标题： 二维空间中定义在$SBD^p$上泛函的积分表示 显示英文标题

标题： Integral representation for functionals defined on $SBD^p$ in dimension two

摘要： 我们证明了一个积分表示结果，适用于在空间$SBD^p(\Omega)$上具有使泛函强制的生长条件的泛函，对于$\Omega\subset\mathbb{R}^2$。 分布应变是$L^p$部分与在有限$\mathcal{H}^{1}$维测度集上支撑的有界测度之和的函数空间$SBD^p$自然出现在断裂和损伤模型的研究中。 我们的结果基于用$W^{1,p}$函数进行局部逼近的构造。 我们还在$SBD^p$情况下得到了 Korn 不等式的推广。 显示英文摘要

摘要： We prove an integral representation result for functionals with growth conditions which give coercivity on the space $SBD^p(\Omega)$, for $\Omega\subset\mathbb{R}^2$. The space $SBD^p$ of functions whose distributional strain is the sum of an $L^p$ part and a bounded measure supported on a set of finite $\mathcal{H}^{1}$-dimensional measure appears naturally in the study of fracture and damage models. Our result is based on the construction of a local approximation by $W^{1,p}$ functions. We also obtain a generalization of Korn's inequality in the $SBD^p$ setting.