Abstract We show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower semicontinuity and relaxation theorems for BV, BD, and for more general first-order linear PDE side constrains. Our proofs are based on recent progress in the understanding of singularities of measure solutions to linear PDEs and of the generalized convexity notions corresponding to these PDE constraints.

中文翻译：

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PDE 约束下线性增长积分泛函的下半连续性和松弛

摘要 我们展示了线性增长积分函数的一般下半连续性和松弛定理，这些函数定义在满足线性 PDE 边约束（任意阶数）的向量测度上。这些结果概括了 BV、BD 和更一般的一阶线性 PDE 侧约束的几个已知的下半连续性和松弛定理。我们的证明基于对线性偏微分方程的度量解奇点的理解以及与这些偏微分方程约束对应的广义凸性概念的最新进展。