Inequalities in Mechanics and Physics (Grundlehren der mathematischen Wissenschaften, 219)

1. We begin by giving a simple example of a partial differential inequality that occurs in an elementary physics problem. We consider a fluid with pressure u(x, t) at the point x at the instant t that 3 occupies a region Q oflR bounded by a membrane r of negligible thickness that, however, is semi-permeable, i. e., a membrane that permits the fluid to enter Q freely but that prevents all outflow of fluid. One can prove then (cf. the details in Chapter 1, Section 2.2.1) that au (aZu azu aZu) (1) in Q, t>o, -a - du = g du = -a z + -a z + -a z t Xl X X3 z l g a given function, with boundary conditions in the form of inequalities u(X, t{raquo}o => au(x, t)/an=O, XEr, (2) u(x, t)=o => au(x, t)/an?:O, XEr, to which is added the initial condition (3) u(x, O)=uo(x). We note that conditions (2) are non linear; they imply that, at each fixed instant t, there exist on r two regions r~ and n where u(x, t) =0 and au (x, t)/an = 0, respectively. These regions are not prescribed; thus we deal with a "free boundary" problem Front Matter....Pages I-XVI Problems of Semi-Permeable Media and of Temperature Control....Pages 1-76 Problems of Heat Control....Pages 77-101 Classical Problems and Problems with Friction in Elasticity and Visco-Elasticity....Pages 102-196 Unilateral Phenomena in the Theory of Flat Plates....Pages 197-227 Introduction to Plasticity....Pages 228-277 Rigid Visco-Plastic Bingham Fluid....Pages 278-327 Maxwell’s Equations. Antenna Problems....Pages 328-381 Back Matter....Pages 382-400