Suppose $S_0,S\subseteq\mathbb{R}^3$ are embedded surfaces. For a smooth map $\phi:S_0\rightarrow S$, define an energy $E[\phi]$ by $$E[\phi]:=\int_{\Sigma_0}F[\Lambda(d\phi_p)]\,dA(p).$$ Here, $\Lambda(d\phi_p)\equiv (\sigma_1(d\phi_p),\sigma_2(d\phi_p))$ is the set of singular values of the Jacobian of $\phi$ at $p\in S_0$. As an example, if we define $F[\sigma_1,\sigma_2]:=\sigma_1^2+\sigma_2^2$, then $E[\phi]$ is the Dirichlet energy of $\phi$.
Here's my question: Given a smooth map $\phi_0:S_0\rightarrow S$, is there a sufficient condition on $F:\mathbb{R}^2\rightarrow\mathbb{R}$ guaranteeing existence of a map $\phi:S_0\rightarrow S$ in the homotopy class of $\phi$ that locally minimizes $E[\cdot]$?
What I have in mind is the gradient flow of Eells and Sampson, which proves existence of harmonic maps when $S$ has negative Gaussian curvature by starting with an arbitrary $\phi_0$ and flowing along the gradient of the Dirichlet energy to a local optimum. This is an elegant construction, but the drawback is that not all target surfaces $S$ admit a harmonic map $\phi:S_0\rightarrow S$.
My intuition is that Eells and Sampson's construction fails in the presence of positive curvature because $F[\sigma_1,\sigma_2]=\sigma_1^2+\sigma_2^2$ "wants to" pinch points, i.e. reach a Jacobian with as small singular values as possible. But perhaps objectives like the symmetric Dirichlet energy, which appears in computer graphics applications, which looks like $F[\sigma_1,\sigma_2]:=\sigma_1^2+\sigma_2^2+\sigma_1^{-2}+\sigma_2^{-2}$, would have better properties since it has an asymptote whenever $\sigma_i=0$ and is minimized when $\sigma_1=\sigma_2=1$.