By Georges Duvaut, Jacques Louis Lions (auth.)

1. we start via giving an easy instance of a partial differential inequality that happens in an user-friendly physics challenge. We contemplate a fluid with strain u(x, t) on the element x on the immediate t that three occupies a zone Q oflR bounded by means of a membrane r of negligible thickness that, even if, is semi-permeable, i. e., a membrane that enables the fluid to go into Q freely yet that stops all outflow of fluid. you can actually turn out then (cf. the main points in bankruptcy 1, part 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 functionality, with boundary stipulations within the kind of inequalities u(X,t»o => au(x,t)/an=O, XEr, (2) u(x,t)=o => au(x,t)/an?:O, XEr, to that's additional the preliminary (3) u(x,O)=uo(x). We word that stipulations (2) are non linear; they indicate that, at every one mounted speedy t, there exist on r areas r~ and n the place u(x, t) =0 and au (x, t)/an = zero, respectively. those areas aren't prescribed; therefore we care for a "free boundary" problem.

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12) u>h => ou/on=O u~h => ou/on= -k(u-h). 3. If k, the conductivity of the wall, is zero, the fluid can neither enter nor leave and the pressure u is the solution of the classical Neuman problem, stationary or instantaneous as the case may be. 4. 11). Actually, we will show 3 that if Uk is the solution corresponding to the conductivity k, then Uk tends, in a certain sense, toward u as k tends toward + 00; here, u denotes the solution of the problem: "thin semi-permeable wall". 0 Let us point out two further natural variants of the preceding problems.

Therefore u is continuous from [0, T] --+ V. f weakly star in Loo(O. f(t), rp(t))dt V rpE V(O, T; H). 49 5. Solution of the Variational Inequalities of Evolution of Section 3 We introduce I/I i as follows (Fig. 17) I/Ij()') = 0, )'~hl +gt/j, hI +gt/j~)'~hl' hI ~)'~h2' ij()'-h2)2, h2~)'~h2+1, j()'-h2)-j/2, h2+1~)'. Then we define 'P j by slope j Ctlj o Fig. 4) holds, for example with CPj=O Vj. 5). From compactness theorems (see for example Lions-Magenes [1], Chap. 1). 19) Vj-+v in L2(1:) strongly.

Generalization. 18a) -ou/on = ([>(u) , where the function ([> is a continuous and increasing function of u or, still more generally, where ([> is a multiple-valued, increasing function with maximal graph (that is to say that the graph is a continuous curve in lR 2). 2. Temperature Control Through the Interior, Regulated by the Temperature in the Interior We prescribe two temperatures h1(x) and h2(x) for xEQ, hl(X)~h2(X), and we ask that the temperature u(x), xEQ, deviates as little as possible from the interval (hl,h2)' For this purpose, we set up volume sources of heat (in the algebraic sense).