The critical case of clamped thermoelastic systems with interior point control: Optimal interior and boundary regularity results

In the case of clamped thermoelastic systems with interior point control defined on a bounded domain Q, the critical case is n = dim Omega = 2. Indeed, an optimal interior regularity theory was obtained in [R. Triggiani, Sharp regularity of hyperbolic-dominated thermoelastic systems with point control: The clamped case, Discrete Contin. Dyn. Syst. (Suppl.) (2007) 993-1004], for n = 1 and n = 3. However, in this reference, an 'epsilon-loss' of interior regularity has occurred due to a peculiar pathology: the incompatibility of 3 3 the B.C. of the spaces H-0(3/2) (Omega) and H-00(3/2) (Q). The present paper manages to establish that, indeed, one can take c = 0, thus obtaining an optimal interior regularity theory also for the case n = 2. The elastic variables have the same interior regularity as in the corresponding elastic problem [R. Triggiani, Regularity with interior point control, Part If: Kirchhoff equations, J. Differential Equations 103 (1993) 394-421] (Kirchhoff). Unlike [R. Triggiani, Sharp regularity of hyperbolic-dominated thermoelastic systems with point control: The clamped case, Discrete Contin. Dyn. Syst. (Suppl.) (2007) 993-1004], the present paper establishes the sought-after interior regularity of the thermoelastic problem through a technical analysis based on sharp boundary (trace) regularity theory of Kirchhoff and wave equations. In the process, a new boundary, regularity result, not contained in [R. Triggiani, Sharp regularity of hyperbolic-dominated thermoelastic systems with point control: The clamped case, Discrete Contin. Dyn. Syst. (Suppl.) (2007) 993-1004], is obtained for the elastic displacement of the thermoelastic system. (c) 2008 Elsevier Inc. All rights reserved.