of the positive part of a W1,1 function in a smooth domain [H. Brezis, A.C. Ponce, Kato's inequality when Δu is a measure, C. R. Acad. Sci. Paris Sér. I 338 (2004) 599-604; H. Brezis, A.C. Ponce, Kato's inequality up to the boundary, Commun. Contemp. Math. 10 (2008) 1217-1241]. Using potential theoretic methods we answer here some questions raised in [H. Brezis, A.C. Ponce, Kato's inequality up to the boundary, Commun. Contemp. Math. 10 (2008) 1217-1241] about the relations between the normal derivative of a function u and the normal derivative of its positive part u₊. The results apply to a large class of domains and elliptic operators in divergence form and finally an expression of the normal derivative of a function of u is given. In the final appendix, H. Brezis solves an old question of J. Serrin about pathological solutions of certain elliptic equations [J. Serrin, Pathological solutions of elliptic differential equations, Ann. Sc. Norm. Super. Pisa (3) 18 (1964) 385-387]. This is used in the paper to extend the first version of our main result.     

On spline collocation methods for boundary integral equations in the plane

Nowadays boundary elemen; methods belong to the most popular numerical methods for solving elliptic boundary value problems. They consist in the reduction of the problem to equivalent integral equations (or certain generalizations) on the boundary Γ of the given domain and the approximate solution of these boundary equations. For the numerical treatment the boundary surface is decomposed into a finite number of segments and the unknown functions are approximated by corresponding finite elements and usually determined by collocation and Galerkin procedures. One finds the least difficulties in the theoretical foundation of the convergence of Galerkin methods for certain classes of equations, whereas the convergence of collocation methods, which are mostly used in numerical computations, has yet been proved only for special equations and methods. In the present paper we analyse spline collocation methods on uniform meshes with variable collocation points for one-dimensional pseudodifferential equations on a closed curve with convolutional principal parts, which encompass many classes of boundary integral equations in the plane. We give necessary and sufficient conditions for convergence and prove asymptotic error estimates. In particular we generalize some results on nodal and midpoint collocation obtained in [2], [7] and [8]. The paper is organized as follows. In Section 1 we formulate the problems and the results, Section 2 deals with spline interpolation in periodic Sobolev spaces, and in Section 3 we prove the convergence theorems for the considered collocation methods.     

On some conformally invariant fully nonlinear equationsSur certaines équations completement nonlinéaires invariantes par transformation conforme

We outline proofs of our results in [7] on Liouville type theorems, Harnack type inequalities, and existence and compactness of solutions to some conformally invariant fully nonlinear elliptic equations of second order on locally conformally flat Riemannian manifolds. Details will appear in [7]. To cite this article: A. Li, Y.Y. Li, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 305–310.     

随机定向收缩及其应用*

作为Altman的定向收缩理论[4,5]和Lee,Padgett的随机收缩理论[1,2]的推广,本文对非线性集值随机算子引入了随机定向收缩概念,利用这一新概念和超限归纳法,我们证明了非线性集值随机算子方程随机解的几个存在性定理.这些定理分别改进和推广了[1,2,4,5,11]中相应的结果.其次,给出了我们的结果对非线性随机积分和微分方程的某些应用.     

Iterationsverfahren für monotone,nicht notwendig Lipschitz-beschränkte Operatoren im Hilbert-Raum

Summary We state three theorems concerning iterative methods for solving equations containing monotone, but not necessarily uniformly monotone operators in Hilbert space, which may be not Lipschitz bounded, thus extending results obtained by Sibony [9, 10]. We apply the method to a nonlinear elliptic Dirichlet problem of laminar flow theory of non Newtonian fluids, thus improving the rather experimental treatment given in Ames [1] and removing the condition of nonzero difference quotients assumed by Cryer [3].     

Permanence under strong aggressions is possible

This paper analyzes the limiting behavior of the positive solutions of a general class of sublinear elliptic weighted mixed boundary value problems as the amplitude of the positive part of the lower order terms of the differential operator blows up to infinity. The main result establishes that the positive solutions approximate zero within the support of the positive part of the potential, whereas they stabilize to the positive solution of a certain elliptic mixed boundary value problem on its complement. Further, we use this result for deriving some general principles in competing species dynamics. Precisely, we shall show that in the presence of a refuge region two competing species must coexist if their reproduction rates are sufficiently large, independently of the strength of the competition. It should be emphasized that the abstract theory developed here allows measuring how large the reproduction rates should be for being permanent, providing us, simultaneously, with the limiting behavior of each of the species separately. Basically, when the pressure from the competitor grows the tackled species concentrates within its refuge. The results of this paper are substantial extensions of some pioneer results found by one of the authors in [16, Section 4]. The main ingredients in deriving the main results of this paper are the continuous dependence of the principal eigenvalue with respect to a general class of perturbations of the domain around its Dirichlet boundary – very recent result coming from [6] – and the continuous dependence of the positive solutions of the sublinear problem – coming from [7].     

一类二阶时滞微分方程的振动性判据

本文从不同的角度研究一类二阶时滞微分方程的振动性，得到了一切解振动性的充分条件。改善了文［１］、［２］的工作．     

Existence of solutions for elliptic equations with discontinuous nonlinearities strong resonance at infinity

In the present paper, our main purposes are to study nonlinear elliptic equations with strong resonance at infinity. Some existence theorems for nontrivial solutions are obtained by using some nonsmooth critical point theorems in [N. C. Kourogenis, N. S. Papageorgiou, Nonsmooth critical point theory and Nonlinear elliptic equations at resonance, J. Austral. Math Soc. (Ser. A) 69 (2000) 245–271]. The two of our theorems generalize Theorems 0.1 and 5.2 in [P. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Anal. TMA 7 (1983) 981–1012] to nonsmooth cases. Another theorem is new even if for the smooth case.     

On the Geometric Measure of Nodal Sets of Solutions

We present some general methods for the estimation of the local Hausdorff measure of nodal sets of solutions to elliptic and parabolic equations. Our main results (Theorems 3.1 and 4.1) improve previous results of Lin Fanghua in [1].