Abstract
The first section in this chapter is based on the following considerations. Obstacle problems such as (4.44) and (4.48) can be formulated even when the operator L is of the nonvariational type; candidates as solutions are those functions u whose first and second derivatives are defined a.e. in Ω, so that Lu certainly makes sense. We can still avail ourselves of existence, uniqueness, and regularity results for v.i.’s if the leading coefficients of L are smooth. If not, we can approximate L by a sequence of operators to which variational tools do apply.
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Bibliographical Notes
Nonvariational obstacle problems were introduced by A. Friedman [55] and A. Bensoussan and J. L. Lions [12] as auxiliary tools in the theory of stochastic control, for the case when the dynamic system at hand is governed by a merely continuous diffusion term. Among the subsequent contributions to the subject we mention the papers by G. M. Troianiello [147–149], P. L. Lions [106], and M. G. Garroni and M. A. Vivaldi [59, 60], all dealing with linear operators, and the papers by G. M. Troianiello [150, 151] and M. G. Garroni and M. A. Vivaldi [61], where nonlinear operators are taken up; for a class of degenerate problems see I. Capuzzo Dolcetta and M. G. Garroni [34].
The presentation provided here is largely taken from the author’s articles. In particular, the notion of a generalized solution and its applications to the study of implicit unilateral problems are based on G. M. Troianiello [148].
Theorems 5.6 and 5.7 are based on their variational counterparts, respectively studied by O. Nakoulima [125] and J. L. Joly and U. Mosco [82]. Theorem 5.8 extends a result previously proven, with different techniques, by M. G. Garroni and M. A. Vivaldi [61].
The results of Section 5.2.1 are due to L. Nirenberg [129]. Lemma 5.10 is based on H. Amman [5] (see also H. Amman and M. G. Crandall [6] and K. Inkmann [79]). The proof of Lemma 5.11, due to the present author, makes a crucial use of some techniques by O. A. Ladyzhenskaya, V. Solonnikov, and N. N. Ural’tseva [96] as well as of some by J. Frehse [51].
Step 2 of the proof of Theorem 5.12 utilizes an idea in an article by K. Akô [4], which also contains the example of Section 5.3.2. Theorem 5.14 extends previous results of H. Amann and M. G. Crandall [6] and J. L. Kazdan and R. J. Kramer [85].
In a variational setting implicit unilateral problems enter the theory of quasivariational inequalities, introduced by A. Bensoussan and J. L. Lions [11]: see A. Bensoussan and J. L. Lions [13], J. L. Joly and U. Mosco [82], C. Baiocchi and A. Capelo [8] as well as, for what concerns in particular the impulse control problem, J. L. Joly, U. Mosco, and G. M. Troianiello [83], I. Capuzzo Dolcetta and M. A. Vivaldi [35], B. Hanouzet and J. L. Joly [71, 73], L. Caffarelli and A. Friedman [26], U. Mosco [120], and A. Bensoussan, J. Frehse, and U. Mosco [14].
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© 1987 Springer Science+Business Media New York
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Troianiello, G.M. (1987). Nonvariational Obstacle Problems. In: Elliptic Differential Equations and Obstacle Problems. University Series in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-3614-1_5
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DOI: https://doi.org/10.1007/978-1-4899-3614-1_5
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