Korevaar, N.

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AstérisqueN. KOREVAARThe normal variations technique for studyingthe shape of capillary surfacesAstérisque, tome 118 (1984), p. 189-195<http://www.numdam.org/item?id=AST_1984__118__189_0>© Société mathématique de France, 1984, tous droits réservés.L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/Publications/Asterisque/) implique l’accord avec les conditions générales d’uti-lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ouimpression systématique est constitutive d’une infraction pénale. Toute copieou impression de ce fichier doit contenir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/Société Mathématique de France Astérisque 118 (1984) p.189 à 195. THE NORMAL VARIATIONS TECHNIQUE FOR STUDYING THE SHAPE OF CAPILLARY SURFACES by N. KOREVAAR (University of Kentucky) A capillary surface S is the (equilibrium) interface between two adjacent fluids that are also contacting rigid walls. In this paper we study the case for which S is the interface between two fluids in a vertical capillary tube, in the presence of a downward pointing gravitational field. S is the graph of a function u whose domain is the (arbitrary) horizontal cross section ft of the tube. Because is in equilibrium i t s mean curvature is proportional to i t s height above a fixed reference plane, and the contact angle between (the normals of) and the tube 3ft x H is physically prescribed. Specifically, for ftcnRn , u€C2(ft) 0 C1 (ft) is a classical solution to the capillary problem if (Figure 1): div Tu = Ku in ft Tu = vu / l + |Vu|2 K > 0 (1) Tu • n = cosy on 3ft , 0 < y < IT , n = exterior normal to 3ft . We often write H(x,u(x)) for div Tu. Geometrically H(x,u(x)) is n times the mean curvature of , at the point Figure 1: Configuration Su: (z-u(x)} (n.0j_ Tu, A+|Vu|2. -1 189 N. KOREVAAR (x,u(x)). The equations (1) arise from the energy functional (2) E(f) = | [Vl + |Df|2 + | f2) - | uf / P = cosy . The aim of this note is to indicate a natural way of studying the steepness of S^. (Full details will appear elsewhere C4D) . I t is well understood how to estimate the height of by constructing comparison surfaces; studying the steepness i s the next logical step. (Actually, several impressive techniques for estimating |Vu| are known, developed because of the important role that a priori gradient bounds play in regularity theory. The methods involved are fairly technical, however, and obscure the geometric content of the problem). Intuitively, if a capillary surface is too steep i t won't minimize (2) and will seek to reduce i t s energy by perturbing i tself (perhaps as indicated by the shaded region in Figure 2). This intuition can be made rigorous: Perturb­ing a small (0(e)) amount along i t s (downward) normal produces a compar ison surface S whose mean curvature v can be estimated in terms of the or i ­ginal surface and the perturbation. If for appropriate perturbations and after raising an amount also 0(e), one can conclude that the lifted S l ies above S (via a v u comparison principle), then one is in business: as e 0 this yields a bound on |Vu| (bottom of Figure 2). The formula relating the mean Figure 2: Intuition 190 THE SHAPE OF CAPILLARY SURFACES curvatures of S and S can be u v derived in a straightforward way. Let f : ]Rn+1 •> IR be the perturbation function, V the downward normal to Su« Perturb (x,u(x)) €Su by the vector efv. For small £ there is the correspondence (Figure 3): (3) (x,v(x)) = (x,u(x)) +£fV(x,u(x)) . Using (3), the inverse function theorem (for 3x/3x ) and the chain rule one can Figure 3: Normal perturbations differentiate v. Using tangential coordinates yields (4) H(x,v(x)) = H(x,u(x)) -e(f||D2u||2 + ATf-fvH(x,u(x)) + 0(e2) D u = "tangential Hessian" = Hessian of the function parameterizing above i t s tangential plane ATf = "tangential Laplacian" = Af - f fv = Vf • v . Lifting Sv an amount 0(e) gives a surface which will be compared to via the well known: Comparison Principle (CP.) ; Let Ocfi y n = exteviov normal to 9©, u,w€C (0). Suppose (i) Vx€© s . t . w(x)<u(x), H(x,w(x)) < H(x,u(x)) (ii) Vx£30 s . t . w(x)£u(x), Yw £ Yu ( i .e . Tw • n >̂  Tu • n) . Then, in fact w(x) >. u(x) Vx£0 . (x,u(x» ^ \efv fíí.v(x)) 191 N. KOREVAAR The easiest gradient estimate is the a priori interior bound for the capillary problem: Theorem 1. Let u satisfy H(x,u(x)) = Ku in BR = {x € 3Rn |x| <R}, K> 0. Then 3c = C(R,K) s . t . |Vu(x)I < C . R - | x | Proof. Pick f(x,z) = f(x) as indicated in Figure 4 and so that | f 1 ^ <_ 1 . Perturb S to S . From (4), 3 M s . t . u v (5) H(x,v(x)) <_ H(x,u(x) ) + £M . (We used the well-known height estimates that imply u is bounded in BR so that Ku is too). Writing w(x) = v(x) + + £(1 + (M/K)) one verifies that H(x,w(x) ) <_ Kw(x) , from which condition (i) of CP. follows. Condition (ii) is true t r iv ia l ly . Hence w >̂ u in BR : w(x) >_ u(x) Figure 4: Interior gradient bound in gravity u(x) - ef ' l+ |Vu|2 + e 1 + M > u(x) + efVu • Vu 1+ |Vu|2 + 0(e2) 1 + M K > f |Vu|2 1+ |Vu|2 q.e.d. For the general prescribed mean curvature equation Cl, 2, 6, 91 f(x,z) must be chosen more carefully. After l ift ing Sv a large enough 0(e) to get , the only places where (i) must be checked are where |Vu| is large. Hence the tangent plane is almost vertical. By introducing exponential growth in f(x,z) in the z direction (and making other modifications too) one can make and hence ATf large enough to dominate the other terms that arise in comparing H(x,w(x)) to H(x,u(x)) . r _ü.-fifíiii_ v x •"ST* 192 THE SHAPE OF CAPILLARY SURFACES To estimate |Vu| near 3ft f must be non-zero there. Hence Sv may not l ie above ft (Figure 5). The fact that Tu • n is prescribed, however, forces Sv to l ie (within 0 ( e ) of being) above ft£ , the domain whose boundary 3ft£ is gotten by perturbing 3ft an amount efcosy along n. Hence by following the normal perturbation with a change of x-coordinates one can return S to S~ , a surface above ft. If one v v chooses f and the return transformation carefully the contact angle condition (ii) of CP. can be controlled. In order to keep H(x,v(x)) <_H(x,u(x)) + eM for condition ( i ) , the return transformation must be chosen carefully (Figure 6). (For global estimates f is never zero and one can just shove everything back along n ) . One gets: Figure 5: Normal perturbations near 3ft Theorem 2 C3, 7, 8, 10]. Let u€C (Oflft) 3ft n 0 smooth (C3) , u satisfying (1) in 0 PI ft and x £ 0 n ft . Then there is an a priori bound |Vu(x) | _< C(x,3ft fl 0,K,y) . Remark. G. Lieberman has recently been able to use maximum principle ideas to get global (but not local) gradient estimates for a class of operators and boundary conditions generalizing (1) C5D. There are more applications of the normal variations technique for studying Vu. Monotonicity results can be proven in domains whose shape leads one to expect monotonicity. Lipschitz continuity of solutions in singular domains of the correct Figure 6: Returning ft£ to ft v X r a x r r 3 V lef cos Y 3ß X X 193 N. KOREVAAR geometry can be shown. Mean curvature problems other than those in capillarity can be studied. This work was partially supported by the U.S. Army under contract no. DAAG29-80-C-0041 and by the National Science Foundation under grants no. MCS-7927062, Mod. 1 and no. MCS-8301906. REFERENCES [1] E. BOMBIERI, E. DE GIORGI, M. MIRANDA, Una maggiorazione a priori relativa alle ipersuperfici minimali nonparametriche, Arch. Rat. Mech. Anal. 32 (1969), 255-269. [2] R. FINN, New estimates for equations of minimal surface type, Arch. Rat. Mech. Anal. 14 (1963), 337-375. [3] C. GERHARDT, Global regularity of solutions to the capillary problem, Ann. Scuola Norm. Sup. Pisa 3 (1976), 157-175. [4] N. KOREVAAR, The normal variations technique for studying the steepness of capillary surfaces, in preparation. [5] G. LIEBERMAN, Gradient estimates for the capillary problem via the maximum principle, preliminary version. [6] L. SIMON, Interior gradient bounds for non-uniformly e l l ip t ic equations, Indiana U. Math. J . 25 #9 (1976), 821-855. [7] L. SIMON and J. SPRUCK, Existence and regularity of a capillary surface with a prescribed contact angle. Arch. Rat. Mech. Anal. 61 (1976), 19-34. [8] J. SPRUCK, On the existence of a capillary surface with a prescribed angle of contact. Comm. Pure Appl. Math. 28 (1975), 189-200. [9] N. TRUDINGER, A new proof of the interior gradient bound for the minimal sur­face equation in n-dimensions, Proc. Nat. Acad. Sci. USA 69 (1972), 166-175. 194 THE SHAPE OF CAPILLARY SURFACES [10] N.N. URAL'TCEVA, Solution to the capillary problem, Vestnik Leningrad Univ. 19 (1973), 54-64; (English translation: Vestnik Leningrad Univ. Math. 6 (1979), 363-375). Nicholas KOREVAAR Department of Mathematics University of Kentucky Lexington, KY 40506 U.S.A. 195 

The normal variations technique for studying the shape of capillary surfaces

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The normal variations technique for studying the shape of capillary surfaces

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