PDE Seminar: Mui -- Upper bounds imply lower bounds

Date: Monday, 22 September 2025.  
Time: 11:00am-12:00pm.  
Venue: The Access Grid Room, Carslaw Room 829.  

Speaker: Dr.  Jonathan Mui (University of Wuppertal) 
Title: Upper bounds imply lower bounds 
Abstract: Suppose $u = u(t;u_0)$ solves the heat equation $u_t - \Delta u = 0$
with initial condition $u_0 \ge 0$ and Dirichlet boundary conditions in a bounded,
sufficiently smooth domain (say $C^2$).  Then for each $t>0$ there exist constants $0 <
c_t < C_t$ such that $c_t \varphi \le u(t) \le C_t \varphi$, where $\varphi$ denotes the
principal eigenfunction of the Dirichlet Laplacian (a.k.a the `ground state’).  Such a
result is well-known in the study of diffusion equations (see e.g.  Theorem 4.2.5 in the
classic book on heat kernels by Davies).  Typically, the upper bound is established
first, and then there are various methods to obtain the lower bound based on spectral
theory and properties of the heat kernel.  

In this talk, we present a functional analytic approach to this phenomenon of lower
bounds obtained from upper bounds.  Our proof relies solely on properties of
positivity-preserving operators, and thus shows that the phenomenon occurs in far
greater generality than is considered in the classical case.  

This is ongoing work with Daniel Daners and Jochen Glück.