Abstract
Given an open domain (possibly unbounded) Ω⊂R n, we prove that uniformly elliptic second order differential operators, under nontangential boundary conditions, generate analytic semigroups in L 1(Ω). We use a duality method, and, further, give estimates of first order derivatives for the resolvent and the semigroup, through properties of the generator in Sobolev spaces of negative order.
Similar content being viewed by others
References
Amann, H.: Dual semigroups and second order linear elliptic boundary value problems. Isr. J. Math. 45, 225–254 (1983)
Amann, H., Escher, J.: Strongly continuous dual semigroups. Ann. Mat. Pura Appl. 171, 41–62 (1996)
Angiuli, L.: Short-time behavior of semigroups and functions of bounded variation. PhD Thesis, Università del Salento (2008)
Angiuli, L., Miranda, M., Pallara, D., Paronetto, F.: BV functions and parabolic initial boundary value problems on domains. Ann. Mat. Pura Appl. 188, 297–311 (2009)
Cannarsa, P., Vespri, V.: Generation of analytic semigroups in the L p topology by elliptic operators in R n. Isr. J. Math. 61, 235–255 (1988)
Di Blasio, G.: Analytic semigroups generated by elliptic operators in L 1 and parabolic equations. Osaka J. Math. 28, 367–384 (1991)
Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. Springer, Berlin (2000)
Guidetti, D.: On elliptic systems in L 1. Osaka J. Math. 30, 397–429 (1993)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lectures Notes in Mathematics, vol. 840. Springer, Berlin (1981)
Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969)
Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel (1995)
Lunardi, A., Vespri, V.: Hölder regularity in variational parabolic non-homogeneous equations. J. Differ. Equ. 94, 1–40 (1991)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, Berlin (1983)
Schechter, M.: Negative norms and boundary problems. Ann. Math. (2) 72, 581–593 (1960)
Schechter, M.: On L p estimates and regularity. I. Am. J. Math. 85, 1–13 (1963)
Seeley, R.: Interpolation in L p with boundary conditions. Stud. Math. 44, 47–60 (1972)
Stewart, H.B.: Generation of analytic semigroups by strongly elliptic operators. Trans. Am. Math. Soc. 199, 141–162 (1974)
Stewart, H.B.: Generation of analytic semigroups by strongly elliptic operators under general boundary conditions. Trans. Am. Math. Soc. 259, 299–310 (1980)
Tanabe, H.: Functional Analytic Methods in Partial Differential Equations. Dekker, New York (1997)
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library, vol. 18. North-Holland, Amsterdam (1978)
Vespri, V.: Analytic semigroups generated in H −m,p by elliptic variational operators and applications to linear Cauchy problems. In: Semigroup Theory and Applications. Lecture Notes in Pure and Appl. Math., vol. 116, pp. 419–431. Dekker, New York (1989)
Vespri, V.: Abstract quasilinear parabolic equations with variable domains. Differ. Integral Equ. 4, 1041–1072 (1991)
Vespri, V.: Analytic semigroups generated by ultraweak operators. Proc. R. Soc. Edinb. A 119, 87–105 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Rainer Nagel.
Rights and permissions
About this article
Cite this article
Angiuli, L., Pallara, D. & Paronetto, F. Analytic semigroups generated in L 1(Ω) by second order elliptic operators via duality methods. Semigroup Forum 80, 255–271 (2010). https://doi.org/10.1007/s00233-009-9200-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-009-9200-y