×
Liskevich, Vitali; Sobol, Zeev; Vogt, Hendrik

On the \(L_{p}\)-theory of \(C_{0}\)-semigroups associated with second-order elliptic operators. II. (English) Zbl 1020.47029

J. Funct. Anal. 193, No. 1, 55-76 (2002).
[For Part I, see the preceding review)].
The authors continue to study the \(L_p\) properties of second-order elliptic differential operators corresponding to expressions \(\Lambda=-\nabla.(a\nabla) + b.\nabla+ \nabla.c+V\) with singular measurable coefficients \(a,b,c,V\). Here, they consider the case of uniformly elliptic operators. They obtain the interval \((p_{\min}, p_{\max})\) in the \(L_p\)-scale, where the associated quasi-contractive \(C_0\)-semigroups can be defined on \(L_p\) spaces.
The Sharpness of the main result is proven and the analyticity of the generators are given for \(p\) where the semigroup is defined.
Reviewer: Miklavž Mastinšek (Maribor)

Cited in 45 Documents

MSC:

47D06 One-parameter semigroups and linear evolution equations
47D09 Operator sine and cosine functions and higher-order Cauchy problems

Keywords:

\(L_p\) properties of second-order elliptic differential operators; quasi-contractive \(C_0\)-semigroups; uniformly elliptic operators
Cite Review PDF
Full Text: DOI

References:

[1] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York, San Francisco · Zbl 0186.19101
[2] Beliy, A.; Semenov, Yu., On the \(L^p\)-theory of the Schrödinger semigroups, II, Siberian Math. J., 31, 16-26 (1990) · Zbl 0815.47050
[3] Coulhon, Th., Dimensions of continuous and discrete semigroups on the \(L^p\)-spaces, Semigroup Theory and Evolution Equations, 135 (1991), Marcel Dekker: Marcel Dekker New York, p. 93-99 · Zbl 0744.47032
[4] Davies, E. B., Heat Kernels and Spectral Theory (1989), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0699.35006
[5] Davies, E. B., \(L^p\) spectral independence and \(L^1\) analyticity, J. London Math. Soc. (2), 52, 177-184 (1995) · Zbl 0913.47032
[6] Davies, E. B., Uniformly elliptic operators with measurable coefficients, J. Funct. Anal., 132, 141-169 (1995) · Zbl 0839.35034
[7] Fabes, E. B.; Stroock, D. W., The \(L^p\)-integrability of Green’s functions and fundamental solutions for elliptic and parabolic equations, Duke Math. J., 51, 997-1016 (1984) · Zbl 0567.35003
[8] Hempel, R.; Voigt, J., The spectrum of a Schrödinger operator on \(L_p(R^ν)\) is \(p\)-independent, Comm. Math. Phys., 104, 243-250 (1986) · Zbl 0593.35033
[9] Hieber, M., Gaussian estimates and holomorphy of semigroups on \(L^p\) spaces, J. London Math. Soc. (2), 54, 148-160 (1996) · Zbl 0893.47024
[10] Kato, T., Perturbation Theory of Linear Operators (1980), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0435.47001
[11] Kovalenko, V.; Perelmuter, M., Yu. Semenov, Schrödinger operators with \(L^{1/2}_w (R^{l\) · Zbl 0463.47027
[12] Kovalenko, V.; Semenov, Yu., \(C_0\)-semigroups in \(L^p(R^{d\) · Zbl 0778.47031
[13] Krylov, N., Lectures on Elliptic and Parabolic Equations in Hölder Spaces (1996), American Mathematical Society: American Mathematical Society Providence · Zbl 0865.35001
[14] P. C. Kunstmann, and, H. Vogt, Weighted norm estimates and \(L_p\); P. C. Kunstmann, and, H. Vogt, Weighted norm estimates and \(L_p\) · Zbl 1126.35034
[15] Liskevich, V., On \(C_0\)-semigroups generated by elliptic second order differential expressions on \(L^p\)-spaces, Differential Integral Equations, 9, 811-826 (1996) · Zbl 0852.47018
[16] Liskevich, V.; Vogt, H., On \(L^p\)-spectra and essential spectra of second order elliptic operators, Proc. London Math. Soc., 80, 590-610 (2000) · Zbl 1053.35114
[17] Ouhabaz, E. M., Gaussian estimates and holomorphy of semigroups, Proc. Amer. Math. Soc., 123, 1465-1474 (1995) · Zbl 0829.47032
[18] Schreieck, G., \(L_p\)-Eigenschaften der Wärmeleitungshalbgruppe mit singulärer Absorption (1996), Shaker Verlag: Shaker Verlag Aachen · Zbl 0857.47026
[19] Schreieck, G.; Voigt, J., Stability of the \(L_p\)-spectrum of Schrödinger operators with form small negative part of the potential, (Bierstedt; Pietsch; Ruess; Vogt, Functional Analysis (1994), Marcel-Dekker: Marcel-Dekker New York) · Zbl 0817.35065
[20] Semenov, Yu., Stability of \(L^p\)-spectrum of generalized Schrödinger operators and equivalence of Green’s functions, Internat. Math. Res. Notices, 12, 574-593 (1997) · Zbl 0905.47031
[21] Yu. Semenov, Solvability, analyticity and \(L^p\); Yu. Semenov, Solvability, analyticity and \(L^p\)
[22] Yu. Semenov, Semigroups generated by 2nd order elliptic non-divergence form operators in \(R^d\); Yu. Semenov, Semigroups generated by 2nd order elliptic non-divergence form operators in \(R^d\)
[23] Simon, B., Schrödinger semigroups, Bull. Amer. Math. Soc. (N. S.), 7, 447-526 (1982) · Zbl 0524.35002
[24] Sobol, Z.; Vogt, H., On \(L_p\)-theory of \(C_0\)-semigroups associated with second order elliptic operators. I, J. Funct. Anal., 193, 24-54 (2002) · Zbl 1020.47028
[25] Stroock, D. W., Estimates on the heat kernel for second order divergence form operators, Probability Theory (Singapore, 1989) (1992), de Gruyter: de Gruyter Berlin, p. 29-44 · Zbl 0779.60065
[26] H. Vogt, Analyticity of semigroups on spaces of exponentially bounded volume, preprint.; H. Vogt, Analyticity of semigroups on spaces of exponentially bounded volume, preprint.
[27] Voigt, J., Absorption semigroups, their generators and Schrödinger semigroups, J. Funct. Anal., 67, 167-205 (1986) · Zbl 0628.47027
[28] Voigt, J., One-parameter semigroups acting simultaneously on different \(L^p\) spaces, Bull. Soc. Roy. Sci. Liège, 61, 465-470 (1992) · Zbl 0805.47035
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.