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.
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