Abstract
Strongly elliptic differential operators with (possibly) unbounded lower order coefficients are shown to generate analytic semigroups of linear operators onL p(R n), 1≦p≦∞. An explicit characterization of the domain is given for 1<p<∞. An application to parabolic problems is also included.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Agmon,On the eigenfunctions and the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math.15 (1962), 119–147.
H. Amann,Dual semigroup and second order linear elliptic boundary value problems, Isr. J. Math.45, (1983), 225–254.
P. Cannarsa, B. Terreni and V. Vespri,Analytic semigroups generated by non-variational elliptic systems of second order under Dirichlet boundary conditions, J. Math. Anal. Appl.112 (1985), 56–103.
P. Cannarsa and V. Vespri,On maximal L p regularity for the abstract Cauchy problem, Boll. Un. Mat. Ital.5-B (1986), 165–175.
P. Cannarsa and V. Vespri,Generation of analytic semigroups by elliptic operators with unbounded coefficients, SIAM J. Math. Anal.18 (1987), 857–872.
P. Cannarsa and V. Vespri,Existence and uniqueness of solutions to a class of stochastic partial differential equations, Stochastic Anal. Appl.3 (1985), 315–339.
P. Cannarsa and V. Vespri,Existence and uniqueness results for a non-linear stochastic partial differential equation, inStochastic Partial Differential Equations and Applications, Trento, 1985, Lecture Notes in Mathematics,1236, pp. 1–24.
E. B. Davies,Some norm bounds and quadratic form inequalities for Schroedinger operators II, J. Oper. Theory12 (1984), 177–196.
E. B. Davies and B. Simon,L 1 properties of intrinsic Schroedinger semigroups, J. Funct. Anal.65 (1966), 126–146.
S. D. Eidel’mann,Parabolic Systems, Nauka, Moscow, 1964; English translation: North-Holland, Amsterdam, 1969.
R. S. Freeman and M. Schechter,On the existence, uniqueness and regularity of solutions to general elliptic boundary value problems, J. Differential Equations,15 (1974), 213–246.
Y. Higouchi,A priori estimates and existence theorems on elliptic boundary value problems for unbounded domains, Osaka J. Math.5 (1968), 103–135.
O. A. Ladyzhenskaja, N. N. Ural’ceva and V. A. Solonnikov,Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs 23, AMS, Providence, 1968.
J. L. Lions and E. Magenes,Problèmes aux limites non homogènes et application II, Dunod, Paris, 1968.
N. Okazawa,An L p theory for Schroedinger operators with nonnegative potentials, J. Math. Soc. Japan36 (1984), 675–688.
A. Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
J. Voigt,Absorption semigroups, their generators and Schroedinger operators, J. Funct. Anal.67 (1986), 167–205.
W. von Wahl,The equation u′+A(t)u=f in a Hilbert space and L p estimates for parabolic equations, J. London Math. Soc.25 (1982), 483–497.
Author information
Authors and Affiliations
Additional information
This work has been partially supported by the Research Funds of the Ministero della Pubblica Istruzione.
The authors are members of GNAFA (Consiglio Nazionale delle Ricerche).
Rights and permissions
About this article
Cite this article
Cannarsa, P., Vespri, V. Generation of analytic semigroups in theL p topology by elliptic operators inR n . Israel J. Math. 61, 235–255 (1988). https://doi.org/10.1007/BF02772570
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02772570