Search results for "Free boundary problem"

On a singular boundary value problem for a second order ordinary differential equation

2000

A free boundary problem stemmed from combustion theory. Part II: Stability, instability and bifurcation results

2002

AbstractWe deal with a free boundary problem, depending on a real parameter λ, in a infinite strip in R2, providing stability, instability and bifurcation.

Bifurcation of traveling waves in a Keller–Segel type free boundary model of cell motility

2018

We study a two-dimensional free boundary problem that models motility of eukaryotic cells on substrates. This problem consists of an elliptic equation describing the flow of cytoskeleton gel coupled with a convection-diffusion PDE for the density of myosin motors. The two key properties of this problem are (i) presence of the cross diffusion as in the classical Keller-Segel problem in chemotaxis and (ii) nonlinear nonlocal free boundary condition that involves curvature of the boundary. We establish the bifurcation of the traveling waves from a family of radially symmetric steady states. The traveling waves describe persistent motion without external cues or stimuli which is a signature of …

Trapping Horizons as inner boundary conditions for black hole spacetimes

2007

We present a set of inner boundary conditions for the numerical construction of dynamical black hole space-times, when employing a 3+1 constrained evolution scheme and an excision technique. These inner boundary conditions are heuristically motivated by the dynamical trapping horizon framework and are enforced in an elliptic subsystem of the full Einstein equation. In the stationary limit they reduce to existing isolated horizon boundary conditions. A characteristic analysis completes the discussion of inner boundary conditions for the radiative modes.

Drops moving in flow with chernical reaction

1994

We propose a free boundary model described by coupled Navier-Stokes and chemical reaction equations with discontinuous coefRcients to simulate the chemical re- ¿ctions in viscous drops moving in a viscous incompressible ûuid. Approximation of the solution by a special ñnite element method (FEM) with a method of mapping is discussed. Several numerical resulùs åre presented.

Minimizers for the Thin One‐Phase Free Boundary Problem

2021

We consider the “thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in urn:x-wiley:00103640:media:cpa22011:cpa22011-math-0001 plus the area of the positivity set of that function in urn:x-wiley:00103640:media:cpa22011:cpa22011-math-0002. We establish full regularity of the free boundary for dimensions urn:x-wiley:00103640:media:cpa22011:cpa22011-math-0003, prove almost everywhere regularity of the free boundary in arbitrary dimension, and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight. While our results are typical for…

\( L^{1} \) existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions

2007

Abstract In this paper we study the questions of existence and uniqueness of weak and entropy solutions for equations of type − div a ( x , D u ) + γ ( u ) ∋ ϕ , posed in an open bounded subset Ω of R N , with nonlinear boundary conditions of the form a ( x , D u ) ⋅ η + β ( u ) ∋ ψ . The nonlinear elliptic operator div a ( x , D u ) is modeled on the p-Laplacian operator Δ p ( u ) = div ( | D u | p − 2 D u ) , with p > 1 , γ and β are maximal monotone graphs in R 2 such that 0 ∈ γ ( 0 ) and 0 ∈ β ( 0 ) , and the data ϕ ∈ L 1 ( Ω ) and ψ ∈ L 1 ( ∂ Ω ) .

Morse-Smale index theorems for elliptic boundary deformation problems.

2012

AbstractMorse-type index theorems for self-adjoint elliptic second order boundary value problems arise as the second variation of an energy functional corresponding to some variational problem. The celebrated Morse index theorem establishes a precise relation between the Morse index of a geodesic (as critical point of the geodesic action functional) and the number of conjugate points along the curve. Generalization of this theorem to linear elliptic boundary value problems appeared since seventies. (See, for instance, Smale (1965) [12], Uhlenbeck (1973) [15] and Simons (1968) [11] among others.) The aim of this paper is to prove a Morse–Smale index theorem for a second order self-adjoint el…

Renormalized solutions for degenerate elliptic–parabolic problems with nonlinear dynamical boundary conditions and L1-data

2008

Abstract We consider a degenerate elliptic–parabolic problem with nonlinear dynamical boundary conditions. Assuming L 1 -data, we prove existence and uniqueness in the framework of renormalized solutions. Particular instances of this problem appear in various phenomena with changes of phase like multiphase Stefan problems and in the weak formulation of the mathematical model of the so-called Hele–Shaw problem. Also, the problem with non-homogeneous Neumann boundary condition is included.

Existence of a traveling wave solution in a free interface problem with fractional order kinetics

2021

Abstract In this paper we consider a system of two reaction-diffusion equations that models diffusional-thermal combustion with stepwise ignition-temperature kinetics and fractional reaction order 0 α 1 . We turn the free interface problem into a scalar free boundary problem coupled with an integral equation. The main intermediary step is to reduce the scalar problem to the study of a non-Lipschitz vector field in dimension 2. The latter is treated by qualitative topological methods based on the Poincare-Bendixson Theorem. The phase portrait is determined and the existence of a stable manifold at the origin is proved. A significant result is that the settling time to reach the origin is fin…