We provide the proof of an existence and uniqueness theorem for weak solutions of the equilibrium problem in linear dilatational strain gradient elasticity for bodies occupying, in the reference configuration, Lipschitz domains with edges. The considered elastic model belongs to the class of so-called incomplete strain gradient continua whose potential energy density depends quadratically on linear strains and on the gradient of dilatation only. Such a model has many applications, e.g., to describe phenomena of interest in poroelasticity or in some situations where media with scalar microstructure are necessary. We present an extension of the previous results by Eremeyev et al. (2020 Z angew Math Phys 71(6): 1–16) to the case of domains with edges and when external line forces are applied. Let us note that the interest paid to Lipschitz polyhedra-type domains is at least twofold. First, it is known that geometrical singularity of the boundary may essentially influence singularity of solutions. On the other hand, the analysis of weak solutions in polyhedral domains is of great significance for design of optimal computations using a finite-element method and for the analysis of convergence of numerical solutions.

我们提供了线性膨胀应变梯度弹性平衡问题的弱解的存在性和唯一性定理的证明，用于在参考配置中占据具有边缘的 Lipschitz 域的物体。所考虑的弹性模型属于所谓的不完全应变梯度连续体类，其势能密度二次取决于线性应变和仅取决于膨胀梯度。这样的模型有很多应用，例如，描述多孔弹性中感兴趣的现象，或者在需要具有标量微观结构的介质的某些情况下。我们提出了 Eremeyev 等人先前结果的扩展。（2020 年Z angew 数学物理 71(6): 1-16) 到具有边缘的域和施加外线力时的情况。让我们注意到支付给 Lipschitz 多面体类型域的利息至少是两倍。首先，已知边界的几何奇异性可能在本质上影响解的奇异性。另一方面，多面体域弱解的分析对于使用有限元方法的最优计算设计和数值解的收敛性分析具有重要意义。