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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 $y''(0)=-H$, $u_y=D_nu$, $u_{xx}=-\lambda_1 u -D_{nn}u$; so you get what you wanted.