The comparison of capillary surfaces heights in case of small gravity

✍ Scribed by A.A Kosmodem'yanskii Jr

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 60 KB
• Volume: 43
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(99)00252-7

✦ Synopsis

We consider Finn question [2]: does a liquid in "wide" capillary tube lift lower than in "narrow" one. This question is equivalent to the following problem: let u i and u e be solutions of the equation div Tu = ku

(1) in convex domains D i and D e (D i ⊂ D e ) with boundary conditions on boundaries i and e

(Tu; n) = cos ;

(2)

where Tu = ∇u= 1 + |∇u| 2 and n is the unit vector of outward normal to . Is it right that u i ¿ u e ?

We assume that D is convex domain and belongs C 2; . Denote by |D| the square of D; | | the perimeter of and K the curvature of .

Later on we denote by C (with indexes and without them) the distinct constants depending on geometrical characteristics of domain D.

It is evident that in convex domain D Poincarà e inequality holds D u 2 d x dy ≤ 1 |D| D u d x dy 2 + D |∇u| 2 d x dy:

The estimates of gradients of solutions (1) and (2) were received in [5,[7][8][9]. We base the result of Theorem 2 from [9]: there exists the total estimate of gradient capillary