The Optimal Interior Regularity for the Critical Case of a Clamped Thermoelastic System with Point Control Revisited

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 [Triggiani, Discrete Contin. Dyn. Syst., 2007] for n = 1 and n = 3. However, in this reference, an ‘є-loss’ of interior regularity has occurred due to a peculiar pathology: the incompatibility of the B.C. of the spaces H3/2 0 (Ω) and H3/2 00 (Ω). This problem for n = 2 was rectified in [Triggiani, J. Differential Equations, 2008]: this 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. As an additional bonus, a sharp boundary regularity of the elastic displacement is also obtained. In the present paper, we revisit that problem using a technique developed by these authors to circumvent the pathology of the incompatible boundary conditions. This yields a more direct proof of the optimal interior regularity (but not of the boundary regularity).