A necessary condition of solvability

for the capillarity boundary

of Monge-Ampere equations in two dimensions

Author:
Ma Xi-Nan

Journal:
Proc. Amer. Math. Soc. **127** (1999), 763-769

MSC (1991):
Primary 35J25, 35J60, 35J65; Secondary 53C45

MathSciNet review:
1487323

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider a class of Monge-Ampere equations with a prescribed contact angle boundary value problem on a bounded strictly convex domain in two dimensions. The purpose is to give a sharp necessary condition of solvability for the above mentioned equations. This is achieved by using the maximum principle and introducing a curvilinear coordinate system for Monge-Ampere equations in two dimensions. An interesting feature of our necessary condition is the need for a certain strong restriction between the curvature of the boundary of domain and the boundary condition, which does not appear in the Dirichlet and Neumann boundary values.

**1.**L. Caffarelli, L. Nirenberg, and J. Spruck,*Correction to: “The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation” [Comm. Pure Appl. Math. 37 (1984), no. 3, 369–402; MR0739925 (87f:35096)]*, Comm. Pure Appl. Math.**40**(1987), no. 5, 659–662. MR**896772**, 10.1002/cpa.3160400508**2.**P.-L. Lions, N. S. Trudinger, and J. I. E. Urbas,*The Neumann problem for equations of Monge-Ampère type*, Comm. Pure Appl. Math.**39**(1986), no. 4, 539–563. MR**840340**, 10.1002/cpa.3160390405**3.**Ma Xi-nan,*Isoperimetric bounds for Monge-Ampere equations in two dimensions*, to appear in Analysis.**4.**René P. Sperb,*Maximum principles and their applications*, Mathematics in Science and Engineering, vol. 157, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR**615561****5.**Neil S. Trudinger,*On degenerate fully nonlinear elliptic equations in balls*, Bull. Austral. Math. Soc.**35**(1987), no. 2, 299–307. MR**878440**, 10.1017/S0004972700013253**6.**John I. E. Urbas,*The oblique derivative problem for equations of Monge-Ampère type in two dimensions*, Miniconference on geometry and partial differential equations, 2 (Canberra, 1986) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 12, Austral. Nat. Univ., Canberra, 1987, pp. 171–195. MR**924435****7.**John Urbas,*Nonlinear oblique boundary value problems for Hessian equations in two dimensions*, Ann. Inst. H. Poincaré Anal. Non Linéaire**12**(1995), no. 5, 507–575. MR**1353259**

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Additional Information

**Ma Xi-Nan**

Affiliation:
Department of Mathematics, East China Normal University, Shanghai 200062, People’s Republic of China

Email:
xnma@math.ecnu.edu.cn

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-04750-4

Keywords:
Monge-Ampere equation,
maximum principle,
curvilinear coordinate system,
contact angle boundary

Received by editor(s):
June 16, 1997

Communicated by:
Peter Li

Article copyright:
© Copyright 1999
American Mathematical Society