A NEW PROOF OF THE GRADIENT ESTIMATE FOR GRAPHS OF PRESCRIBED MEAN-CURVATURE

The aim of this paper is to give a new proof of the gradient estimate for graphs of prescribed mean curvature H = H(x, y, z). Similarly as in [2] where the case H = H(x, y) is studied, we introduce conformal parameters for the surface. Then we employ the differential equation for the unit normal of the surface derived in [3] Satz 1. By this method, which is contained in [4] Satz 4, we prove the following. Theorem (Assumptions): I. We prescribe the function H = H(x, y, z) : IR3 --> IR is-an-element-of C1 + alpha (IR3) with alpha is-an-element-of (0, 1) satisfying (1) [GRAPHICS] with the constants h0 is-an-element-of [0, + infinity) and h1 is-an-element-of [0, + infinity) and require (2) partial derivative with respect to z H(x, y, z) greater-than-or-equal-to 0 in IR3. II. On the disc B(R): = {x, y) is-an-element-of IR2\x2 + y2 less-than-or-equal-to R2} with radius R > 0 we consider a solution z = zeta(x, y) : B(R) --> IR is-an-element-of C3 + alpha(B(R)) of the nonparametric equation (3) (1 + zeta(y)2)zeta(xx) - 2-zeta(x)zeta(y)zeta(xy) + (1 + zeta(x)2)zeta(yy) = 2H(x, y, zeta(x,y))(W(x, y))3 in B(R) of prescribed mean curvature H(x, y, z) abbreviating W(x, y) : = square-root 1 + \del-zeta(x, y)\2.