{"product_id":"inequalities-in-mechanics-and-physics-9783642661679","title":"Inequalities in Mechanics and Physics","description":"\u003cp\u003e • Author(s): G. Duvant\u003cbr\u003e • Publisher: Springer\u003cbr\u003e • Publisher Imprint: Springer\u003cbr\u003e • BISAC: General\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003e1. 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.\u003c\/p\u003e","brand":"Springer","offers":[{"title":"Paperback","offer_id":45284139958423,"sku":"9783642661679","price":8814.0,"currency_code":"INR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9783642661679.webp?v=1769280298","url":"https:\/\/atlanticbooks.com\/products\/inequalities-in-mechanics-and-physics-9783642661679","provider":"Atlantic Books","version":"1.0","type":"link"}