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

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

MathSciNet review:
1487323

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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,*The Dirichlet Problem for Nonlinear Second-Order Elliptic Equations I.Monge-Ampere Equation*, Comm. Pure App. Math.**37**(1984), 369-402. MR**88k:35073****2.**P.L.Lions, N.S.Trudinger and J.Urbas,*The Neumann problem for equations of Monge-Ampere type*, Comm. Pure Appl. Math.**39**(1986), 539-563. MR**87j:35114****3.**Ma Xi-nan,*Isoperimetric bounds for Monge-Ampere equations in two dimensions*, to appear in Analysis.**4.**R.P.Sperb,*Maximum Principles and Their Applications*, Academic Press,New York, 1981. MR**84a:35033****5.**N.S.Trudinger,*On degenerate fully nonlinear ellptic equations in balls*, Bull. Aust. Math. Soc.**111**(1987), 299-307. MR**88b:35083****6.**J.Urbas,*The oblique derivative problem for equations of Monge-Ampere type*, Proceeding of the Centre for Mathematical Analysis, Australian National University**12**(1987), 171-195. MR**89b:35045****7.**J.Urbas,*Nonlinear oblique boundary value problem for Hessian equations in two dimensions*, Ann. Inst. Henri Poincare**12**(1995), 507-575. MR**96h:35071**

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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:
https://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