论文标题
非本地的Lagrange乘数和运输密度
Nonlocal Lagrange multipliers and transport densities
论文作者
论文摘要
我们证明了Monge-Kantorovich方程的一般解决方案，具有分数$ s $ - 级别约束，$ 0 <s <1 $，与该类型的一般，可能是脱位的线性分数操作员相关， d \ cdot d^su+c \，u，u，\ end {equation*}带有可集成的数据，在空间中，在$λ^{s，p} _0（ω）$中，这是在$ l^p $ $ lies $ $ $ $ $ $ $ $ $ $ d^$ d^$ d的平滑函数中完成的平滑函数。 $ s = 1 $，$ d^1 = d $是经典梯度）。传输密度作为$ l^\ infty（\ r^d）$的双重空间中的广义拉格朗日乘数出现，并且与约束$ | d^su | \ leq g $下的相应运输电位的变异不平等相关联。通过对线性运算符的约束和非线性正则化$ \ MATHSCR l^su $的约束和非线性正则化，通过近似变异不平等来显示它们的存在。为此，我们还开发了空间的一些相关属性$λ^{s，p} _0（ω）$，包括限制案例$ p = \ infty $和连续嵌入$λ^{s，q} _0（q} _0（ω）\ subsetλ^s，p} {s，p} _0（s，p} _0（s，p} _0（s，p} _0（yef）$ 1 $ $ 1 c $ $ pe fe。我们还显示了非局部问题的本地化（$ 0 <s <1 $），当$ s \ rightArrow1 $时，与经典梯度约束的本地极限问题，对于一般的，可能是简单的，可能是变性的部分差异操作员$ \ mathscr l^1U $，仅具有集成的系数和范围的始终约束。
We prove the existence of generalised solutions of the Monge-Kantorovich equations with fractional $s$-gradient constraint, $0<s<1$, associated to a general, possibly degenerate, linear fractional operator of the type, \begin{equation*} \mathscr L^su=-D^s\cdot(AD^su+\bs b\,u)+\bs d\cdot D^su+c\,u , \end{equation*} with integrable data, in the space $Λ^{s,p}_0(Ω)$, which is the completion of the set of smooth functions with compact support in a bounded domain $Ω$ for the $L^p$-norm of the distributional Riesz fractional gradient $D^s$ in $\R^d$ (when $s=1$, $D^1=D$ is the classical gradient). The transport densities arise as generalised Lagrange multipliers in the dual space of $L^\infty(\R^d)$ and are associated to the variational inequalities of the corresponding transport potentials under the constraint $|D^su|\leq g$. Their existence is shown by approximating the variational inequality through a penalisation of the constraint and nonlinear regularisation of the linear operator $\mathscr L^su$. For this purpose, we also develop some relevant properties of the spaces $Λ^{s,p}_0(Ω)$, including the limit case $p=\infty$ and the continuous embeddings $Λ^{s,q}_0(Ω)\subset Λ^{s,p}_0(Ω)$, for $1\le p\le q\le\infty$. We also show the localisation of the nonlocal problems ($0<s<1$), to the local limit problem with classical gradient constraint when $s\rightarrow1$, for which most results are also new for a general, possibly degenerate, partial differential operator $\mathscr L^1u$ only with integrable coefficients and bounded gradient constraint.
