$H^1$ Inner Product for vector valuled functions

Question

I am not sure how to find the $H^{1}(D)$ inner product for two functions $u,v: D \rightarrow \mathbb{R}^2$, ($D \subset \mathbb{R}^2$). The inner product for scalar functions is defined as:

$\int_{D} f \; g \; dx + \int_{D} \nabla f \cdot \nabla g \; dx$

For extending this definition to vector valued functions, I found this link (Inner product for vector - valued functions) but it treats only the first term ($L^2$ norm). For the second term ($H^1$ seminorm), I tried to look up definitions of inner product for matrices but found multiple answers. Can someone please tell me which is the correct way to compute this?

Edit: I need to compute it for calculating the Gramian Matrix for a finite set of vector valued functions, with respect to the $H^1(D)$ norm. Is this the right way to do it?

Thank you!

Answer 1

$$H^1(D;\mathbb{R}^2)=\{u \in L^2(D;\mathbb{R}^2) : \nabla u \in L^2(D;\mathbb{R}^{2 \times 2}) \}$$

Let $u,v :D \to \mathbb{R}^2$. Let us denote by $u\cdot v=\sum_{i} u_i v_i$ the vector scalar product and by $A:B=\sum_{i,j} a_{ij} b_{ij}$ the matrix scalar product. The inner product of $H^1(D;\mathbb{R}^2)$ is given by

$$(u,v)_{H^1}=\int_D u \cdot v + \nabla u : \nabla v \text{ d}x.$$