{"product_id":"inequalities-in-mechanics-and-physics-von-g-duvant-j-l-lions","title":"Inequalities in Mechanics and Physics","description":"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\u0026gt;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»o =\u0026gt; au(x,t)\/an=O, XEr, (2) u(x,t)=o =\u0026gt; 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.\u003cdiv class=\"aw-variant-hidden-subtitle-div\" id=\"aw-variant-subtitle-9783642661679\"\u003e\u003ch3\u003e\u003c\/h3\u003e\u003c\/div\u003e","brand":"Libri","offers":[{"title":"Softcover - 9783642661679","offer_id":39444916568157,"sku":"9783642661679","price":128.39,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0940\/0622\/files\/52689302-7608-40c3-b79e-19acd0015b9d.jpg?v=1772174205","url":"https:\/\/shop.autorenwelt.de\/en\/products\/inequalities-in-mechanics-and-physics-von-g-duvant-j-l-lions","provider":"Autorenwelt Shop","version":"1.0","type":"link"}