Functional spaces
Definition 1: Functional spaces over a one-dimensional area
For \( \Omega = \left(x_l, x_r\right) \subset {\cal R} \)
\( L^2(\Omega) = \{ v:\Omega \rightarrow {\cal R} : \int_{\Omega} v(x)^2 dx < \infty \} \)
along with the norm \( \|u\|_{L^2(\Omega)}^2 = \int_{\Omega} u(x)^2 dx \)
and the dot product \( \left(u,v\right) L^2(\Omega) =\int_{\Omega} u(x)v(x) dx \)
\( H^1(\Omega) = \{v:\Omega \rightarrow {\cal R} : \int_{\Omega} v(x)^2+\frac{dv(x)}{dx}^2 dx < \infty \} \)
where the derivative \( \frac{dv(x)}{dx} \) it is understood in a weak sense, that is, we define it in terms of integration by parts \( \int_{\Omega} u\frac{dv(x)}{dx} dx = - \int_{\Omega} \frac{du(x)}{dx} v dx \) for all "smooth" functions \( u \in C^\infty_0(\Omega) = \{v:\Omega \rightarrow {\cal R} : \int_{\Omega} v(x)^2+\frac{dv(x)}{dx}^2 dx < \infty \} \) infinitely many differentiable times defined on closed prefixes \( (a,b) \subset \Omega \).
In space \( H^1(\Omega) \) we define the norm \( \|u\|_{H^1(\Omega)}^2 = \int_{\Omega} u(x)^2+\frac{du(x)}{dx}^2 dx \) and scalar product \( \left(u,v\right)_{H^1(\Omega)} =\int_{\Omega} u(x)v(x)+\frac{du(x)}{dx}\frac{dv(x)}{dx} dx \)
\( H^1_0(\Omega) = \{v:\Omega \rightarrow {\cal R} : \int_{\Omega} v(x)^2+\frac{dv(x)}{dx}^2 dx < \infty; v(0)=0 \} \)
Along with the norm \( \|u\|_{H_0^1(\Omega)}^2 = \int_{\Omega} \frac{du(x)}{dx}^2 dx \)
and the dot product \( \left(u,v\right) H_0^1(\Omega) =\int_{\Omega} \frac{du(x)}{dx}\frac{dv(x)}{dx} dx \)
Definition 2: Functional spaces on a two-dimensional area
For \( \Omega \subset {\cal R}^2 \)
\( L^2(\Omega) = \{ v:\Omega \rightarrow {\cal R} : \int_{\Omega} v(x_1,x_2)^2 dx_1dx_2 < \infty \} \)
along with the norm \( \|u\|^2_{L^2(\Omega)} = \int_{\Omega} v(x_1,x_2)^2 dx_1dx_2 \)
and the dot product \( \left(u,v\right)_{L^2(\Omega)} = \int_{\Omega} u(x_1,x_2)v(x_1,x_2) dx_1dx_2 \)
\( H^1(\Omega) = \{v:\Omega \rightarrow {\cal R} : \int_{\Omega} (v(x_1,x_2)^2+\frac{dv(x_1,x_2)}{dx_1}^2+\frac{dv(x_1,x_2)}{dx_2}^2 )dx_1dx_2 < \infty \} \) where derivatives
\( \frac{dv(x_1,x_2)}{dx_i},i=1,2 \) we understand in a weak sense
\( \int_\Omega u(x_1,x_2) \frac{dv(x_1,x_2)}{dx_i} dx_1 dx_2= - \int_\Omega \frac{du(x_1,x_2)}{dx_i} v(x_1,x_2) dx_1 dx_2 \). In other words, they are defined by integration over parts with smooth functions \( u \in C^\infty_0(\Omega) \) infinitely many times differentiable defined over a compact area contained in \( \Omega \).
In space
\( H^1(\Omega) \) we define the norm \( \|u\|^2_{H^1(\Omega)} = \int_{\Omega} \left( u(x_1,x_2)^2+\frac{du(x_1,x_2)}{dx_1}^2+\frac{du(x_1,x_2)}{dx_2}^2\right) dx_1dx_2 \)
and the dot product \( \left(u,v\right)_{H^1(\Omega)} = \int_{\Omega} \left( u(x_1,x_2)v(x_1,x_2) +\frac{du(x_1,x_2)}{dx_1}\frac{dv(x_1,x_2)}{dx_1}+\frac{du(x_1,x_2)}{dx_2}\frac{dv(x_1,x_2)}{dx_2}\right)dx_1dx_2 \).
\( H^1_0(\Omega) = \{v:\Omega \rightarrow {\cal R} : \int_{\Omega} (v(x_1,x_2)^2+\frac{dv(x_1,x_2)}{dx_1}^2+\frac{dv(x_1,x_2)}{dx_2}^2) dx_1dx_2 < \infty, tr v(x_1,x_2)=0 \} \)
along with the norm \( \|u\|^2_{H^1_0(\Omega)} = \int_{\Omega} \left( \frac{du(x_1,x_2)}{dx_1}^2+\frac{du(x_1,x_2)}{dx_2}^2\right) dx_1dx_2 \)
and the dot product \( \left(u,v\right)H^1_0(\Omega) = \int_{\Omega} \left( \frac{du(x_1,x_2)}{dx_1}\frac{dv(x_1,x_2)}{dx_1}+\frac{du(x_1,x_2)}{dx_2}\frac{dv(x_1,x_2)}{dx_2}\right)dx_1dx_2 \)
where the trace operator means determining the value of the function on the boundary, which in the case of functions that are elements of the Sobolev space requires introducing classes of equivalence of functions equal to the integral value.