  In this paper, we investigate Monge-Kantorovich problems for which the
absolute continuity of marginals is relaxed. For $X,Y\subseteq\mathbb{R}^{n+1}$
let $(X,\mathcal{B}_X,\mu)$ and $(Y,\mathcal{B}_Y,\nu)$ be two Borel
probability spaces, $c:X\times Y\to\mathbb{R}$ be a cost function, and consider
the problem \begin{align*}\tag{MKP}\label{MKPEQ} \inf\left\{\int_{X\times Y}
c(x,y)\,d\lambda\ :\ \lambda \in\Pi(\mu,\nu) \right\}. \end{align*} Inspired by
the seminal paper \cite{GANGBOMCCANN2} with applications in shape recognition
problem, we first consider \eqref{MKPEQ} for the cost $c(x,y)=h(x-y)$ with $h$
strictly convex defined on the multi-layers target space \begin{align*}
X=\overline{X}\times\{\overline{x}\},\quad\text{and}\quad Y=\bigcup_{k=1}^K
\left(\overline{Y}_{k}\times \{\overline{y}_k\}\right), \end{align*} where
$\overline{X}, \overline{Y}_{k}\subseteq \mathbb{R}^{n}$ for $k\in
\{1,\ldots,K\},$ $\overline{x}\in \mathbb{R}$, and $\{\overline{y}_1,...,
\overline{y}_K\}\subseteq \mathbb{R}$. Here, we assume that
$\mu|_\overline{X}\ll\mathcal{L}^n$ (the Lebesgue measure on $\mathbb{R}^n$),
but $\mu$ is singular w.r.t. $\mathcal{L}^{n+1}$. When $K=1$, this translates
to the standard \eqref{MKPEQ} for which the unique solution is concentrated on
a map. We show that for $K\geq 2,$ the solution is still unique but it
concentrates on the graph of several maps. Next, we study \eqref{MKPEQ} for a
closed subset $X\subseteq \mathbb{R}^{n+1}$ and its $n$-dimensional submanifold
$X_0$ with the first marginal of the form \begin{align*} \int_X
f(x)\,d\mu(x)=\int_X f(x)\alpha(x)\,d\mathcal{L}^{n+1}(x)+\int_{X_0} f(x_0)\,d
S(x_0),\ \ \forall f\in C_b(X). \end{align*} Here, $S$ is a measure on $X_0$
such that $S\ll \mathcal{L}^{n}$ on each coordinate chart of $X_0$. This can be
seen as a two-layers problem as the measure $\mu$ charges both $n$- and
$n+1$-dimensional subsets.
