A unified approach to embedding theorems for Sobolev type spaces of vector-valued functions, defined via their symmetric gradient, is proposed. The Sobolev spaces in question are built upon general rearrangement-invariant norms. Optimal target spaces in the relevant embeddings are determined within the class of all rearrangement-invariant spaces. In particular, all symmetric gradient Sobolev embeddings into rearrangement-invariant target spaces are shown to be equivalent to the corresponding embeddings for the full gradient built upon the same spaces. A sharp condition for embeddings into spaces of uniformly continuous functions, and their optimal targets, are also exhibited. By contrast, these embeddings may be weaker than the corresponding ones for the full gradient. Related results, of independent interest in the theory of symmetric gradient Sobolev spaces, are established. They include global approximation and extension theorems under minimal assumptions on the domain. A formula for the K-functional, which is pivotal for our method based on reduction to one-dimensional inequalities, is provided as well. The case of symmetric gradient Orlicz-Sobolev spaces, of use in mathematical models in continuum mechanics driven by nonlinearities of non-power type, is especially focused.

提出了一种统一的方法来嵌入向量值函数的 Sobolev 类型空间的定理，通过它们的对称梯度定义。所讨论的 Sobolev 空间建立在一般重排不变范数上。相关嵌入中的最佳目标空间是在所有重排不变空间的类中确定的。特别是，所有对称梯度 Sobolev 嵌入到重排不变目标空间中都被证明等效于构建在相同空间上的完整梯度的相应嵌入。还展示了嵌入均匀连续函数空间的尖锐条件及其最佳目标。相比之下，这些嵌入可能比完整梯度的相应嵌入要弱。相关结果，建立了对对称梯度 Sobolev 空间理论的独立兴趣。它们包括在域的最小假设下的全局近似和扩展定理。一个公式还提供了 K泛函，这对于我们基于归约到一维不等式的方法至关重要。对称梯度 Orlicz-Sobolev 空间的情况在由非幂类型的非线性驱动的连续介质力学中的数学模型中使用，尤其受到关注。