A study of existence and uniqueness of weak solutions to boundary value problems describing an elastic body with weakly nonlocal surface elasticity is presented. The chosen model incorporates the surface strain energy as a quadratic function of the surface strain tensor and the surface deformation gradients up to Nth order. The virtual work principle, extended for higher‐order strain gradient media, serves as a basis for defining the weak solution. In order to characterize the smoothness of such solutions, certain energy functional spaces of Sobolev type are introduced. Compared with the solutions obtained in classical linear elasticity, weak solutions for solids with surface stresses are smoother on the boundary; more precisely, a weak solution belongs to $H^1(V)\cap H^N(S_s)$ where $S_s\subset S\equiv \partial V$ and $V\subset {\mathbb{R}}^3$.

中文翻译：

---

关于N阶表面弹性内边值问题的弱解。

提出了对描述具有弱非局部表面弹性的弹性体的边值问题的弱解的存在性和唯一性的研究。选择的模型将表面应变能量作为表面应变张量的二次函数，并且将表面变形梯度提高到N阶。虚拟工作原理扩展到高阶应变梯度介质，作为定义弱解的基础。为了表征这种解决方案的平滑性，引入了Sobolev型的某些能量功能空间。与经典线性弹性所获得的解相比，具有表面应力的固体的弱解在边界上更平滑；更确切地说，弱解决方案属于$H^{\mathrm{1个}}（\mathrm{伏特}）\cap H^{ñ}（{\mathrm{小号}}_s）$ 在哪里 ${\mathrm{小号}}_s\subset \mathrm{小号}\equiv \partial \mathrm{伏特}$ 和 $\mathrm{伏特}\subset {\mathbb{[R}}^3$。