共查询到20条相似文献,搜索用时 15 毫秒
1.
Pierre Dreyfuss 《Potential Analysis》2007,26(2):101-119
We prove that under some global conditions on the maximum and the minimum eigenvalue of the matrix of the coefficients, the gradient of the (weak) solution of some degenerate elliptic equations has higher integrability than expected. Technically we adapt the Giaquinta–Modica regularity method in some degenerate cases. When the dimension is two, a consequence of our result is a new Hölder continuity result for the weak solution. 相似文献
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We give examples of discontinuous solutions and of unbounded solutions of linear isotropic degenerate elliptic equations. Discontinuous solutions exist even when both the maximum eigenvalue and the inverse of the minimum eigenvalue of the matrix of the coefficients are in the intersection of all the Lp spaces. 相似文献
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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. 相似文献
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利用Moser-Nash迭代和稠密引理,得到了在自然增长下的非线性退化椭圆方程有界弱解具有某一Hlder指数的正则性;在已知数据的进一步正则性下,建立了具有任意γ满足0≤γ<κ的优化Hlder连续性指数,其中κ是A-调和函数的局部Hlder连续指数. 相似文献
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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. 相似文献
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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. 相似文献
10.
N. V. Krylov 《偏微分方程通讯》2015,40(8):1393-1407
We show that there exists an algebraic rate of convergence of solutions of finite-difference approximations for uniformly elliptic Isaacs equations in smooth bounded domains. 相似文献
11.
In this paper, we prove higher integrability results for the gradient of the solutions of some elliptic equations with degenerate
coercivity whose prototype is
where for example, a(x,u)=(1+|u|)−θ with θ ∈ (0,1). We study the same problem for minima of functionals closely related to the previous equation. 相似文献
12.
In this paper we prove convergence rates for the problem of approximating functions f by neural networks and similar constructions. We show that the rates are the better the smoother the activation functions are, provided that f satisfies an integral representation. We give error bounds not only in Hilbert spaces but also in general Sobolev spaces Wm, r(Ω). Finally, we apply our results to a class of perceptrons and present a sufficient smoothness condition on f guaranteeing the integral representation. 相似文献
13.
Nadir Arada Eduardo Casas Fredi Tröltzsch 《Computational Optimization and Applications》2002,23(2):201-229
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. 相似文献
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讨论了退化中立型微分方程的周期解问题,给出了周期解存在性的条件和二维退化中立型微分方程周期解存在的代数判据,并且举例说明了其应用. 相似文献
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O. Mali 《Numerical Functional Analysis & Optimization》2017,38(1):58-79
In this article, functional type a posteriori error estimates are presented for a certain class of optimal control problems with elliptic partial differential equation constraints. It is assumed that in the cost functional the state is measured in terms of the energy norm generated by the state equation. The functional a posteriori error estimates developed by Repin in the late 1990s are applied to estimate the cost function value from both sides without requiring the exact solution of the state equation. Moreover, a lower bound for the minimal cost functional value is derived. A meaningful error quantity coinciding with the gap between the cost functional values of an arbitrary admissible control and the optimal control is introduced. This error quantity can be estimated from both sides using the estimates for the cost functional value. The theoretical results are confirmed by numerical tests. 相似文献
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得到一类退化椭圆型方程弱解梯度在其拟线性系数矩阵$A(\cdot,u)$对任意$u$关于$x$一致满足VMO条件下在Morrey空间$L^{p,\lambda}$的内部正则性. 相似文献
18.
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. 相似文献
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利用临界点理论中的山路引理,证明一类含退化椭圆算子的Kirchhoff型方程在适当的假设条件下解的存在性,所得结论丰富和发展了已有文献的相关结果. 相似文献
20.
We obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation/numerical schemes for parabolic Hamilton-Jacobi-Bellman equations by introducing a new notion of consistency. Our results are robust and general - they improve and extend earlier results by Krylov, Barles, and Jakobsen. We apply our general results to various schemes including Crank-Nicholson type finite difference schemes, splitting methods, and the classical approximation by piecewise constant controls. In the first two cases our results are new, and in the last two cases the results are obtained by a new method which we develop here.