Kazuaki Taira

A Priori Estimates

This Chapter 6 and the next Chapter 7 are devoted to the proof of Theorem 1.2. In this chapter we study the operator Ap, and prove a priori estimates for the operator Ap − λI (Theorem 6.3) which will play a fundamental role in the next chapter. In the proof we make good use of Agmon’s method (Proposition 6.4). This is a technique of treating a spectral parameter λ as a second-order, elliptic differential operator of an extra variable and relating the old problem to a new problem with the additional variable

BOUNDARY VALUE PROBLEMS FOR ELLIPTIC PSEUDO-DIFFERENTIAL OPERATORS

The purpose of this paper is to study boundary value problems for elliptic pseudo-differential operators which originate from the problem of ex- istence of Markov processes in probability theory. We prove existence and uniqueness theorems for these boundary value problems in the framework of Sobolev spaces. Our approach permits us to interpret sufficient conditions for the unique solvability of boundary value problems in terms of Markovian mo- tion.

Logistic Dirichlet problems with discontinuous coefficients

Abstract This paper is devoted to the study of the existence of positive solutions of semilinear Dirichlet eigenvalue problems for diffusive logistic equations with discontinuous coefficients which model population dynamics in environments with spatial heterogeneity. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of singular integral operators. Moreover, we make use of an Lp variant of an estimate for the Green operator of the Dirichlet problem introduced in the study of Feller semigroups.

A mathematical analysis of thermal explosions

This paper is devoted to the study of semilinear degenerate elliptic boundary value problems arising in combustion theory which obey the simple Arrhenius rate law and a general Newton law of heat exchange. We prove that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless rate of heat production. 2000 Mathematics Subject Classification. 35J65, 80A25. 1. Introduction and main results. In a reacting material undergoing an exothermic reaction in which reactant consumption is neglected, heat is being produced in accor- dance with Arrhenius rate law and Newtonian cooling. Thermal explosions occur when the reactions produce heat too rapidly for a stable balance between heat production and heat loss to be preserved. In this paper, we are concerned with the localization of the values of a dimensionless heat evolution rate at which such critical phenomena as ignition and extinction occur. For detailed studies of thermal explosions, the reader might be referred to Aris (3, 4), Bebernes-Eberly (5), Boddington-Gray-Wake (6), and Warnatz-Maas-Dibble (22). Let D be a bounded domain of Euclidean space R N , N ≥ 2, with smooth bound- ary ∂D; its closure D = D ∪ ∂D is an N-dimensional, compact smooth manifold with boundary. We let

Introduction to diffusive logistic equations in population dynamics

The purpose of this paper is to provide a careful and accessible exposition of diffusive logistic equations with indefinite weights which model population dynamics in environments with strong spatial heterogeneity. We prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment. Moreover we prove that a population will grow exponentially until limited by lack of available resources if the diffusion rate is below some critical value; this idea is generally credited to Thomas Malthus. On the other hand, if the diffusion rate is above this critical value, then the model obeys the logistic equation introduced by P. F. Verhulst.

Elements of Probability Theory

This chapter is intended as a brief introduction to probability theory. Especially, we introduce the general theory of conditional probabilities and conditional expectations which plays a vital role in the study of Markov processes in Chap. 9.

Theory of Semigroups

This chapter is devoted to the general theory of semigroups. These topics form the necessary background for the proof of Theorems 1.2 and 1.3. In Sects. 4.1–4.3 we study Banach space valued functions, operator valued functions and exponential functions, generalizing the numerical case. Section 4.4 is devoted to the theory of contraction semigroups. This question is answered by the Hille–Yosida theorem.

Elements of Functional Analysis

This chapter is devoted to a review of standard topics from functional analysis such as quasinormed and normed linear spaces and closed, compact and Fredholm linear operators on Banach spaces. These topics form a necessary background for what follows. In Sects. 3.1–3.3 we study linear operators and functionals, quasinormed and normed linear spaces. In a normed linear space we consider continuous linear functionals as generalized coordinates of the space. In Sect. 3.4 we prove the Riesz–Markov representation theorem which describes an intimate relationship between Radon measures and non-negative linear functionals on the spaces of continuous functions. This fact constitutes an essential link between measure theory and functional analysis, providing a powerful tool for constructing Markov transition functions in Chap. 9.

Theory of Distributions

This chapter is a summary of the basic definitions and results from the theory of distributions or generalized functions which will be used in subsequent chapters. Distribution theory has become a convenient tool in the study of partial differential equations. Many problems in partial differential equations can be formulated in terms of abstract operators acting between suitable spaces of distributions, and these operators are then analyzed by the methods of functional analysis. The virtue of this approach is that a given problem is stripped of extraneous data, so that the analytic core of the problem is revealed.

Introduction and Main Results

In this monograph we solve the problem of the existence of Feller semigroups associated with strong Markov processes. More precisely, we prove the unique solvability of boundary value problems for Waldenfels integro-differential operators with general Ventcel’ (Wentzell) boundary conditions, and construct Feller semigroups corresponding to the diffusion phenomenon where a Markovian particle moves chaotically in the state space, incessantly changing its direction of motion until it “dies” at the time when it reaches the set where the particle is definitely absorbed. This monograph provides a careful and accessible exposition of the functional analytic approach to the problem of constructing strong Markov processes with Ventcel’ boundary conditions in probability. Our approach here is distinguished by the extensive use of ideas and techniques characteristic of recent developments in the theory of partial differential equations.