Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Applications of the duality between the Complex Monge-Amp\`ere Equation
  and the Hele-Shaw flow   We give two applications of the the duality between the complex Homogeneous
Monge-Amp\`ere Equation (HMAE) and the Hele-Shaw flow. First, we prove
existence of smooth boundary data for which the weak solution to the Dirichlet
problem for the HMAE over $\mathbb P^1\times \overline{\mathbb D}$ is not twice
differentiable at a given collection of points, and also examples that are not
twice differentiable along a set of codimension one in $\mathbb{P}^1\times
\partial \mathbb{D}$. Second, we produce explicit families of smooth geodesic
rays in the space of K\"ahler metrics on $\mathbb P^1$ and on the unit disc
$\mathbb D$ that are constructed from an exhausting family of increasing
smoothly varying simply connected domains.

31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
15
The categories of this paper are: nucl-ex
14
The categories of this paper are: math-ph
22
The categories of this paper are: chao-dyn
5
The categories of this paper are: gr-qc
21
The categories of this paper are: adap-org
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat