Bernoulli Free-Boundary Problems
Eugene Shargorodsky · John F. Toland
jan. 2008 · American Mathematical Soc.
5,0star
1 anmeldelsereport
E-bog
70
Sider
reportBedømmelser og anmeldelser verificeres ikke Få flere oplysninger
Om denne e-bog
When a domain in the plane is specified by the requirement that there exists a harmonic function which is zero on its boundary and additionally satisfies a prescribed Neumann condition there, the boundary is called a Bernoulli free boundary. (The boundary is 'free' because the domain is not known a priori and the name Bernoulli was originally associated with such problems in hydrodynamics.) Questions of existence, multiplicity or uniqueness, and regularity of free boundaries for prescribed data need to be addressed and their solutions lead to nonlinear problems.In this paper an equivalence is established between Bernoulli free-boundary problems and a class of equations for real-valued functions of one real variable. The authors impose no restriction on the amplitudes or shapes of free boundaries, nor on their smoothness. Therefore the equivalence is global, and valid even for very weak solutions. An essential observation here is that the equivalent equations can be written as nonlinear Riemann-Hilbert problems and the theory of complex Hardy spaces in the unit disc has a central role. An additional useful fact is that they have gradient structure, their solutions being critical points of a natural Lagrangian. This means that a canonical Morse index can be assigned to free boundaries and the Calculus of Variations becomes available as a tool for the study. Some rather natural conjectures about the regularity of free boundaries remain unresolved.
Bedømmelser og anmeldelser
5,0
1 anmeldelse
Bedøm denne e-bog
Fortæl os, hvad du mener.
Oplysninger om læsning
Smartphones og tablets
Installer appen Google Play Bøger til Android og iPad/iPhone. Den synkroniserer automatisk med din konto og giver dig mulighed for at læse online eller offline, uanset hvor du er.
Bærbare og stationære computere
Du kan høre lydbøger, du har købt i Google Play via browseren på din computer.
e-læsere og andre enheder
Hvis du vil læse på e-ink-enheder som f.eks. Kobo-e-læsere, skal du downloade en fil og overføre den til din enhed. Følg den detaljerede vejledning i Hjælp for at overføre filerne til understøttede e-læsere.
