标题： 在变分问题中边界数据的取得具有线性增长 显示英文标题

标题： On the attainment of boundary data in variational problems with linear growth

摘要： 众所周知，具有线性增长的凸变分问题和狄利克雷边界条件，如果边界条件没有适当放松，可能没有极小值。我们证明，对于广泛的被积函数，包括最小梯度问题和非参数化普拉托问题，在边界具有适当的平均凸性条件下，如果边界数据属于$BV$或$W^{\alpha,p}$且$\alpha p\geq 2$，则松弛问题的极小值在迹的意义下满足边界数据，而无需任何连续性假设。与之前的工作不同，我们的方法还能在被积函数具有一定准各向同性假设的情况下处理系统。我们进一步表明，在没有这种准各向同性假设的情况下，均匀凸域上存在光滑的反例。还给出了对极小值唯一性的应用以及关于带狄利克雷边界条件的ROF泛函和最小梯度函数的迹空间的开放问题。 显示英文摘要

摘要： It is well-known that convex variational problems with linear growth and Dirichlet boundary conditions might not have minimizers if the boundary condition is not suitably relaxed. We show that for a wide range of integrands, including the least gradient problem and the non-parametric Plateau problem, and under suitable mean-convexity conditions of the boundary, minimizers of the relaxed problem attain the boundary data in the trace sense if it lies in $BV$ or $W^{\alpha,p}$ with $\alpha p\geq 2$ without any kind of continuity assumption. Unlike previous works, our methods are also able to treat systems under a certain quasi-isotropy assumption on the integrand. We further show that without this quasi-isotropy assumption, smooth counterexamples on uniformly convex domains exist. Further applications to the uniqueness of minimizers and to open problems about the ROF functional with Dirichlet boundary conditions, and to the trace space of functions of least gradient are given.