# Variational problems living in two different Sobolev spaces

Is there a general reference concerning variational problems living in $$W^{h,p}\times W^{k,p}$$, with $$h, k\in\mathbb{N}_0$$ not coinciding? I'm thinking to problems of type: $$\inf_{u,v}\int_{\Omega} f_1\left(\frac{\partial^{|\alpha|} u}{\partial x_1^{\alpha_1}\dots \partial x_n^{\alpha_n}}\right)+f_2\left(\frac{\partial^{|\beta|} v}{\partial x_1^{\beta_1}\dots \partial x_n^{\beta_n}}\right)+ f_3\left(u-v\right)+V(x,u,v),$$ where $$\Omega$$ is a compact domain in $$\mathbb{R}^n$$ ($$n\ge 1$$), $$|\alpha|=h$$ and $$|\beta|=k. You can take $$f_{1,2,3}$$ and $$V$$ as nice as you wish ($$V$$ can also be assumed to depend only on $$v$$). You can include terms depending on other partial derivatives. What's important for me is that there's a term depending on $$u-v$$.

In the simplest one-dimensional cases (e.g., $$h=1$$, $$k=0$$, $$f_{1,2,3}$$ and $$V(x,v)$$ smooth), the term $$u-v$$ can be usually exploited to gain $$H^1$$ regularity for $$v$$ if, for instance, $$f_1$$ and $$f_3$$ are convex. Is any generalization of this known?

A further question: I encountered problems of this kind while studying (nonlinear) deformations of homogenized microstructured continua. Where else may variational problem of this kind arise?