Understanding the Hot Spots conjecture for planar domains

The proposed research lies in the common grounds of two active areas of research in pure mathematics - spectral geometry and analysis of partial differential equations. We are particularly interested in the analysis of Laplace eigenfunctions on domains in the plane. A second Neumann eigenfunction u of the Laplacian approximates the temperature distribution on an insulated domain for large times. The hot spots conjecture, due to J. Rauch (1974), is the assertion that u does not assume its maximum value in the interior of the domain. The conjecture is known to be false for non-simply connected domains in the plane. The objective of the proposed research is to understand this conjecture for some classes of domains in the plane. We have three different objectives: Centrally symmetric convex domains: Although it is known, from the work of Jerison and Nadirashvili (2000), that convex domains in the plane with bi-axial symmetry satisfy the hot spots conjecture, the same conclusion for centrally symmetric convex domains is not yet known. In their work, Jerison-Nadirashvili make the hot spots conjecture for this class of domains. Split domains: Roughly speaking, a split domain in the plane is a domain with Lipschitz boundary from which a straight line is removed. We will study hot spots conjecture for split domains with small slits. Burdzy (2005) gave a counterexample to hot spots conjecture which is an annulus. We will investigate what happens to this counterexample when a slit is introduced to turn the annulus into a simply connected domain. Convex domains in the plane with Robin boundary conditions: If one considers Robin boundary conditions then, for certain values of the Robin parameter, it follows from the maximum principle, that there are no hot spots for second Robin eigenfunctions. We will investigate how hot spots may appear as the Robin parameter goes to zero to provide Neumann boundary condition. The community of mathematicians working in the area of spectral geometry and analysis of PDE in UK alone is significantly large. Mathematicians in many UK universities, including University of Bristol, University of Reading, University of Leeds, Kings College London, University College London, University of Edinburgh, Durham University, are working in these fields and are very active. Outside of the UK, there are mathematicians at Indiana University (Bloomington, IN, USA), University of Washington (Seattle, USA), MIT (USA), Perdue University (USA), Simon Fraser University (Canada), University of Montreal (Canada), Institut Denis Poisson (France), Max Planck Institute, Bonn (Germany) who are working in these areas. Needless to say that the proposed research will be of interest to many of these mathematicians and many more.