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Maximum principles are bedrock results in the theory of second order elliptic equations. This principle, simple enough in essence, lends itself to a quite remarkable number of subtle uses when combined appropriately with other notions. Intended for a wide audience, the book provides a clear and comprehensive explanation of the various maximum principles available in elliptic theory, from their beginning for linear equations to recent work on nonlinear and singular equations.



Chapter 1. Introduction and Preliminaries

The maximum principles of Eberhard Hopf are classical and bedrock results of the theory of second order elliptic partial differential equations. They go back to the maximum principle for harmonic functions, already known to Gauss in 1839 on the basis of the mean value theorem. On the other hand, they carry forward to the maximum principles of Gilbarg, Trudinger and Serrin, and the maximum principles for singular quasilinear elliptic differential inequalities, a theory initiated particularly by Vázquez and Diaz in the 1980s, but with earlier intimations in the work of Benilan, Brezis and Crandall. The purpose of the present work is to provide a clear explanation of the various maximum principles available for second-order elliptic equations, from their beginnings in linear theory to recent work on nonlinear equations, operators and inequalities. While simple in essence, these results lend themselves to a quite remarkable number of subtle uses when combined appropriately with other notions.

Chapter 2. Tangency and Comparison Theorems for Elliptic Inequalities

We begin with the classical maximum principle due to E. Hopf [46], together with an extended commentary and discussion of Hopf’s original paper by J. Serrin [97].

Chapter 3. Maximum Principles for Divergence Structure Elliptic Differential Inequalities

For a large number of divergence structure equations, including equations which involve the important p-Laplacian operator Δp, there is a further series of maximum principles. In particular, in this chapter we study the differential inequality
$$ divA\left( {x,u,Du} \right) + B\left( {x,u,Du} \right) \geqslant 0in\Omega , $$
where Ω is a bounded domain in ℝn (unless otherwise stated explicitly), and
$$ A\left( {x,z,\xi } \right):\Omega \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n ,B\left( {x,z,\xi } \right):\Omega \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}. $$

Chapter 4. Boundary Value Problems for Nonlinear Ordinary Differential Equations

Here we begin the study of the strong maximum principle and the compact support principle for divergence structure inequalities, especially of the canonical form
$$ div\left\{ {A\left( {\left| {Du} \right|} \right)\left. {Du} \right)} \right\} - f\left( u \right) \leqslant 0,u \geqslant 0. $$
In general, the results described cannot be obtained from the nonlinear theorems of the previous chapters, since equation (4.1.1) has specialized properties which are crucially used.

Chapter 5. The Strong Maximum Principle and the Compact Support Principle

With the work of the preceding Chapter 4 available, we can turn to the proofs of the Strong Maximum Principle, Theorem 1.1.1, and the Compact Support Principle, Theorem 1.1.2, stated in the Introduction.

Chapter 6. Non-homogeneous Divergence Structure Inequalities

We consider the quasilinear differential inequality
$$ divA\left( {x,u,Du} \right) + B\left( {x,u,Du} \right) \geqslant 0in\Omega , $$
where Ω is a bounded domain in ℝn, and A and B satisfy the generic assumptions of Section 3.1. Here we shall extend the validity of Theorems 3.2.1 and 3.2.2 to the case when (6.1.1) is inhomogeneous, that is, there are constants a2, b1, b2, a, b ≥ 0 such that for all (x, z, ξ) Ω × ℝ+ × ℝn there holds, for p > 1,
$$ \begin{gathered} \left\langle {A\left( {x,z,\xi } \right),\xi } \right\rangle \geqslant \left| \xi \right|^p - a_2 z^p , \hfill \\ B\left( {x,z,\xi } \right) \leqslant b_1 \left| \xi \right|^{p - 1} + b_2 z^{p - 1} + b^{p - 1} , \hfill \\ \end{gathered} $$
while for p = 1,
$$ \left\langle {A\left( {x,z,\xi } \right),\xi } \right\rangle \geqslant \left| \xi \right| - a_2 z - a,B\left( {x,z,\xi } \right) \leqslant b $$
(in (6.1.3) we write b for b2 and discard the terms b1|ξ|p−1, bp−1). As in Section 3.1 the domain Ω is assumed to be bounded. This condition can be removed if Ω has finite measure and the boundary condition for |x| → ∞ is taken in the form (3.2.12).

Chapter 7. The Harnack Inequality

The ideas of Section 6.2 have far-reaching extensions to questions of local boundedness of solutions of the inequality (6.1.1) and to both weak and strong Harnack-type theorems.1 These results have already seen application in Section 2.5, but are crucial as well for regularity and existence theory for quasilinear elliptic equations.

Chapter 8. Applications

A Cauchy-Liouville type theorem is a statement that under appropriate circumstances an entire solution (a solution defined over ℝn) of an elliptic equation must be constant.1 For the Laplace equation in particular, it is enough that a solution u should be bounded, or even, at a minimum, that u(x) = o(|x|) as |x| → ∞. For quasilinear equations, and even for semilinear equations of the form Δu + B(u, Du) = 0, x ∈ ℝn, (8.1.1) the same question is more delicate than might at first be expected, since a number of different kinds of behavior can be seen even for relatively simple examples.


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