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 $|\beta|=k<h$. 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?



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