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

Abstract

In the case of clamped thermoelastic systems with interior point control defined on a bounded domain Ω, the critical case is n = dim Ω = 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 ‘ϵ-loss’ of interior regularity has occurred due to a peculiar pathology: the incompatibility of the B.C. of the spaces H 0 3 2 ( Ω ) and H 00 3 2 ( Ω ) . The present paper manages to establish that, indeed, one can take ϵ = 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 II: 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.