# Analysis of Heat Equations on Domains. (LMS-31)

El Maati Ouhabaz
Pages: 296
https://www.jstor.org/stable/j.ctt7s45z

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. Preface
(pp. ix-xii)
4. Notation
(pp. xiii-xiv)
5. Chapter One SESQUILINEAR FORMS, ASSOCIATED OPERATORS, AND SEMIGROUPS
(pp. 1-42)

LetHbe a Hilbert space over$\mathbb{K} = \mathbb{C}$or$\mathbb{R}$. We denote by (.;.) the inner product ofHand by$\left\| . \right\|$the corresponding norm. Letabe a sesquilinear form onH, i.e.,ais an application from$H \times H$into$\mathbb{K}$such that for every$\alpha \in \mathbb{K}$and$u,v,h \in H$:

$a\left( {\alpha u + v,h} \right) = \alpha a\left( {u,h} \right) + a\left( {v,h} \right)$and$a\left( {u,\alpha v + h} \right) = \bar \alpha a\left( {u,v} \right) + a\left( {u,h} \right)$. (1.1)

Here$\bar \alpha$denotes the conjugate number ofα. Of course,$\bar \alpha = \alpha$if$\mathbb{K} = \mathbb{R}$and in this case the formais then bilinear. For simplicity, we will not distinguish the two cases$\mathbb{K} = \mathbb{R}$and$\mathbb{K} = \mathbb{C}$. We We will use the sesquilinear term in both cases and also write...

6. Chapter Two CONTRACTIVITY PROPERTIES
(pp. 43-78)

LetHbe a Hilbert space over$\Bbbk = \mathbb{R}$or$\mathbb{C}$. Denote byAa sesquilinear form onH. We assume thatais densely defined, accretive, continuous, and closed (see (1.2)―(1.5)). Denote byAits associated operator. We have seen in the previous chapter that ―Agenerates a strongly continuous semigroup${\left( {{e^{ - tA}}} \right)_{t \ge 0}}$onH. Assume now that$H = {L^2}\left( {X,\mu ,\mathbb{C}} \right)$,where$\left( {X,\mu } \right)$is a σ-finite measure space. Several properties of the semigroup like positivity,${L^p}$-contractivity, domination, and so on can be characterized in terms of the operatorA. However, in most applications, one does not precisely know the operatorA. Typical situations where...

7. Chapter Three INEQUALITIES FOR SUB-MARKOVIAN SEMIGROUPS
(pp. 79-98)

In this chapter we prove certain inequalities for sub-Markovian semigroups and their generators. A relationship between the sub-Markovian property and Kato type inequalities for the generator is established. This completes some of the results of Section 2.2. Note that the operators in consideration here are not necessarily associated with sesquilinear forms.

Starting with a symmetric sub-Markovian semigroup on some${L^2}$-space, we have seen in the previous chapter that it induces strongly continuous semigroups on the scale of${L^p}$-spaces. We prove some inequalities for the corresponding generator on${L^p}$. As a consequence, we obtain the holomorphy of such semigroups on${L^p},1 < p < \infty$,...

8. Chapter Four UNIFORMLY ELLIPTIC OPERATORS ON DOMAINS
(pp. 99-142)

This chapter is devoted to second-order uniformly elliptic operators considered on domains of${\mathbb{R}^d}$and subject to various boundary conditions. We apply the criteria of positivity,${L^\infty }$-contractivity, and domination of semigroups, stated in an abstract setting in the previous chapters, to concrete situations of differential operators. The operators under consideration here may have real- or complex-valued measurable coefficients. The aim is to describe precisely how the above properties of the semigroup depend on the boundary conditions and on the coefficients of the operator.

Let Ω be an open subset of${\mathbb{R}^d}\left( {d \geqslant 1} \right)$, endowed with the Lebesgue measure$dx$. We write L...

9. Chapter Five DEGENERATE-ELLIPTIC OPERATORS
(pp. 143-154)

We have studied in the last chapter contractivity properties of semigroups associated with second-order uniformly elliptic operators. In particular, we have seen that it is possible in several cases to extend the semigroup initially defined on${L^2}\left( {\Omega ,\mathbb{C}} \right)$to${L^p}\left( {\Omega ,\mathbb{C}} \right)$for$p\not = 2$. In the present chapter, we study similar questions for second-order degenerate-elliptic operators. More precisely, we consider operators of the type

$Au = - \sum\limits_{k,j = 1}^d {{D_j}\left( {{a_{kj}}{D_k}u} \right)} + \sum\limits_{k = 1}^d {{b_k}{D_k}u + {a_0}u}$, (5.1)

where the matrix${\left( {{a_{kj}}} \right)_{k,j}}$satisfies the following weaker assumption than (4.2):

$\sum\limits_{j,k = 1}^d {{a_{kj}}\left( x \right){\xi _j}{\xi _k} \ge 0}$for all$\xi \in {\mathbb{R}^d}$, a.e.$x \in \Omega$. (5.2)

In this case, any realization of the operatorAis called a degenerate-elliptic operator. For such operators, several difficulties...

10. Chapter Six GAUSSIAN UPPER BOUNDS FOR HEAT KERNELS
(pp. 155-192)

Let$\left( {X,\mu } \right)$be a σ-finite measure space and let${L^p}: = {L^p}\left( {X,\mu } \right)$be the corresponding complex or real Lebesgue spaces. In this section we study${L^2} - {L^\infty }$estimates of the type:

$\left\| {{e^{ - tA}}} \right\|L({L^2},{L^\infty }) \le c{t^{d/4}}$for all$t > 0$. (6.1)

Since the operators under consideration in this book are defined by sesquilinear forms, we shall focus on characterizations of (6.1) in terms of forms. We mainly concentrate on the case of symmetric forms and assume that the associated semigroup is${L^\infty }$-contractive. The latter property can be removed in the statements. It will be assumed only for simplicity.

We start with the following extrapolationresult. LEMMA 6.1 Let${{\rm{(T(t))}}_{t > 0}}$...

11. Chapter Seven GAUSSIAN UPPER BOUNDS AND ${L^p}$-SPECTRAL THEORY
(pp. 193-252)

The present chapter is devoted to applications of Gaussian upper bounds to${L^p}$-spectral theory. Although this monograph is mainly concerned with second-order differential operators on domains of the Euclidean space, this chapter is written in a general setting of operators on metric spaces. The framework includes Laplacians on Riemannian manifolds or arbitrary domains of manifolds as well as some higher-order differential operators on domains of Euclidean space.

Let$\left( {X,\rho ,\mu } \right)$be a metric σ-finite measured space. Denote by$B\left( {x,r} \right)$the open ball inXfor the distanceσ, of center$x \in X$and radius$r > 0$, that is,

$B(x,r): = \{ y \in X,p(x,y) < r\}$,

and by${\rm{V(x,r)}}$its...

12. Chapter Eight A REVIEW OF THE KATO SQUARE ROOT PROBLEM
(pp. 253-264)

T. Kato asked in the early 1960’s the question whether for a divergence form uniformly elliptic operatorAthe domain of the square root${A^{1/2}}$coincides with the domain of the sesquilinear form ofA. The question was solved in few particular cases but the general case was open for several decades, and was known as the Kato square root problem. The problem was solved in the case of dimension one by Coifman, McIntosh, and Meyer [CMM82]. The case of arbitrary dimension has been solved only recently by Hofmann, Lacey, and McIntosh [HLM02] for second-order divergence form operators whose heat...

13. Bibliography
(pp. 265-282)
14. Index
(pp. 283-284)