scholarly journals A minimization problem with free boundary related to a cooperative system

2018 ◽  
Vol 167 (10) ◽  
pp. 1825-1882 ◽  
Author(s):  
Luis A. Caffarelli ◽  
Henrik Shahgholian ◽  
Karen Yeressian
Keyword(s):  
Free Boundary ◽  
Minimization Problem ◽  
Cooperative System ◽  
Problem With Free Boundary

Regularity results for solutions to the c-Plateau problem with free boundary having small singular set, and generally in codimension zero

2016 ◽  
Vol 9 (3) ◽  
pp. 259-282 ◽  
Author(s):  
Leobardo Rosales
Keyword(s):  
Free Boundary ◽  
Minimization Problem ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Plateau Problem ◽  
Singular Set ◽  
Codimension Two ◽  
Rectifiable Currents ◽  
Problem With Free Boundary ◽  
Regularity Results

AbstractWe prove two results for the c-Plateau problem, introduced in [17], which is a minimization problem for integer rectifiable currents. First, we prove there is no solution to the c-Plateau problem with free boundary having singular set of finite Hausdorff codimension two measure and with regular part having constant mean curvature. Second, we prove regularity up to Hausdorff codimension seven of the free boundary of top-dimensional solutions to the c-Plateau problem.

The Two-dimensional Problem with Free Boundary in Problems of Medicine

2021 ◽  
Author(s):  
Fatimat Kh. Kudayeva ◽  
Aslan Kh. Zhemukhov ◽  
Aslan L. Nagorov ◽  
Arslan A. Kaygermazov ◽  
Diana A. Khashkhozheva ◽  
...  
Keyword(s):  
Free Boundary ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Problem With Free Boundary

An explicit example of Hopf bifurcation in fluid mechanics

1983 ◽  
Vol 390 (1799) ◽  
pp. 293-320 ◽  
Keyword(s):  
Hopf Bifurcation ◽  
Phase Space ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Fluid Mechanics ◽  
Invariant Subspace ◽  
Normal Modes ◽  
Free Boundary Conditions ◽  
Wave Numbers ◽  
Problem With Free Boundary

It is observed that a complete and explicit example of Hopf bifurcation appears not to be known in fluid mechanics. Such an example is presented for the rotating Bénard problem with free boundary conditions on the upper and lower faces, and horizontally periodic solutions. Normal modes are found for the linearization, and the Veronis computation of the wave numbers is modified to take into account the imposed horizontal periodicity. An invariant subspace of the phase space is found in which the hypotheses of the Joseph-Sattinger theorem are verified, thus demonstrating the Hopf bifurcation. The criticality calculations are carried through to demonstrate rigorously, that the bifurcation is subcritical for certain cases, and to demonstrate numerically that it is subcritical for all the cases in the paper.

A problem with free boundary

1989 ◽  
Vol 45 (3) ◽  
pp. 1163-1172
Author(s):  
E. V. Remizova
Keyword(s):  
Free Boundary ◽  
Problem With Free Boundary
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