Dynamic boundary value problems and evolution inequalities

Abstract

In the first two chapters, a thorough background for boundary value and variational problems, refering to the origin of certain mathematical models, can be found. Excerpts from the literature are included. There are some references to work done in subsequent chapters. The approach differs from the usual in some cases, in order to extend some principles. Such an extension can be found on pages 3, 4 and 11. Thus the connection between boundary value and variational problems for elliptic operators becomes clear. In paragraph 2.5.1 relations on the boundary in the form of inequalities are explained by the approach in [43], where the boundary is considered a medium, with its own conservation laws. In § 2.11.3 the result of the theorem on page 38, is restricted to a one-dimensional problem, giving a one-dimensional mathematical model for linear elasticity with friction, in terms of a well-known variational inequality (at the end of § 2.11.3). The aim of chapter two is to introduce the concept of variational in- equalities and to discuss some standard examples. In chapter 3 the concept of monotone operators in Banach spaces, is used to obtain existence theorems for variational inequalities. A new concept, ultimately positive operator, is introduced. An approach of Browder [10], based on a lemma of Minty [38] , is fo11 owed and extended. The final result, Theorem 3.1.10, is comparable to the known result {Theorem 2.10.3) of Browder, but the proofs differ totally. The weak formulation of problems, suggested in chapter 1 (§ 1.3.11.b), is used here. Inner products of equation 1.2 with functions L¢¢EC00(G) and L a second order elliptic operator are used, whereas 0 the usual weak formulation is obtained by using inner products with 00 functions¢ EC (G). It becomes necessary to introduce two Hilbert 0 spaces V 0 and V 1 (§ 3. 2) and some important properties are derived. Lemma 3.2.4 and § 3.2.5.a, b, show why these spaces, instead of the usual H2 (G), are used. In § 3.3 two examples are given and two regularity theorems (§ 3.4) conclude the chapter. In the first theorem the convergence of the differential quotient, ohu (§ 3.4.2.c) is studied and in the second, a sequence {u£} c V1 is constructed, to approximate the solution in the space V 0 In chapter four free boundary problems are studied in an abstract sense. Certain variational inequalities are interpreted in an abstract sense, see for example Theorem 4.7.2. This theorem is needed in the proof of Theorem 4.7.5, which interprets the examples in § 3.3.2. Theorem 4.7.2 is also applicable to problems in the spaces V0 and V1 (the examples in 4.7.7 and the results on page 123). In chapter five, two approaches in connection with evolution inequalities are followed. In the first, the existence of time derivitives, is assured in a weak sense. In the second, a much stronger result is obtained, so much so that the existence of time derivitives in 5.6.1 in a strong sense, can be proved by using the second approach (5.7.14 ). A regularity theorem for 5.6.1.c is given. The solution is approximated by a sequence {u } c V. 1 Theorem 4.7.2 is used in Theorem 5.9 in order to interpret the evolution inequalities concerned. It is shown that dynamic conditions on the boundary are needed.