Concentration phenomena in variational problems
Graduate Seminar on PDEs S4B2
Prof. Dr. J. J. L. Velázquez, Dr. Y. Seki
The goal of this seminar is to study several elliptic problems whose solutions can be obtained by finding the critical points of some suitable functionals. The specific feature of the considered problems is that some of the nonlinear terms cannot be estimated by means of regularizing terms in the functional. More precisely, the nonlinearities fail to be estimated by the regularizing terms because they are in the critical exponent in which the Sobolev inequalities required to estimate the nonlinearities fail.
These problems are often characterized by the onset of concentration of some functions of the solutions to Dirac masses. Very often topological constraints are needed in order to prove the existence of solutions for the problems under consideration. To determine the position of the points where the concentrations take place requires to solve some delicate properties of the Green's functions associated to some elliptic equations.
The goal of this seminar is to study several of the papers where these concentration phenomena had been considered. One of the aims is to explore the possibility of obtaining inequalities which could be useful to study evolution problems. Some of the papers which will be studied are listed below.
- H. Brezis, F. Merle. Uniform estimates and blow-up behavior for solutions of –Δu=V(v)·e^u in two dimensions. Comm. Partial Differential Equations 16(8-9), 1223–1253, 1991.
- M. Grossi, F. Takahashi. Nonexistence of multi-bubble solutions to some elliptic equations on convex domains. J. Funct. Anal. 259(4), 904–917, 2010.
- F. Robert, M. Struwe. Asymptotic profile for a fourth order PDE with critical exponential growth in dimension four. Adv. Nonlinear Stud 4(4), 397–415, 2004.
- O. Rey. Proof of two conjectures of H. Brezis and L. A. Peletier. Manuscripta Math 65(1), 19–37, 1989.
- M. Struwe. A global compactness result for elliptic boundary value problems involving limit nonlinearities. Math. Z. 187(4), 511–517, 1984.
- M. Struwe, G. Tarantello. On multivortex solutions in Chern-Simons gauge theory. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 8(1), 109–121, 1998.