• DOI:10.1090/S0002-9947-1974-0358067-4
• Corpus ID: 53659690

Generation of analytic semigroups by strongly elliptic operators

@article{Stewart1974GenerationOA,
  title={Generation of analytic semigroups by strongly elliptic operators},
  author={H. Bruce Stewart},
  journal={Transactions of the American Mathematical Society},
  year={1974},
  volume={199},
  pages={141-162}
}

• H. Stewart
• Published 1 November 1974
• Mathematics
• Transactions of the American Mathematical Society

Strongly elliptic operators are shown to generate analytic semigroups of evolution operators in the topology of uniform convergence, when realized under general boundary conditions on (possibly) unbounded domains. An application to the existence and regularity of solutions to parabolic initial-boundary value problems is indicated. Introduction. Extending the results of a previous paper [19], we propose to prove a theorem on the generation of analytic semigroups by strongly elliptic operators A of order 2m in the topology of uniform convergence, under more general boundary conditions. As before, there is a direct application to parabolic initial-boundary value problems: we give an existence and uniqueness theorem for such problems, using the Kato-Tanabe theory for temporally inhomogeneous evolution equations du/dt + A(t)u = / The topology of uniform convergence in the space variables yields classical solutions of this parabolic problem which are analytic in time at each space point; furthermore, the initial values are assumed in the pointwise continuous sense. Since we treat general boundary conditions, our parabolic problem is roughly comparable to the one discussed in Arima [4], where existence and uniqueness are treated (using a fundamental solution of the parabolic problem), but not analyticity. The semigroup generation theorem for strongly elliptic operators is by now well established in the Lp spaces. The L2 case was treated by Browder in [5]; shortly thereafter Agmon, in [1], gave a method of proving the basic a priori estimate \\u\\ < (M/\z\)\\iA + z)u\\, |arg _| < \m + e, (E) in Lp, and this method has been filled out with existence theorems and used by several authors, including Friedman [8], Higuchi [9], Lau [14], and Freeman and Schechter [7]. The work of Lau offers an attractive combination of full development and general hypotheses, so we shall borrow Lp results from [14]. We recall that the Kato-Tanabe theorem applied in the Lp topology gives solutions which are Received by the editors June 25, 1979. AMS (MOS) subject classifications (1970). Primary 35K35, 47D05. 'This work was supported in part by the U. S. Department of Energy under contract EY-76-C-02-0016. Accordingly, the U. S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U. S. Government purposes. © 1980 American Mathematical Society 0002-9947/80/0000-02 1 8/$04.00 299 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use