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ÌýThe $k$-Hessian operator $\sigma_k$ is the $k$-th elementaryÌýsymmetric function of the eigenvalues of the Hessian. It is known that theÌý$k$-Hessian equation $\sigma_k(D^2 u)=f$ with Dirichlet boundary condition$u=0$ is variational; indeed, this problem can be studied by means of theÌý$k$-Hessian energy $\int -u \sigma_k(D^2 u)$. We construct a naturalÌýboundary functional which, when added to the $k$-Hessian energy, yields asÌýits critical points solutions of $k$-Hessian equations with generalÌýnon-vanishing boundary data. As a consequence, we prove a sharp SobolevÌýtrace inequality for $k$-admissible functions $u$ which estimates theÌý$k$-Hessian energy in terms of the boundary values of $u$. This is jointÌýwork with Jeffrey Case.