This book is devoted to the study of maximum principles in partial differential equations. I t contains a wealth of material much of which is presented for the first time in a book form. An attractive feature of the book is that it is completely elementary and thus accessible to a wide audience of readers. The book has four chapters. Chapter I deals with the one dimensional maximum principle. The discussion of this very simple model of a maximum principle forms a good introduction to the general theory. Various applications of the principle are given to show that it is a very useful tool even in the study of ordinary differential equations. As an example, it is shown that many oscillation and comparison results in the Sturm-Liouville theory could be deduced most easily by a maximum principle argument. The proper discussion of maximum principles in partial differential equations begins in Chapter II . This chapter, which is the backbone of the book, is devoted to elliptic equations. The material covered in this chapter includes the E. Hopf maximum principle and its generalizations; the Phragmèn-Lindelöf principle for solutions of elliptic equations; Serrin's version of the Harnack inequality for solutions of general elliptic equations in two variables (this is probably the most difficult result discussed in the book) ; various versions of the Hadamard three circles theorems for solutions of elliptic equations; applications of the maximum principle to nonlinear equations and to problems of fluid flow. Chapter III is devoted to parabolic equations. The plan of this chapter parallels that of Chapter II . The topics discussed include the L. Nirenberg strong maximum principle; a three curves theorem with an interesting application to the Tikhonov uniqueness theorem; a Phragmèn-Lindelöf principle for parabolic equations with applications to uniqueness results; nonlinear operators; a maximum principle for certain parabolic systems. The fourth and the last chapter is devoted to hyperbolic equations. The results in this chapter are somewhat special since a maximum principle in the proper sense does not hold for solutions of hyperbolic equations. Nevertheless, solutions of certain hyperbolic equations

Maximum Principles In Differential Equations

A coupled bulk-surface model for cell polarisation.

We consider a coupled bulk-surface system of partial differential equations with nonlinear coupling modelling receptor-ligand dynamics. The model arises as a simplification of a mathematical model for the reaction between cell surface resident receptors and ligands present in the extra-cellular medium. We prove the existence and uniqueness of solutions. We also consider a number of biologically relevant asymptotic limits of the model. We prove convergence to limiting problems which take the form of free boundary problems posed on the cell surface. We also report on numerical simulations illustrating convergence to one of the limiting problems as well as the spatio-temporal distributions of the receptors and ligands in a realistic geometry.

Coupled bulk-surface free boundary problems arising from a mathematical model of receptor-ligand dynamics

We analyze a certain class of coupled bulk–surface reaction–drift–diffusion systems arising in the modeling of signalling networks in biological cells. The coupling is by a nonlinear Robin-type boundary condition for the bulk variable and a corresponding source term on the cell boundary. For reaction terms with at most linear growth and under different regularity assumptions on the data we prove the existence of weak and classical solutions. In particular, we show that solutions grow at most exponentially with time. Furthermore, we rigorously derive an asymptotic reduction to a non-local reaction–drift–diffusion system on the membrane in the fast-diffusion limit.

Well-posedness and fast-diffusion limit for a bulk–surface reaction–diffusion system

Effect of complex landscape geometry on the invasive species spread: Invasion with stepping stones.

We consider a reaction-diffusion system where some components react and diffuse on the boundary of a region, while other components diffuse in the interior and react with those on the boundary through mass transport. We establish local well-posedness and global existence of solutions for these systems using classical potential theory and linear estimates for initial boundary value problems.

Global Existence of Solutions to Reaction-Diffusion Systems with Mass Transport Type Boundary Conditions

We consider reaction diffusion systems where components diffuse inside the domain and react on the surface through mass transport type boundary conditions. Under reasonable hypotheses, we establish the existence of component wise non-negative global solutions which are uniformly bounded in the sup norm.

Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions

We consider reaction diffusion systems where some components react and diffuse on the surface, and others diffuse inside the domain and react with surface components through dynamic boundary conditions. Under reasonable hypotheses, we establish the existence of componentwise non-negative global solutions.

Global existence of solutions to volume-surface reaction diffusion systems with dynamic boundary conditions

We consider reaction-diffusion systems where some components react and diffuse on the boundary of a region, while other components diffuse in the interior and react with those on the boundary through mass transport. We establish criteria guaranteeing that solutions are uniformly bounded in time.

Uniform bounds for solutions to volume-surface reaction diffusion systems

We consider a question of global existence for two component volume-surface reaction-diffusion systems. The first of the components diffuses in a region, and then reacts on the boundary with the second component, which diffuses on the boundary. We show that if the first component is bounded a priori on any time interval, and the kinetic terms satisfy a generalized balancing condition, then both solutions exist globally. We also pose an open question in the opposite direction, and give some a priori estimates for associated m component systems.

Vandana Sharma

Papers

Global Existence of Solutions to Reaction-Diffusion Systems with Mass Transport Type Boundary Conditions

Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions

Global existence of solutions to volume-surface reaction diffusion systems with dynamic boundary conditions

Uniform bounds for solutions to volume-surface reaction diffusion systems

Martin’s Problem for Volume-Surface Reaction-Diffusion Systems