Analyticity, and lack thereof, of thermo-elastic semigroups

Applied Mathematics, Thornton Hall University of Virginia Charlottesville, VA 22903

Abstract

We consider (linear) thermo-elastic plate equations under five sets of canonical B.C., including two cases where the mechanical and the thermal variables are coupled on the boundary. The challenging so-called free B.C. case of [Lag], [A-L] is included. The main results are as follows. If rotational forces are not accounted for, then the resulting s.c. contraction semigroup is, moreover, analytic on the natural (energy) space under all such canonical B.C. By contrast, if rotational forces are accounted for, then the corresponding s.c. contraction semigroup has a structural property that makes it more akin to a s.c. group (at least in the mechanical part); a fortiori, it is neither compact, nor differentiable, nor uniformly continuous for all t > 0. Analyticity of the s.c. thermo-elastic semigroup, particularly in the difficult case of free B.C., has been an open problem for some time in specialized circles. Similarly, a general description of the cases where analyticity fails has been the object of inquiries.