非凸泛函高阶奇异摄动的局部插值技术：自由间断问题
Local interpolation techniques for higher-order singular perturbations of non-convex functionals: free-discontinuity problems
AI总结：
本文提出基于局部插值不等式的一般方法，分析高阶奇异摄动非凸梯度泛函，刻画自由间断问题中的跳跃能量密度，并给出 Mumford-Shah 与 Blake-Zisserman 泛函逼近。
摘要：
我们利用局部插值不等式，发展了一种处理依赖梯度、并受到阶数 $k\ge 2$ 导数奇异摄动的非凸积分泛函的一般方法。当应用于产生自由间断能量的泛函时，此类方法允许在定义跳跃部分密度函数的问题中改变直至 $k-1$ 阶导数的边界值，从而能够证明最优轮廓公式，并推出紧性和下界。作为应用，我们证明：对于依赖梯度且在无穷远处表现为常数的能量，其 $k$ 阶摄动的跳跃能量密度等于常数 $m_k$ 乘以跳跃大小的 $k$ 次方根。该结果首先针对截断二次能量密度并在一维情形下得到证明，随后可通过切片技术获得一般的高维情形。一大类非凸能量都可以作为这些特殊能量的包络来研究。最后，我们指出，令 $k$ 趋于无穷大可得到 Mumford-Shah 泛函的一种逼近。我们还导出了 Blake-Zisserman 泛函的一种新逼近。
英文摘要：
We develop a general approach, using local interpolation inequalities, to non-convex integral functionals depending on the gradient with a singular perturbation by derivatives of order $k\ge 2$. When applied to functionals giving rise to free-discontinuity energies, such methods permit to change boundary values for derivatives up to order $k-1$ in problems defining density functions for the jump part, thus allowing to prove optimal-profile formulas, and to deduce compactness and lower bounds. As an application, we prove that for $k$-th order perturbations of energies depending on the gradient behaving as a constant at infinity, the jump energy density is a constant $m_k$ times the $k$-th root of the jump size. The result is first proved for truncated quadratic energy densities and in the one-dimensional case, from which the general higher-dimensional case can be obtained by slicing techniques. A wide class of non-convex energies can be studied as an envelope of these particular ones. Finally, we remark that an approximation of the Mumford-Shah functional can be obtained by letting $k$ tend to infinity. We also derive a new approximation of the Blake-Zisserman functional.
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