Does such a Sobolev space exist in literature?

Question

I'm working on a system of PDEs over $\mathbb{R}^3$, that has a dependent variable $u(x,y,z,t)$. The system can be reduced to something like the following: $$u + \partial^2_{x_1x_1}u=f \quad \text{on } \Omega \subset \mathbb{R}^3.$$ The variables $x_2$ and $x_3$ are referred to by another PDE and in some boundary conditions, so I cannot just treat this PDE solely one direction.

Does anyone know of any literature that deals with something like this, where the Sobolev space would be something like $H^{2,0,0}$? Is it possible to endow this Sobolev space with some norm, possibly such as $\|u\|_0 + \|\partial_1u\|_0 + \|\partial^2_1u\|_0$?

Answer 1

You're looking for something called an anisotropic Sobolev space. Unfortunately there are many ways to get anisotropy (different number of $L^p$ derivatives in each direction? Same number of derivatives but different $L^p$ in each direction? Both?) so you're going to need to sift through some noise. From this MO post you may be interested in Lions & Magenes' book "Non-Homogeneous Boundary Value Problems and Applications" (Ch 4 Vol 2)

2. The Spaces $H^{r, s}(Q) .$ Trace Theorems. Compatibility Relations

$$\text{2.1 ${H}^{r, s}$-Spaces}$$

Let $r$ and $s$ be two non-negative real numbers. For $\Omega$ an open set in $\mathbf{R}^{n},$ we define: $$(2.1) \quad H^{r, s}(Q)=H^{0}\left(0, T ; H^{r}(\Omega)\right) \cap H^{s}\left(0, T ; H^{0}(\Omega)\right), \quad(Q=\Omega \times] 0, T[)$$ which is a Hilbert space with the norm $$ \left(\int_{0}^{T}\|u(t)\|_{H^{r}(\Omega)}^{2} d t+\|u\|_{H^{s\left(0, T ; H^{0}(\Omega)\right)}}^{2}\right)^{1 / 2} $$ In $(2.1)$, $ H^{\circ}\left(0, T ; H^{r}(\Omega)\right)=L^{2}\left(0, T ; H^{r}(\Omega)\right),$ where $H^{r}(\Omega)$ is defined in Chapter $1,$ Section $9,$ and $H^{s}\left(0, T ; H^{0}(\Omega)\right)$ is defined in the same way as $H^{s}(0, T) ;$ see Chapter $1,$ Section $9 ;$ for example, if $X$ is a Hilbert space, $$ H^{s}(0, T ; X)=\left[H^{m}(0, T ; X), H^{0}(0, T ; X)\right]_{\theta}, \quad(1-\theta) m=s $$ integer $m>s,$ and $$ H^{m}(0, T ; X)=\left\{v \mid v, v^{\prime}, \ldots, v^{(m)} \in L^{2}(0, T ; X)\right\} . $$