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## Über dieses Buch

​In the present book the conditions are studied for the semi-boundedness of partial differential operators which is interpreted in different ways. Nowadays one knows rather much about L2-semibounded differential and pseudo-differential operators, although their complete characterization in analytic terms causes difficulties even for rather simple operators. Until recently almost nothing was known about analytic characterizations of semi-boundedness for differential operators in other Hilbert function spaces and in Banach function spaces. The goal of the present book is to partially fill this gap. Various types of semi-boundedness are considered and some relevant conditions which are either necessary and sufficient or best possible in a certain sense are given. Most of the results reported in this book are due to the authors.

## Inhaltsverzeichnis

### Chapter 1. Preliminary Facts on Semi-boundedness of Forms and Operators

Let us recall that, in the classical theory of Hilbert spaces, the class of symmetric operators A, semi-bounded below is selected by the inequality
$$(Au,u)\geqslant c\parallel u \parallel^2$$
valid for any u belonging to D(A), the domain of A $$A(c\in\mathbb{R})$$.

### Chapter 2. $$L^p$$ -dissipativity of Scalar Second-order Operators with Complex Coefficients

In this chapter we start dealing with concrete problems. Let us consider a scalar second-order operator with complex coefficients
$$Au\;=\;\mathrm{div}(\mathcal{A}\nabla u)\;+\;\boldsymbol{b}\nabla u\;+\;\mathrm{div}(\boldsymbol{c}u)\;+\;au.$$
.

### Chapter 3. Elasticity System

Let us consider the classical operator of linear elasticity
$$Eu\;=\;\Delta {u}\;+\;(1\;-\;2\nu)^{-1}\nabla \mathrm{div}\;u$$
where ν is the Poisson ratio.

### Chapter 4. $$L^p$$ -dissipativity for Systems of Partial Differential Operators

This chapter is devoted to systems of partial differential operators. After some auxiliary results in Section 4.1, we give an algebraic necessary condition for the $$L^p$$-dissipativity of a general system in the two-dimensional case (Section 4.2). Several results are stated in terms of eigenvalues of the coefficient matrix of the system.

### Chapter 5. The Angle of $$L^p$$ -dissipativity

The analyticity of a contractive semigroup $$\left\{T(t)\right\}$$ is closely connected with the possibility of extending $$\left\{T(t)\right\}$$ to a contractive semigroup $$\left\{T(z)\right\}(z\;\in\;\mathbb{C})$$ in an angle, called angle of dissipativity.

### Chapter 6. Higher-order Differential Operators in $$L^p$$

In previous sections we have found conditions for the $$L^p$$-dissipativity of secondorder scalar equations and systems. One can ask whether these results hold for higher-order operators. In this chapter it is proved that the answer is negative.

### Chapter 7. Weighted Positivity and Other Related Results

Most of the results in the present chapter concern the $$L^2$$-weighted positivity of different operators. In the case of functions taking scalar values, by this positivity we mean the inequality
$$Re\int_\Omega\langle Lu,u\rangle\Psi dx\geqslant 0,\quad \forall u \in C^\infty_0(\Omega)$$
where
$$\Psi$$
is a weight.

### Backmatter

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