相关论文: Laplacian growth in the half plane

The Laplacian growth problem in the limit of zero surface tension is proved to be equivalent to finding a particular solution to the dispersionless Toda lattice hierarchy. The hierarchical times are harmonic moments of the growing domain.…

A new class of solutions to Laplacian growth with zero surface tension is presented and shown to contain all other known solutions as special or limiting cases. These solutions, which are time-dependent conformal maps with branch cuts…

The Laplacian growth (the Hele-Shaw problem) of multi-connected domains in the case of zero surface tension is proven to be equivalent to an integrable systems of Whitham equations known in soliton theory. The Whitham equations describe…

We study the integrable structure of the 2D Laplacian growth problem with zero surface tension in an infinite channel with periodic boundary conditions in the transverse direction. Similar to the Laplacian growth in radial geometry, this…

Filtration combustion is described by Laplacian growth without surface tension. These equations have elegant analytical solutions that replace the complex integro-differential motion equations by simple differential equations of pole motion…

I show that the evolution of a two dimensional surface in a Laplacian field can be described by Hamiltonian dynamics. First the growing region is mapped conformally to the interior of the unit circle, creating in the process a set of…

We present a new class of exact solutions for the so-called {\it Laplacian Growth Equation} describing the zero-surface-tension limit of a variety of 2D pattern formation problems. Contrary to common belief, we prove that these solutions…

Some physical problems as flame front propagation or Laplacian growth without surface tension have nice analytical solutions which replace its complex integro-differential motion equations by simple differential equations of poles motion in…

We discuss the recently established equivalence between the Laplacian growth in the limit of zero surface tension and the universal Whitham hierarchy known in soliton theory. This equivalence allows one to distinguish a class of exact…

We prove the existence of ground state solutions to critical growth $p$-Laplacian and fractional $p$-Laplacian problems that are nonresonant at zero.

The problem of Laplacian growth is considered within the Loewner-equation framework. A new method of deriving the Loewner equation for a large class of growth problems in the half-plane is presented. The method is based on the…

We consider minimal graphs u(x,y)>0 over unbounded domains D (with u vanishing on the boundary of D). Assuming D contains a sector properly containing a halfplane, we obtain estimates on growth and provide examples illustrating a range of…

We report a new algorithm to generate Laplacian Growth Patterns using iterated conformal maps. The difficulty of growing a complete layer with local width proportional to the gradient of the Laplacian field is overcome. The resulting growth…

It had been conjectured that Diffusion Limited Aggregates and Laplacian Growth patterns (with small surface tension) are in the same universality class. Using iterated conformal maps we construct a 1-parameter family of fractal growth…

Harmonic moments are integrals of integer powers of z = x+iy over a domain. Here the domain is an exterior of a bubble of air growing in an oil layer between two horizontal closely spaced plates. Harmonic moments are a natural basis for…

Applying the method of moving planes in integral forms, we establish radial symmetry for positive solutions to a class of semilinear equations involving the fractional Laplacian in the unit ball and obtain Liouville type theorems concerning…

Let $0<\alpha<2$ be any real number. In this paper, we investigate the following semilinear equations involving the fractional Laplacian \begin{equation}(-\bigtriangleup)^{\alpha/2} u(x)=f(u),\end{equation} on $\mathbb{R}^n$ and…

A nested family of growing or shrinking planar domains is called a Laplacian growth process if the normal velocity of each domain's boundary is proportional to the gradient of the domain's Green function with a fixed singularity on the…

A one-parametric stochastic dynamics of the interface in the quantized Laplacian growth with zero surface tension is introduced. The quantization procedure regularizes the growth by preventing the formation of cusps at the interface, and…

Laplacian growth is the study of interfaces that move in proportion to harmonic measure. Physically, it arises in fluid flow and electrical problems involving a moving boundary. We survey progress over the last decade on discrete models of…