Abstract
It is shown that general second order elliptic boundary value problems on bounded domains generate analytic semigroups onL 1. The proof is based on Phillips’ theory of dual semigroups. Several sharp estimates for the corresponding semigroups inL p, 1≦p<∞, are given.
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References
R. A. Adams,Sobolev Spaces, Academic Press, New York, 1975.
H. Amann,Nonlinear elliptic equations with nonlinear boundary conditions, inNew Developments in Differential Equations (W. Eckhaus, ed.), North-Holland Math. Studies 21, 1976, pp. 43–63.
Ph. Bénilan,Opérateurs accrétifs et semi-groupes dans les espaces L p (1≦p≦∞), inFunctional Analysis and Numerical Analysis, Japan-France Seminar, Tokyo and Kyoto, 1976 (H. Fujita, ed.), Japan Society for the Promotion of Science, 1978, pp. 15–53.
J. Bergh and J. Löfström,Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976.
J.-M. Bony,Principe du maximum dans les espaces de Sobolev, C.R. Acad. Sci. Paris, Sér. A,265 (1967), 333–336.
H. Brézis and W. A. Strauss,Semi-linear second-order elliptic equations in L 1, J. Math. Soc. Jpn.25 (1973), 565–590.
P. L. Butzer and H. Berens,Semi-Groups of Operators and Approximation, Springer-Verlag, Berlin, 1967.
E. B. Davies,One-Parameter Semigroups, Academic Press, London, 1980.
N. Dunford and J. T. Schwartz,Linear Operators. Part I, Interscience, New York, 1957.
H. O. Fattorini,On the angle of dissipativity of ordinary and partial differential operators, preprint, 1982.
A. Friedman,Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
G. Greiner,Zur Perron-Frobenius-Theorie stark stetiger Halbgruppen, Math. Z.177 (1981), 401–423.
G. Greiner, J. Voigt and M. Wolff,On the spectral bound of the generator of semigroups of positive operators, J. Operator Theory5 (1981), 245–256.
E. Hille and R. S. Phillips,Functional Analysis and Semi-Groups, Amer. Math. Soc., Providence, R. I., 1957.
T. Kato,Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.
M. A. Krasnoselskii, P. P. Zabreiko, E. I. Pustylnik and P. E. Sobolevskii,Integral Operators in Spaces of Summable Functions, Noordhoff, Leyden, 1976.
J.-L. Lions,Théorèmes de trace et d’interpolation (IV), Math. Ann.151 (1963), 42–56.
J.-L. Lions et E. Magenes,Problemi ai limiti non omogenei (III), Ann. Scuola Norm. Sup. PisaXV (1961), 41–103.
J.-L. Lions et E. Magenes,Non-Homogeneous Boundary Value Problems and Applications I, Springer-Verlag, Berlin, 1972.
C. Miranda,Partial Differential Equations of Elliptic Type, Springer-Verlag, New York, 1970.
J. Nečas,Les méthodes directes en théorie des équations elliptiques, Academia, Prague, 1967.
A. Pazy,On the differentiability and compactness of semi-groups of linear operators, J. Math. Mech.17 (1968), 1131–1141.
A. Pazy,Semi-Groups of Linear Operators and Applications to Partial Differential Equations, Univ. of Maryland, Lecture Notes #10, 1974.
R. S. Phillips,The adjoint semi-group, Pac. J. Math.5 (1955), 269–283.
M. H. Protter and H. F. Weinberger,Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, N.J., 1967.
F. Rothe,Uniform bounds from bounded L p -functionals in reaction-diffusion equations, J. Differ. Equ.45 (1982), 207–233.
H. H. Schaefer,Topological Vector Spaces, Springer-Verlag, New York, 1971.
E. Sinestrari,Accretive differential operators, Boll. Unione Mat. Ital. (5)13-A (1976), 19–31.
E. M. Stein,Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970.
H. B. Stewart,Generation of analytic semigroups by strongly elliptic operators, Trans. Am. Math. Soc.199 (1974), 141–162.
H. B. Stewart,Generation of analytic demigroups by strongly elliptic operators under general boundary conditions, Trans. Am. Math. Soc.259 (1980), 299–310.
H. Tanabe,On semilinear equations of elliptic and parabolic type, inFunctional Analysis and Numerical Analysis, Japan-France Seminar, Tokyo and Kyoto, 1976 (H. Fujita, ed.), Japan Society for the Promotion of Science, 1978, pp. 455–463.
H. Tanabe,Equations of Evolution, Pitman, London, 1979.
H. Triebel,Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.
F. B. Weissler,Semilinear evolution equations in Banach spaces, J. Funct. Anal.32 (1979), 277–296.
F. B. Weissler,Local existence and nonexistence for semilinear parabolic equations in L p, Indiana Univ. Math. J.29 (1980), 79–102.
K. Yosida,Functional Analysis, Springer-Verlag, Berlin, 1965.
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Amann, H. Dual semigroups and second order linear elliptic boundary value problems. Israel J. Math. 45, 225–254 (1983). https://doi.org/10.1007/BF02774019
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DOI: https://doi.org/10.1007/BF02774019