Nonlinear half-dirichlet problems for first order elliptic equations in the unit ball of Rm(m≥3)

On the basis of an a priori estimate for solutions of a linear half-Dirichlet problem in the unit ball first order equations with the Di-rac operator as principal part for Clifford algebravalued functions an existence and uniqueness result is proved for a related nonlinear problemAn approximation procedure is used and an error estimate given.     

The monotonicity condition and a priori estimates of the Hölder norm for a class of nondiagonal elliptic systems with quadratic nonlinearity

The Dirichlet problem for quasilinear elliptic systems of equations with nondiagonal principal matrix and additional terms with strong nonlinearity in the gradient is considered. An a priori condition on the behavior of solutions in a neighborhood of a fixed point is stated, which allows us to estimate locally the Hölder norm of a solution in terms of its energy norm. Owing to the monotonicity inequality proved for one class of such systems, the a priori assumption on the behavior of solutions is reduced to an optimal one while estimating the Hölder norm of the solution. Bibliography: 27 titles.     

一类渗流方程解在边界上的不稳定性和Blow-up

对一类具有非线性第二、第三边值条件的非线性渗流方程,证明了解的先验的界可以用初值和解在区域边界上的积分来估计和控制．这一先验估计是通过迭代技巧来建立的．根据这个估计,解可能在边界上爆破(Blow-up)从而解有渐近不稳定性．     

Weighted estimate for the convergence rate of a projection difference scheme for a quasilinear parabolic equation

A novel technique is proposed for analyzing the convergence of a projection difference scheme as applied to the initial value problem for a quasilinear parabolic operator-differential equation with initial data u ₀ ∈ H. The technique is based on the smoothing property of solutions to the differential problem for t > 0. Under certain conditions on the nonlinear term, a new estimate of order \(O(\sqrt \tau + h)\) for the convergence rate in a weighted energy norm is obtained without using a priori assumptions on the additional smoothness of weak solutions.     

A Priori Estimate for the Discontinuous Oblique Derivative Problem for Elliptic Systems

Recently [6] an existence as well as a uniqueness theorem for the discontinuous oblique derivative problem for nonlinear elliptic system of first order in the plane, see [12, 19, 23] was proved, based on some a priori estimate from [20]. This estimate, however, is deduced by reductio ad absurdum. Therefore the constants in this estimate are unknown so that the estimate cannot be used for numerical procedures, e.g. for approximating the solution of a nonlinear problem by solutions of related linear problems, see [24, 3, 4]. In this paper a direct proof of an a priori estimate is given using some variations of results from [14], see also [11], where the constants can explicitely be estimated. For related a priori estimates see [1 – 5, 8, 16, 17, 20, 21, 24 – 26]. A basic reference for the oblique derivative problem is [9].     

Time discretisation of monotone nonlinear evolution problems by the discontinuous Galerkin method

A class of discontinuous Galerkin methods is studied for the time discretisation of the initial-value problem for a nonlinear first-order evolution equation that is governed by a monotone, coercive, and hemicontinuous operator. The numerical solution is shown to converge towards the weak solution of the original problem. Furthermore, well-posedness of the time-discrete problem as well as a priori error estimates for sufficiently smooth exact solutions are studied.     

Nonlinear commutators for the fractional <Emphasis Type="Italic">p</Emphasis>-Laplacian and applications

We prove a nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions for the fractional p-Laplacian. This implies that solutions to certain degenerate nonlocal equations are higher differentiable. Also, weakly fractional p-harmonic functions which a priori are less regular than variational solutions are in fact classical. As an application we show that sequences of uniformly bounded \(\frac{n}{s}\)-harmonic maps converge strongly outside at most finitely many points.     

Symmetric jump processes and their heat kernel estimates

We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes (or equivalently, a class of symmetric integro-differential operators). We focus on the sharp two-sided estimates for the transition density functions (or heat kernels) of the processes, a priori Hölder estimate and parabolic Harnack inequalities for their parabolic functions. In contrast to the second order elliptic differential operator case, the methods to establish these properties for symmetric integro-differential operators are mainly probabilistic.     

Riemann-Stieltjes积分边值问题正解的存在唯一性

利用解的先验估计和极值原理,研究了一类具有Riemann-Stieltjes积分边值问题正解的存在唯一性．     

A priori bounds in the cauchy problem for coupled elliptic systems

By means of the logarithmic convexity of a suitable functional, an a priori inequality is developed for the sum of the squares of the solutions of the following improperly posed Cauchy problem. Consider the coupled elliptic system Lu = aν+f,Lν= bu+g, where L is a uniformly elliptic differential operator, a,b,f and g are bounded integrable functions with |b(x)|≧b₀>0 and ν satisfies a stabilizing condition, and where upper bounds for the error in measurement of the Cauchy data on the initial surface are prescribed. From the a priori estimate uniqueness, stability, and pointwise bounds for the solutions u and n are simultaneously deduced. The bounds are improvable by the Ritz technique. Moreover, the method presented here can be extended to the nonlinear system Lu = f(x, ν), Lν =g(x,u)provided g is a suitable form     

Boundedness vs. blow-up in a chemotaxis system

We determine the critical blow-up exponent for a Keller-Segel-type chemotaxis model, where the chemotactic sensitivity equals some nonlinear function of the particle density. Assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution. Furthermore, we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails.     

A priori estimate of\parallel u\parallel _{c^2 (\overline \Omega )}for convex solutions of the Dirichlet problem for the Monge-Ampere equations

One gives an a priori estimate of \(\parallel u\parallel _{c^2 (\overline \Omega )}\) for the convex solutions of the Dirichlet problem for the Monge-Ampère equation (U_ij=f(x,u,u_x)in a strictly convex bounded domain det,ΩcRⁿ     

The problem of Dirichlet for anisotropic quasilinear degenerate elliptic equations

We consider the Dirichlet problem for a class of anisotropic degenerate elliptic equations. New a priori estimates for solutions and for the gradient of solutions are established. Based on these estimates sufficient conditions guaranteeing the solvability of the problem are formulated. The results are new even in the semilinear case when the principal part is the Laplace operator.     

Finding the Minimal Root of an Equation with the Multiextremal and Nondifferentiable Left-Hand Part

A problem very often arising in applications is presented: finding the minimal root of an equation with the objective function being multiextremal and nondifferentiable. Applications from the field of electronic measurements are given. Three methods based on global optimization ideas are introduced for solving this problem. The first one uses an a priori estimate of the global Lipschitz constant. The second method adaptively estimates the global Lipschitz constant. The third algorithm adaptively estimates local Lipschitz constants during the search. All the methods either find the minimal root or determine the global minimizers (in the case when the equation under consideration has no roots). Sufficient convergence conditions of the new methods to the desired solution are established. Numerical results including wide experiments with test functions, stability study, and a real-life applied problem are also presented.     

一类具偏差变元Rayleigh方程周期解的存在性

作者研究一类具偏差变元Ralyleigh方程周期解的存在性问题,利用重合度拓展定理得到了周期解存在性结论.有意义的是本文周期解先验界估计的方法与已有的工作均不相同.     

Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac equations in one space dimension

The existence of global solutions of the Cauchy problem is proved for the Maxwell-Dirac equations coupled through the standard electromagnetic interaction. The proof depends on the conservation of charge and an a priori estimate on the electromagnetic potential. The technique also applies to the Dirac-Klein-Gordon equations with Yukawa coupling.     

Equivalence of variational inequality problems to unconstrained minimization

In this paper we propose a class of merit functions for variational inequality problems (VI). Through these merit functions, the variational inequality problem is cast as unconstrained minimization problem. We estimate the growth rate of these merit functions and give conditions under which the stationary points of these functions are the solutions of VI. This work was supported by the state key project “Scientific and Engineering Computing”.     

Construction of weak solutions of a certain stochastic Navier–Stokes equation

We prove the existence of weak solutions of stochastic Navier–Stokes equation on a two-dimensional torus, which appears in a certain variational problem. Our equation does not satisfy the coercivity condition. We construct its weak solutions due to an approximation by a sequence of solutions of equations with enlarged viscosity terms and then by showing an a priori estimate for them.     

A class of degenerate parabolic equations with a concentrated nonlinear source

In this paper, we consider the Cauchy problem for a class of degenerate parabolic equations with a concentrated nonlinear source. We obtain the existence of the generalized solutions for the problem based on some a priori estimates on solutions.     

The Global Attractors for the Dissipative Generalized Hasegawa-Mima Equation

The long time behavior of solutions of the generalized Hasegawa-Mima equation with dissipation term is considered. The existence of global attractors of the periodic initial value problem is proved, and the estimate of the upper bound of the Hausdorff and fractal dimensions for the global attractors is obtained by means of uniform a priori estimates method.