Second-Order Necessary Optimality Conditions for Some State-Constrained Control Problems of Semilinear Elliptic Equations

In this paper we are concerned with some optimal control problems governed by semilinear elliptic equations. The case of a boundary control is studied. We consider pointwise constraints on the control and a finite number of equality and inequality constraints on the state. The goal is to derive first- and second-order optimality conditions satisfied by locally optimal solutions of the problem. Accepted 6 May 1997     

Hölder Estimates for A Class of Degenerate Elliptic Equations

In this paper we study the regularity theory of the solutions of a class of degenerate elliptic equations in divergence form. By introducing a proper distance and applying the compactness method we establish the Hölder type estimates for the weak solutions.     

On the Dirichlet Problem for a System of Degenerate at a Point Elliptic Equations in the Class of Bounded Functions

We consider a weakly connected (by the lowest terms) system of elliptic equations of second order with the main part in the form of the Laplace operator, the order of which becomes degenerate at an interior point of the domain. We investigate a Dirichlet-type problem in the class of bounded Hölder vector functions. We obtain sufficient conditions for the existence and uniqueness of a solution.     

退化的非线性椭圆方程的径向解

利用打靶法讨论一类退化非线性椭圆型方程径向解的性态，得出了径向解的间断及不间断的结果．     

半线性非一致椭圆方程的局部最大值原理及其在Hessian方程上的应用

本文获得了一个半线性散度型非一致椭圆方程的局部最大值原理，并由此导出了Hessian方程解的局部C^1,1估计和一个Bernstein型结果。     

Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems

We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The analysis of the approximate control problems is carried out. The uniform convergence of discretized controls to optimal controls is proven under natural assumptions by taking piecewise constant controls. Finally, error estimates are established and some numerical experiments, which confirm the theoretical results, are performed.The first two authors were supported by Ministerio de Ciencia y Tecnología (Spain). The second author was also supported by the DFG research center “Mathematics for key technologies” (FZT86) in Berlin.     

Optimal Control Problems for a Class of Semilinear Multisolution Elliptic Equations

We study the optimal control problem for a class of elliptic problems that may possess multiple solutions. We obtain necessary conditions for optimal control by constructing a related parabolic problem and using known results for the parabolic problem.     

Verification of Second-Order Sufficient Optimality Conditions for Semilinear Elliptic and Parabolic Control Problems

We study optimal control problems for semilinear parabolic equations subject to control constraints and for semilinear elliptic equations subject to control and state constraints. We quote known second-order sufficient optimality conditions (SSC) from the literature. Both problem classes, the parabolic one with boundary control and the elliptic one with boundary or distributed control, are discretized by a finite difference method. The discrete SSC are stated and numerically verified in all cases providing an indication of optimality where only necessary conditions had been studied before.     

退化椭圆问题的最小二乘混合有限元方法

本文研究了电磁场中关于共振现象的一类退化的椭圆问题 ,提出了最小二乘混合有限元方法 .这一方法的好处是可以去掉传统混合元空间的LBB条件所得到的系数矩阵是对称正定的 ,使得法语解更加方便 .本文得到了最小二乘混合有限元方法的L2 和H1估计 .     

Error Estimates for the Numerical Approximation of a Semilinear Elliptic Control Problem

We study the numerical approximation of distributed nonlinear optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. Our main result are error estimates for optimal controls in the maximum norm. Characterization results are stated for optimal and discretized optimal control. Moreover, the uniform convergence of discretized controls to optimal controls is proven under natural assumptions.     

关于一类半线性退化椭圆方程的存在性结果

本文研究下列半线性退化椭圆Ｄｉｒｉｃｈｌｅｔ问题：这里Ｘ＝｛Ｘ１，…，Ｘｍ｝是一组满足Ｈｏｒｍａｎｄｅｒ条件的实光滑向量场．假设它们在区域的边界附近还满足一些附加条件，以及ｆ∈Ｃ∞〔Ω×Ｒ×Ｒｍ），并且　ｚｆ（ｘ，ｚ，ξ）≥０，ｓｉｇｎＸｆ（ｘ，ｚ，０）≥μ＞－∞，ｃ（ｘ）≥ｃ０＞０和ｆ（ｘ，ｚ，ξ）关于变量ξ满足一定的增长条件．我们证明了当边界是无穷可微时，上述岸线性Ｄｉｒｉｃｈｌｅｔ问题的光滑解的存在性和唯一性．     

Backward Stochastic Differential Equations and Dirichlet Problems of Semilinear Elliptic Operators with Singular Coefficients

In this paper, we prove that there exists a unique solution to the Dirichlet boundary value problem for a general class of semilinear second order elliptic differential operators which do not necessarily have the maximum principle and are non-symmetric in general. Our method is probabilistic. It turns out that we need to solve a class of backward stochastic differential equations with singular coefficients, which is of independent interest itself. The theory of Dirichlet forms also plays an important role.     

On the Dirichlet Problem with an Asymptotic Condition for an Elliptic System Strongly Degenerate at a Point

We consider a Dirichlettype problem for a system of elliptic equations of second order with a strong degeneracy at an inner point of the domain, when, in a neighborhood of this point, the principal term of the asymptotics of a solution is additionally given. We prove the existence and uniqueness of a solution of the problem considered in a weighted class of Hölder vector functions.     

高阶半线性椭圆型方程奇摄动边值问题

本文研究了一类高阶半线性椭圆型方程奇摄动边值问题,利用比较定理,证明了渐近解的一致有效性。     

高阶半线性椭圆型方程奇摄动广义Dirichlet边值问题

本文讨论了半线性椭圆型方程奇摄动广义边值问题,在适当的条件下研究了Dirichlet边值问题广义解的存在唯一性及其渐近性态。     

Error Estimates and Superconvergence of RT0 Mixed Methods for a Class of Semilinear Elliptic Optimal Control Problems

In this paper, we will investigate the error estimates and the superconvergence property of mixed finite element methods for a semilinear elliptic control problem with an integral constraint on control. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element and the control variable is approximated by piecewise constant functions. We derive some superconvergence properties for the control variable and the state variables. Moreover, we derive $L^∞$- and $H^{-1}$-error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.     

一类退化椭圆型方程边值问题的适定性

研究一类特殊退化椭圆型方程边值问题的适定性,该类问题与双曲空间中的极小图的Dirichlet问题,曲面的无穷小等距形变刚性问题等等的研究密切相关,而这类方程的特征形式在区域上是变号的,其适定性是值得深入讨论的.最后,得到这类边值问题的H~1弱解的存在性和唯一性.     

L r Estimates for Degenerate Elliptic Problems

We prove L ^r estimates for the Dirichlet problem –div(a(x,u,Du))=f with f in L ^q for 1q+, where the operator satisfies (|s|)|| ^p a(x,s,), with p>1. These estimates are obtained without symmetrization and are sharp in some cases.