On a unilateral problem associated with elliptic operators

Hess, Peter

doi:10.1090/S0002-9939-1973-0328336-7

On a unilateral problem associated with elliptic operators
HTML articles powered by AMS MathViewer

by Peter Hess PDF

Proc. Amer. Math. Soc. 39 (1973), 94-100 Request permission

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: $\mathcal {A}u + \beta (u) \backepsilon f$ on $\Omega ,u = 0$ on $\partial \Omega$, admits a strong solution. It is not required that $\mathcal {A}$ is monotone.

References

Shmuel Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR 0178246

Nouveaux théorèmes de régularité pour les problèmes unilatéraux

H. Brezis, M. G. Crandall, and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Comm. Pure Appl. Math. 23 (1970), 123–144. MR 257805, DOI 10.1002/cpa.3160230107
Felix E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear functional analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1976, pp. 1–308. MR 0405188
Felix E. Browder and Peter Hess, Nonlinear mappings of monotone type in Banach spaces, J. Functional Analysis 11 (1972), 251–294. MR 0365242, DOI 10.1016/0022-1236(72)90070-5
Michael G. Crandall and Amnon Pazy, Semi-groups of nonlinear contractions and dissipative sets, J. Functional Analysis 3 (1969), 376–418. MR 0243383, DOI 10.1016/0022-1236(69)90032-9
Peter Hess, On nonlinear mappings of monotone type homotopic to odd operators, J. Functional Analysis 11 (1972), 138–167. MR 0350525, DOI 10.1016/0022-1236(72)90084-5
Peter Hess, On nonlinear mappings of monotone type with respect to two Banach spaces, J. Math. Pures Appl. (9) 52 (1973), 13–26. MR 636418

Variational inequalities for strongly nonlinear elliptic operators

Peter Hess, A strongly nonlinear elliptic boundary value problem, J. Math. Anal. Appl. 43 (1973), 241–249. MR 318670, DOI 10.1016/0022-247X(73)90272-2
Tosio Kato, Accretive operators and nonlinear evolution equations in Banach spaces. , Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 1, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 138–161. MR 0271782