zbMATH — the first resource for mathematics

Generation of analytic semigroups in the \(L^ p\) topology by elliptic operators in \({\mathbb{R}}^ n\). (English) Zbl 0669.35026
The paper deals with some spectral properties of a strongly elliptic differential operator with unbounded coefficients. Generation of analytic semigroups in \(L^ p({\mathbb{R}}^ n)\) is involved and an application to a Cauchy problem of parabolic type is given.
Reviewer: C.A.Marinov

35J15 Second-order elliptic equations
47D03 Groups and semigroups of linear operators
35A20 Analyticity in context of PDEs
35K15 Initial value problems for second-order parabolic equations
PDF BibTeX Cite
Full Text: DOI
[1] S. Agmon,On the eigenfunctions and the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math.15 (1962), 119–147. · Zbl 0109.32701
[2] H. Amann,Dual semigroup and second order linear elliptic boundary value problems, Isr. J. Math.45, (1983), 225–254. · Zbl 0535.35017
[3] 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. · Zbl 0593.47048
[4] P. Cannarsa and V. Vespri,On maximal L p regularity for the abstract Cauchy problem, Boll. Un. Mat. Ital.5-B (1986), 165–175. · Zbl 0608.35027
[5] P. Cannarsa and V. Vespri,Generation of analytic semigroups by elliptic operators with unbounded coefficients, SIAM J. Math. Anal.18 (1987), 857–872. · Zbl 0623.47039
[6] 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. · Zbl 0622.60068
[7] 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.
[8] E. B. Davies,Some norm bounds and quadratic form inequalities for Schroedinger operators II, J. Oper. Theory12 (1984), 177–196. · Zbl 0567.47006
[9] E. B. Davies and B. Simon,L 1 properties of intrinsic Schroedinger semigroups, J. Funct. Anal.65 (1966), 126–146. · Zbl 0613.47039
[10] S. D. Eidel’mann,Parabolic Systems, Nauka, Moscow, 1964; English translation: North-Holland, Amsterdam, 1969.
[11] 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. · Zbl 0279.35032
[12] Y. Higouchi,A priori estimates and existence theorems on elliptic boundary value problems for unbounded domains, Osaka J. Math.5 (1968), 103–135.
[13] 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.
[14] J. L. Lions and E. Magenes,Problèmes aux limites non homogènes et application II, Dunod, Paris, 1968.
[15] N. Okazawa,An L p theory for Schroedinger operators with nonnegative potentials, J. Math. Soc. Japan36 (1984), 675–688. · Zbl 0556.35032
[16] A. Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. · Zbl 0516.47023
[17] J. Voigt,Absorption semigroups, their generators and Schroedinger operators, J. Funct. Anal.67 (1986), 167–205. · Zbl 0628.47027
[18] 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. · Zbl 0493.35050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.