椭圆拟变分不等式的一种新惩罚方法
A new penalty method for elliptic quasivariational inequalities
AI总结：
本文研究自反 Banach 空间中的椭圆拟变分不等式，基于收敛准则提出一种新惩罚方法，证明无约束变分-半变分不等式解序列收敛，并应用于弹性摩擦接触问题及数值模拟验证。
摘要：
我们考虑自反 Banach 空间 $X$ 中的一类椭圆拟变分不等式，并回顾了文献 [10] 中获得的收敛性准则。该类中的每个不等式 $\cal P$ 均受约束集 $K$ 控制，并具有唯一解 $u\in K$。该准则提供了保证任意序列 $\{u_n\}\subset X$ 收敛于解 $u$ 的充分必要条件。然后，我们考虑由参数序列 $\{λ_n\}\subset\mathbb{R}_+$ 控制的无约束变分-半变分不等式序列 $\{\cal P_n\}$。我们利用该准则推导出：如果对于每个 $n\in\mathbb{N}$，项 $u_n$ 代表问题 $\cal P_n$ 的解，那么当 $λ_n\to 0$ 时，序列 $\{u_n\}$ 收敛于 $u$。我们将抽象结果应用于具有单侧约束的弹性摩擦接触问题的研究中，并给出了相应的力学解释。我们还展示了一个二维示例的数值模拟，作为我们收敛结果的证据。
英文摘要：
We consider a class of elliptic quasivariational inequalities in a reflexive Banach space $X$ for which we recall a convergence criterion obtained in [10]. Each inequality $\cal P$ in the class is governed by a set of constraints $K$ and has a unique solution $u\in K$. The criterion provides necessary and sufficient conditions which guarantee that an arbitrary sequence $\{u_n\}\subset X$ converges to the solution $u$. Then, we consider a sequence $\{\cal P_n\}$ of unconstrained variational-hemivariational inequalities governed by a sequence of parameters $\{λ_n\}\subset\mathbb{R}_+$. We use our criterion to deduce that, if for each $n\in\mathbb{N}$ the term $u_n$ represents a solution of Problem $\cal P_n$, then the sequence $\{u_n\}$ converges to $u$ as $λ_n\to 0$. We apply our abstract results in the study of an elastic frictional contact problem with unilateral constraints and provide the corresponding mechanical interpretations. We also present numerical simulation in the study of a two-dimensional example which represents an evidence of our convergence results.
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