you use a local system of coordinates with x-axis tangent to the boundary and y along the normal vector; let y=y(x) the equation of the curve, so
u(x,y(x))=0 (u=\phi_1).
differentiating this equation twice at x=0 will give
u_xx(0) + y''(0)u_y(0)=0. 
but y''(0)=-H, u_y=Dnu, u_xx=-\lambda_1 u -Dnnu; so you get what you wanted.