scholarly journals On the existence of non-flat profiles for a Bernoulli free boundary problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Giovanni Gravina ◽  
Giovanni Leoni
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Boundary Problem ◽  
Dirichlet Boundary ◽  
Free Boundary Problems ◽  
Energy Functional ◽  
Dirichlet Boundary Conditions ◽  
Boundary Problems ◽  
Global Minimizers ◽  
Flat Profile

AbstractIn this paper, we consider a large class of Bernoulli-type free boundary problems with mixed periodic-Dirichlet boundary conditions. We show that solutions with non-flat profile can be found variationally as global minimizers of the classical Alt–Caffarelli energy functional.

Properties of some ordinary differential equations related to free boundary problems

1986 ◽  
Vol 104 (3-4) ◽  
pp. 217-234 ◽  
Author(s):  
Gunduz Caginalp ◽  
Stuart Hastings
Keyword(s):  
Phase Transition ◽  
Free Boundary ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Phase Field ◽  
Free Boundary Problems ◽  
Second Order ◽  
Boundary Problems ◽  
Field Approach

SynopsisSome second order ordinary differential equations of the form ξ2ϕ″ + ξ2(N − 1)″′/r + ½(ϕ − ϕ3) + ½k = 0 are studied. Properties such as existence and monotonicity of solutions are considered for N ≧ 1, ξ > 0 and two sets of boundary conditions. For N = 1, some explicit results are obtained for small ξ. These ODE's arise from a phase field approach to free boundary problems involving a phase transition.

SINGULAR FREE BOUNDARY PROBLEM FROM IMAGE PROCESSING

2005 ◽  
Vol 15 (05) ◽  
pp. 689-715 ◽  
Author(s):  
A. L. AMADORI ◽  
J. L. VÁZQUEZ
Keyword(s):  
Image Processing ◽  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Free Boundary Problems ◽  
Singular Problem ◽  
Well Posedness ◽  
Boundary Problems ◽  
Nonlinear Parabolic ◽  
Singular Data

We study a degenerate nonlinear parabolic equation with moving boundaries which arises in the study of the technique of contour enhancement in image processing. In order to obtain mass concentration at the contour, singular data are imposed at the free boundary, leading to a nonstandard free boundary problem. Our main results are: (i) the well-posedness for the singular problem, without monotonicity assumptions on the initial datum, and (ii) the convergence of the approximation by means of combustion-type free-boundary problems.

Numerical treatment of a singularity in a free boundary problem

1972 ◽  
Vol 330 (1583) ◽  
pp. 573-580 ◽  
Keyword(s):  
Free Boundary ◽  
Numerical Solution ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Complex Variable ◽  
Free Boundary Problems ◽  
Separation Point ◽  
Free Boundaries ◽  
Boundary Problems ◽  
A Free Boundary Problem

The numerical solution of free boundary problems gives rise to many computational difficulties. One such difficulty is due to the singularity at the separation point between the fixed and free boundaries. A method is suggested which uses complex variable techniques to determine the shape of the free boundary near to the separation point. This complex variable solution is also used to improve the accuracy of the finite-difference solution in the neighbourhood of the singularity. The analytical study was incorporated into an algorithm for the numerical solution of a particular free boundary problem concerning the percolation of a fluid through a porous dam. Some numerical results for this problem are presented.

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