Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On a unilateral problem associated with elliptic operators

Author: Peter Hess
Journal: Proc. Amer. Math. Soc. 39 (1973), 94-100
MSC: Primary 35J60
MathSciNet review: 0328336
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathcal{A}$ be a uniformly elliptic linear differential expression of second order, defined on the bounded domain $ \Omega \subset {R^m}$, and let $ \beta \subset R \times R$ be a maximal monotone graph. Under some growth assumption on $ \beta $ it is shown that for any given $ f \in {L^2}(\Omega )$ the problem: $ \mathcal{A}u + \beta (u) \mathrel\backepsilon f$ on $ \Omega ,u = 0$ on $ \partial \Omega $, admits a strong solution. It is not required that $ \mathcal{A}$ is monotone.

References [Enhancements On Off] (What's this?)

  • [1] S. Agmon, Lectures on elliptic boundary value problems, Van Nostrand Math. Studies, no 2, Van Nostrand, Princeton, N.J., 1965. MR 31 #2504. MR 0178246 (31:2504)
  • [2] H. Brézis, Nouveaux théorèmes de régularité pour les problèmes unilatéraux (to appear).
  • [3] H. Brézis, M. G. Crandall and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Comm. Pure Appl. Math. 23 (1970), 123-144. MR 0257805 (41:2454)
  • [4] F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Sympos. Pure Math., vol. 18, part 2, Amer. Math. Soc., Providence, R.I. (to appear). MR 0405188 (53:8982)
  • [5] F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Functional Analysis 11 (1972), 251-294. MR 0365242 (51:1495)
  • [6] M. G. Crandall and A. Pazy, Semi-groups of nonlinear contractions and dissipative sets, J. Functional Analysis 3 (1969), 376-418. MR 39 #4705. MR 0243383 (39:4705)
  • [7] P. Hess, On nonlinear mappings of monotone type homotopic to odd operators, J. Functional Analysis 11 (1972), 138-167. MR 0350525 (50:3017)
  • [8] -, On nonlinear mappings of monotone type with respect to two Banach spaces, J. Math. Pures Appl. 52 (1973), 13-26. MR 0636418 (58:30529)
  • [9] -, Variational inequalities for strongly nonlinear elliptic operators, J. Math. Pures Appl. (to appear).
  • [10] -, A strongly nonlinear elliptic boundary value problem, J. Math. Anal. Appl. (to appear). MR 0318670 (47:7217)
  • [11] T. Kato, Accretive operators and nonlinear evolution equations in Banach spaces Proc. Sympos. Pure Math., vol. 18, part 1, Amer. Math. Soc., Providence, R.I., 1970, pp. 138-161. MR 42 #6663. MR 0271782 (42:6663)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35J60

Retrieve articles in all journals with MSC: 35J60

Additional Information

Keywords: Uniformly elliptic linear differential operator, maximal monotone mapping, Yosida approximation, homotopy arguments, Sobolev space, strong solution
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society