具有线性增长的变分问题解的唯一性与边界行为
Uniqueness and boundary behaviour of solutions to variational problems with linear growth
AI总结：
针对线性增长变分积分的Dirichlet问题，证明当边界凸点集测度足够大时，松弛问题在有界变差函数空间中存在唯一解，且极小值解内部光滑并在凸点集上达到边界数据。
AI中文摘要：
我们研究密度函数f满足适当椭圆性条件的线性增长变分积分J[u] = ∫_Ω f(∇u) dx的Dirichlet问题。证明若边界∂Ω上凸点集Γ₀的（n-1维）豪斯多夫测度ℋ^{n-1}(Γ₀)足够大（例如ℋ^{n-1}(Γ₀) > (2/3)ℋ^{n-1}(∂Ω)），则松弛问题在有界变差函数空间中存在唯一解u；且极小值解u在Ω内部光滑，至少在Γ₀上按经典意义达到给定边界数据。
英文摘要：
We investigate the Dirichlet problem for the variational integral $J[u] = \int_Ω f(\nabla u) \, dx$ with density $f$ of linear growth satisfying appropriate ellipticity conditions. We show that the relaxed problem admits a unique solution $u$ in the space of functions of bounded variation, if the set $Γ_0$ of convex points $x \in \partialΩ$ is sufficiently large. For example, the inequality $\mathcal{H}^{n-1}(Γ_0) > \frac{2}{3}\mathcal{H}^{n-1}(\partialΩ)$ is sufficient. Moreover, the minimizer $u$ is smooth in the interior of $Ω$ and attains the prescribed boundary data at least on $Γ_0$ in the classical sense.
