标题： 二维弹性接触问题与库仑摩擦的最佳存在结果 显示英文标题

标题： Optimal existence results for the 2d elastic contact problem with Coulomb friction

摘要： 在本文中，揭示了线弹性中具有库仑摩擦的增量准静态接触问题（Signorini-Coulomb问题）的结构，并为最一般的二维问题，具有任意几何形状和弹性模量张量，证明了最优的存在性结果。 该问题被简化为一个涉及非线性算子的变分不等式，该算子处理弹性和摩擦。 该算子被证明属于所谓的Leray-Lions算子类，因此可以引用Brézis的结果来解决变分不等式。 结果表明，Leray-Lions算子定义中的一个性质难以验证，需要证明线性弹性Neumann-to-Dirichlet算子的一个新的精细性质。 这一精细性质仅在二维问题的情况下得到证明，目前将我们的存在性结果限制在该情况下。 在各向同性弹性的情况下，无论是均匀还是非均匀的，都证明了Signorini-Coulomb问题在任意大的摩擦系数下的解的存在性。 在各向异性弹性的情况下，给出了一个大摩擦系数下解不存在的例子，并在摩擦系数的最优条件下证明了解的存在性。 显示英文摘要

摘要： In this article, the structure of the incremental quasistatic contact problem with Coulomb friction in linear elasticity (Signorini-Coulomb problem) is unraveled and optimal existence results are proved for the most general bidimensional problem with arbitrary geometry and elasticity modulus tensor. The problem is reduced to a variational inequality involving a nonlinear operator which handles both elasticity and friction. This operator is proved to fall into the class of the so-called Leray-Lions operators, so that a result of Br\'ezis can be invoked to solve the variational inequality. It turns out that one property in the definition of Leray-Lions operators is difficult to check and requires proving a new fine property of the linear elastic Neumann-to-Dirichlet operator. This fine property is only established in the case of the bidimensional problem, limiting currently our existence result to that case. In the case of isotropic elasticity, either homogeneous or heterogeneous, the existence of solutions to the Signorini-Coulomb problem is proved for arbitrarily large friction coefficient. In the case of anisotropic elasticity, an example of nonexistence of a solution for large friction coefficient is exhibited and the existence of solutions is proved under an optimal condition for the friction coefficient.