By Georges Duvaut, Jacques Louis Lions (auth.)
1. we start by means of giving an easy instance of a partial differential inequality that happens in an ordinary physics challenge. We think about a fluid with strain u(x, t) on the aspect x on the rapid t that three occupies a sector Q oflR bounded via a membrane r of negligible thickness that, despite the fact that, is semi-permeable, i. e., a membrane that enables the fluid to go into Q freely yet that stops all outflow of fluid. it is easy to end up 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 type 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 is additional the preliminary (3) u(x,O)=uo(x). We be aware that stipulations (2) are non linear; they suggest that, at each one fastened immediate 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; hence we care for a "free boundary" problem.
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E. in O}. 37), we obtain the same properties as before. 3. Spaces of Vector-Valued Functions For a precise formulation of the problems of evolution posed in Section 3, we need additional tools which we will now introduce. 40) IlfIILOO(O,T;X) = esssup Ilf(t)llx· te(O,T) Ilflb(o,T;x) < 00 (p# (0), This is a Banach space. 41) ~'(]O, T[;X) = 2'(~(]0, T[;X) where, generally, 2'(Y; X) denotes the space of linear continuous mappings of Y-+X. 42) j(cp) = J~ f(t)cp(t)dt, ([0, T]), (a well defined mapping cp-+j(cp), linear continuous from ~(]O, TD-+X).
These devices have limited power which, consequently, limits the heat flux - g; we assume therefore that the latter remains in the closed intervals [gl,g2] with OE[gl,g2J. 21) g=O; 22 I. Problems of Semi-Permeable Media ii) if u(x,t)~[hl(X),h2(X)], we inject a quantity of heat proportional to the distance between u(x) and the interval [h 1(x),h 2(x)], if that is possible. Hence u(x, t) > h2(x) =:. 22) u(x,t) 62), are continuous on H 1(Q). 16) can be written To make the meaning of the stationary inequalities (Sec. 5) precise, it remains then only to define If'(v) for vEH1(Q). 10). 1, v(x) is defined almost everywhere on r, and therefore also 1/1 (v(x)). 34) E = {AI I/I(A) finite}; the set E is closed and convex (bounded or not). e. for XEr. 36) the set K is closed convex in Hl(Q). 37) the function 1/1 is at most increasing quadratically at infinity on E (this is obviously meaningless when E is bounded, i.