Nonlinear elliptic problems with boundary blow-up
1997 (English)Doctoral thesis, comprehensive summary (Other academic)
A classical problem in differential geometry is the prescribed curvature problemin a bounded domain, where the task is to conformally deform the Euclidean metric to another complete Riemannian metric with a prescribed curvature function.If this function is negative, the problem is equivalent to solving a nonlinear elliptic partial differential equation with boundary blow-up. The problem is solvablein non-tangentially accessible domains, a class including the well-known vonKoch snowflake, and there are estimates of the behaviour of the solution at anypoint close to the boundary in terms of the distance from the point to the boundary. There is a characterization of the negative continuous functions for whichthe prescribed curvature problem is solvable. Another result is the existence ofa unique distribution solution of a particular boundary-blow-up problem relevant in probability theory, in any bounded domain. If the Laplace operator in the original problem is exchanged for the pseudo-Laplace operator or the real Monge-Ampère operator, it is still possible to state necessary and sufficient growth conditions on the nonlinear absorption term, which depends on the solution, for the corresponding problem to be solvable. Blow-up estimates andsome uniqueness results hold. The behaviour of a solution in a ball determineswhether there does or does not exist an entire solution of the equation, i.e. asolution defined throughout the Euclidean space. Existence results and growthestimates for solutions of a generalized mean curvature flow problem follow fromthe general technique which is the foundation of all the results in the thesis.
Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis , 1997. , 43 p.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 6
Mathematics, Boundary blow-up, NTA domain, fractal, p-Laplace
operator, Mange-Ampère operator, viscosity solution, mean curvature flow, gradient estimate, Fefferman's equation, superdiffusion, entire solution, maximum principle, Primary 35J65
Research subject Mathematics
IdentifiersURN: urn:nbn:se:uu:diva-16ISBN: 91-506-1204-2OAI: oai:DiVA.org:uu-16DiVA: diva2:161190
1997-05-16, Room 247, Building 2, Polacksbacken, Uppsala University, Uppsala, 10:15