A-调和方程弱解的边界Lq估计：姚锋平的研究

115 浏览量更新于2024-09-03 收藏 190KB PDF 举报

本文标题"Boundary regularity for weak solutions of A-Harmonic Equations"聚焦于A-调和方程的弱解在边界上的行为。A-调和方程是一种非线性椭圆偏微分方程，其一般形式为div(A(∇u, x)) = div(|f|p-2f)，其中A是一个依赖于梯度和位置坐标的向量场，f是给定的源项，p是固定的有界指数。作者姚锋平来自上海大学理学院，研究的是在系数A满足特定条件下的问题。在文章的核心内容中，姚锋平主要探讨了当系数A具有适当的连续性和可测性时，如何通过弱解的概念在开球B_R（半径为R的开球）的正半空间B_+R中得到边界Lq类型的梯度估计。Lq空间在这里指的是Sobolev空间，它是函数空间的一个重要分支，用于描述函数在空间和梯度上的分布特性。这种估计对于理解非线性偏微分方程的解的行为以及它们与边界的关系至关重要。问题的关键在于找到合适的假设条件，确保A-调和方程的解在边界附近的梯度大小有适当的控制。这涉及到对梯度增长、系数A的结构以及非线性源项f的影响的深入分析。边界条件（0.1）和（0.2）分别规定了解在边界面上的值以及在坐标轴上的边界条件。此外，文章还提到了研究工作的资金支持，表明这项工作得到了一定的资助，可能来自于某个科研机构或项目。这对于理解研究背景和可能的应用范围也是重要的信息。这篇文章是对A-调和方程的边界行为进行理论分析的重要成果，对于深入理解这类非线性偏微分方程的解的性质具有重要意义，也为后续的理论发展和实际应用提供了新的见解。

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

Now we are set to state the main result.

Theorem 0.2. Assume that f ∈ L

). If u ∈ W

1,p

) is a weak solution of (0.1)-(0.2)

with A satisfying (0.3)-(0.6), then we have

∇u ∈ L

) for any q ≥ p

with the estimate



|∇u|

dx ≤ C





|f|

dx +



|u|

dx + 1



, (0.9)

where C is a constant independent from u and f.

Remark 0.3. Actually, similarly to [9–11] we can generalize regularity estimates in Sobolev

spaces to Orlicz spaces



ϕ(|∇u|

) dx ≤ C





ϕ (|f|

) dx +



ϕ (|u|

) dx + 1



. (0.10)

Indeed, if ϕ(t) = |t|

q/p

with q ≥ p, (0.10) is reduced to L

estimates (0.9). Here we omit the

details of the proof since more complicated analysis and computation should be carefully carried

out.

The result still holds when we replace B

, B

by B

, B

. Actually, in [11] we have

studied the local regularity in Orlicz spaces for (0.1) with A satisfying (0.3)-(0.5) and

|A (ξ, x) − A (ξ, y)| ≤ C

w(|x − y|)



1 + |ξ|

p−1



, (0.11)

in which we mainly used an iteration-covering approach based on [12]. We remark the condition

(0.11) is stronger than the condition (0.6). Now the boundary regularity estimates are studied.

Our approach in this work is very much inﬂuenced by [4, 8, 10, 13]. Our basic tools in this

approach are the Vitali covering lemma and the Hardy-Littlewood maximal function. Unlike

[11, 12], we do not need to use an approximation argument to justify a priori estimates and

reach the general case.

This paper will be organized as follows: In Section 2, we give some preliminary tools and

a new normalization method, which are very important to obtain the main result. We ﬁnish

the proof of Theorem 0.2 in Section 3.

1 Preliminary materials

1.1 Preliminary tools.

We use the Hardy-Littlewood maximal function which controls the local behavior of a

function.

- 3 -

剩余13页未读，继续阅读

weixin_38670949

粉丝: 8
资源: 983

A-调和方程弱解的边界Lq估计：姚锋平的研究

Boundary behavior of large solutions for non-divergence structure elliptic equations of second order

Ali-R-Ansari_Numerical-solution-of-a-convection-diffusion-problem-with-Robin-boundary-conditions_Journal-of-Computation-and_Applied-Mathematics_2003

Two-point Boundary Value Problem of Integro-differential Equations of mixed type in Banach Spaces

FEM-Source Integral Boundary Conditions for Computation of the Open-Boundary Electrostatic Fields

Boundary value problems for $2n$-order $p$-Laplacian difference equations containing both advance and retardation

Regularity of solutions of a free boundary problem modeling the growth of solid tumors

Existence of positive solutions for singular higher-order fractional differential equations with infinite-points boundary conditions

Compressible Navier-Stokes-Poisson equations with non-homogeneous boundary conditions

5 Key Properties of Weak Solutions to Partial Differential Equations: Generalized Solutions Beyond ...

可以省略 boundary 吗?multipart/form-data; boundary=--------------------------909512080119736368446920

最新资源