Vol. 69, No. 1, 1977

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Equations of mean curvature type in 2 independent variables

Leon M. Simon

Vol. 69 (1977), No. 1, 245–268

The object of this paper is to develop a regularity theory for equations of mean curvature type in two independent variables. An equation of mean curvature type in two independent variables is defined to be an equation of the form

aij(x,u,Du)Diju = b(x,u,Du )

on a domain Ω R2, where the functions aij, b satisfy special structural conditions. Namely, we require that (i) (1 + |Du|2)12b(x,u,Du) is bounded by a fixed constant (independent of u), and (ii) the quadratic form i,j=12aij(x,u,Du)ξiξj is bounded from above and below in terms of the quadratic form i,j=12gij(Du)ξiξj, where gij(Du) = δij DiuDju∕(1 + |Du|2), i,j = 1,2, are the coefficients of the minimal surface equation.

Mathematical Subject Classification 2000
Primary: 49F10, 49F10
Secondary: 35J20
Received: 16 August 1976
Published: 1 March 1977
Leon M. Simon