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On nonlinear mixed boundary value problems for second order elliptic differential equations on domains with corners

Published online by Cambridge University Press:  14 November 2011

Bernhard Kawohl
Bernhard Kawohl
Affiliation:
Fachbereich Mathematik, Technische Hochschule Darmstadt, Schlossgartenstr. 7, D-6100 Darmstadt, B.R.D.
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Synopsis

We investigate the existence, uniqueness and regularity of solutions to the linear differential equation Lu = f under nonlinear mixed boundary conditions on domains with singular boundary points.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 87 , Issue 1-2 , 1980 , pp. 35 - 51
Copyright
Copyright © Royal Society of Edinburgh 1980

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References

1Attouch, H.. Convergence des fonctions convexes, des sous differentiels et semi-groupes associés. C.R. Acad. Sci. Paris 284A (1977), 539–542.Google Scholar
2Avantaggiatti, A. and Troisi, M.. Spazi di Sobolev con peso e problemi ellitici in un angolo. I. Ann. Mat. Pura Appl. 95 (1973), 361–408; II. Ibid. 97 (1973), 207–252; III. Ibid. 99 (1974), 1–51.CrossRefGoogle Scholar
3Babuška, I. and Rosenzweig, M.. A finite element scheme for domains with corners. Numer. Math. 20 (1972), 1–21.CrossRefGoogle Scholar
4Bailet-Intissar, J.. Problèmes aux limites non lineares avec conditions mêlées. C.R. Acad. Sci. Paris 287A (1978), 1061–1063.Google Scholar
5Benci, V. and Fortunato, D.. Some compact embedding theorems for weighted Sobolev spaces. Boll. Un. Mat. Ital. (5) 13-B (1976), 832–843.Google Scholar
6Brezis, H.. Operateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert (Amsterdam, London: North Holland, 1973).Google Scholar
7Brezis, H.. Problèmes unilatéraux. J. Math. Pures Appl. 51 (1972), 1–168.Google Scholar
8Brezis, H.. Proprietés regularisantes de certains semi-groupes non linéaires. Israel J. Math. 9 (1971), 513–534.CrossRefGoogle Scholar
9Carleman, T.. Über eine nichtlineare Randwertaufgabe bei der Gleichung Δu = 0. Math. Z. 9 (1921), 35–43.Google Scholar
10Carslaw, H. S. and Jaeger, J. C.. Conduction of heat in solids (London: Oxford University Press, 1946).Google Scholar
11Colli-Franzone, P.. Approssimatione mediante il metodo di penalizzazione, di problemi misti di Dirichlet-Neumann per operatori lineari ellitici del secondo ordine. Boll. Un. Mat. Ital. (4) 7 (1973), 229–250.Google Scholar
12Duvaut, G. and Lions, J. L.. Inequalities in Mechanics and Physics (Berlin, Heidelberg, New York: Springer, 1976).Google Scholar
13Fichera, G.. Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. 7 (1964), p. 91–140.Google Scholar
14Frehse, J.. Two dimensional variational problems with thin obstacles, Math. Z. 143 (1975), 279–288.CrossRefGoogle Scholar
15Friedman, J.. Generalized heat transfer between solids and gases under nonlinear boundary conditions. I. Math, and Mech. 8 (1959), 161–183.Google Scholar
16Göpfert, A.. Mathematische Optimierung in allgemeinen Vektorräumen (Leipzig: Teubner, 1973).Google Scholar
17Grisvard, P.. Smoothness of the solution of a monotonie boundary value problem for a second elliptic equation in a general convex domain. Lecture Notes in Mathematics 564, 135–151 (Berlin: Springer, 1977).Google Scholar
18Hess, P.. On some nonlinear elliptic boundary value problems. Lecture Notes in Mathematics 399, 235–241 (Berlin: Springer, 1974).Google Scholar
19Hildebrandt, S. and Nitsche, J. C. C.. On minimal surfaces with free boundaries. Preprint 158 (1977) of Sonderforschungsbereich 72, Bonn.Google Scholar
20Kato, T.. Accretive operators and nonlinear evolution equations in Banach spaces. Nonlinear Functional Analysis (Proc. Symp. Pure Math.) vol. 18, Part 1, pp. 138–161 (Providence, R.I.: Amer. Math. Soc.).Google Scholar
21Kawohl, B.. Coerciveness for second-order elliptic differential equations with unilateral constraints. Nonlinear Anal. 2 (1978), 189–196.CrossRefGoogle Scholar
22Kawohl, B.. Über nichtlineare gemischte Randwertprobleme für elliptische Differentialgleichungen zweiter Ordnung auf Gebieten mit Ecken (TH Darmstadt Doctoral Thesis, 1978).Google Scholar
23Kawohl, B.. On a mixed Signorini problem. Numer. Funct. Anal. Optim. 1 (1979), 633–645.Google Scholar
24Keller, H. B.. Elliptic boundary value problems suggested by nonlinear diffusion processes. Arch. Rational Mech. Anal. 35 (1969), 363–381.Google Scholar
25Kenmochi, N.. Pseudomonotone operators and nonlinear elliptic boundary value problems. J. Math. Soc. Japan 27 (1975), 121–149.Google Scholar
26Kenmochi, N.. Initial boundary value problems for nonlinear parabolic partial differential equations. Boll Un. Mat. Ital. (5) 13-B (1976), 1–18.Google Scholar
27Kondratiev, V. A.. Boundary problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16 (1967), 227–313 or Trudy Moskov. Mat. Obšč. 16 (1967), 209–292.Google Scholar
28Kufner, A.. Lösungen des Dirichletschen Problems für elliptische Differentialgleichungen in Räumen mit Belegungsfunktion. Czechoslovak Math. J. 15 (1965), 621–632.Google Scholar
29Kufner, A., John, O. and Fučik, S.. Function Spaces (Leyden: Noordhoff, 1978).Google Scholar
30Lewy, H.. On a variational problem with inequalities on the boundary. J. Math, and Mech. 17 (1968), 861–884.Google Scholar
31Lions, J. L.. Quelques méthodes de résolution des problèmes aux limites non linéaires (Paris: Dunod Gauthier-Villars, 1969).Google Scholar
32Lions, J. L. and Magenes, E.. Nonhomogeneous boundary value problems and applications I (Berlin, Heidelberg, New York: Springer, 1972).Google Scholar
33Lions, J. L. and Stampacchia, G.. Variational inequalities. Comm. Pure Appl. Math. 20 (1967), 492–519.CrossRefGoogle Scholar
34Mann, W. R. and Wolf, F.. Heat transfer between solids and gases under nonlinear boundary conditions. Quart. Appl. Math. 9 (1952/1953), 163–184.CrossRefGoogle Scholar
35Minty, G. J.. Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29 (1962), 341–346.Google Scholar
36Nashed, M. Z.. Differentiability and related properties on nonlinear operators. In Nonlinear Functional Analysis and Applications, ed. Rall, L. B. (New York, London: Academic Press, 1971).Google Scholar
37Nečas, J.. Les méthodes directes en théorie des équations elliptiques (Paris: Masson, 1967).Google Scholar
38Nitsche, J. C. C.. Variational problems with inequalities as boundary conditions or how to fashion a cheap hat for Giacometti's brother. Arch. Rational Mech. Anal. 35 (1969), 83–113.CrossRefGoogle Scholar
39Panagiotopoulos, P.. Ungleichungsprobleme in der Mechanik (RWTH Aachen Habilitation, 1977).Google Scholar
40Riddell, R. C.. Eigenvalue problems for nonlinear elliptic variational inequalities. Nonlinear Anal. 3 (1979), 1–33.Google Scholar
41Tartar, L.. Variational methods and monotonicity. MRC Technical Report 1571, Madison, Wisconsin, Oct. 1975.Google Scholar
42Veite, W.. Komplementäre Extremalprobleme. In Methoden und Verfahren der mathematischen Physik 15; ed. Brosowski, B., Martensen, E. (Mannheim: Bibliographisches Institut, 1976).Google Scholar
43Weisel, J.. Lösung singulärer Variationsprobleme durch die Verfahren von Ritz und Galerkin mit finiten Elementen — Anwendungen in der konformen Abbildung. Mitt. Math. Sem. Gieβen 138 (1979).Google Scholar
44Wendland, W. L., Stephan, E. and Hsiao, G. C.. On the integral equation method for the plane mixed boundary value problem of the Laplacian. Math. Meth. Appl. Sci. 1 (1979), 265–321.CrossRefGoogle Scholar