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 \({{{\mathcal {P}}}}\). We associated with problem \({{{\mathcal {P}}}}\) a boundary optimal control problem, denoted by \({{{\mathcal {Q}}}}\). For Problem \({{{\mathcal {P}}}},\) we introduce the concept of well-posedness and for Problem \({{{\mathcal {Q}}}}\) we introduce the concept of weakly and weakly generalized well-posedness, both associated with appropriate Tykhonov triples. Our main results are Theorems 5 and 16. Theorem 5 provides the well-posedness of Problem \({{{\mathcal {P}}}}\) and, as a consequence, the continuous dependence of the solution with respect to the data. Theorem 16 provides the weakly generalized well-posedness of Problem \({{{\mathcal {Q}}}}\) 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.

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关于Tykhonov反飞机剪切问题的正确性

我们考虑一个边界值问题，该问题描述了弹性体的摩擦反平面剪切。该过程是静态的，并且使用库仑干摩擦定律的滑动相关版本来模拟摩擦。问题的弱表达是位移场的准变分不等式，表示为\（{{{\ mathcal {P}}}} \\）。我们将问题\（{{{\ mathcal {P}}}} \\）与边界最优控制问题关联，用\（{{{\ mathcal {Q}}}} \\表示。对于问题\（{{{\ mathcal {P}}}}，\），我们介绍适定性的概念，而对于问题\（{{\ mathcal {Q}}}} \\）我们介绍了弱和弱广义适定性的概念，它们都与适当的Tykhonov三元组相关联。我们的主要结果是定理5和16。定理5提供了问题\（{{{\ mathcal {P}}}} \\}的适定性，因此，解决方案相对于数据的连续依赖性。定理16提供了问题\（{{{{数学{Q}}}}} \）的弱广义适定性，在另外的假设下，它的弱适定性。这些定理的证明基于紧性，较低的半连续性，单调性和各种估计的论点。此外，我们提供了对适定性结果的机械解释。