论文标题
关于抗平台剪切问题的Tykhonov良好性
On the Tykhonov Well-posedness of an Antiplane Shear Problem
论文作者
论文摘要
我们考虑了一个边界价值问题，该问题描述了弹性体的摩擦抗平台剪切。该过程是静态的，摩擦是用库仑干摩擦定律的滑动依赖性版本建模的。该问题的薄弱表述是以$ \ cp $表示的排量领域的准级不等式的形式。我们与问题$ \ cp $相关联，边界最佳控制问题，由$ \ cq $表示。对于问题$ \ cp $，我们介绍了适应性良好的概念，对于问题$ \ cq $，我们介绍了弱且弱化的良好性良好的概念，这两者都与适当的Tykhonov Triples相关。我们的主要结果是定理\ ref {t1}和\ ref {t2}。定理\ ref {t1}提供了问题$ \ cp $的适当性，因此，解决方案相对于数据的持续依赖性。定理\ ref {t2}提供了问题$ \ cq $的弱推广，并在其他假设下，其弱势姿势。这些定理的证明是基于紧凑性，较低的半持续性，单调性和各种估计的参数。此外，我们提供了适当的结果的机械解释。
We consider a boundary value problem which describes the frictional antiplane shear of an elastic body. The process is static and friction is modeled with a slip-dependent version of Coulomb's law of dry friction. The weak formulation of the problem is in the form of a quasivariational inequality for the displacement field, denoted by $\cP$. We associated to problem $\cP$ a boundary optimal control problem, denoted by $\cQ$. For Problem $\cP$ we introduce the concept of well-posedness and for Problem $\cQ$ we introduce the concept of weakly and weakly generalized well-posedness, both associated to appropriate Tykhonov triples. Our main result are Theorems \ref{t1} and \ref{t2}. Theorem \ref{t1} provides the well-posedness of Problem $\cP$ and, as a consequence, the continuous dependence of the solution with respect to the data. Theorem \ref{t2} provides the weakly generalized well-posedness of Problem $\cQ$ and, under additional hypothesis, its weakly well posedness. The proofs of these theorems are based on arguments of compactness, lower semicontinuity, monotonicity and various estimates. Moreover, we provide the mechanical interpretation of our well-posedness results.
