PDEsFunctionalAnalysis Sobolev Spaces are Banach spaces of functions whose Weak Derivative are in , : Misplaced &W^{ k,p}(U):&= \{f \in L_\text{loc}^1(U) \space | \space D^\alpha f \in L^p(U) \space \forall \space |\alpha| \leq k \} \\ &= \{f \in L^p(U) \space | \space D^\alpha f \in L^p(U) \space \forall \space |\alpha| \leq k \} \end{aligned}$$ with Sobolev norm $$\begin{aligned} \left|\right|f\left|\right|_{W^{k,p}\left(U\right)}:&=\big{(}\sum_{|\alpha|\leq k}\int_{U}|D^{\alpha} f|^{p}dx\big{)}^{1/p} \\ &\overset{\text{norm}}{=}\sum_{|\alpha|\leq k} \big{|}\big{|} D^{\alpha} f \big{|}\big{|}_{L^p(U)} \end{aligned}$$ for $1 \leq p < \infty$. These represent functions with modest regularity/smoothness. Sobolev Spaces have the following properties: - For $p=1$: $$\begin{aligned} \left|\right|f\left|\right|_{W^{1,p}\left(U\right)} &\overset{\text{norm}}{=} \Big{|}\Big{|} || \nabla f || \Big{|}\Big{|}_{L^p(U)} \\ &:= \left( \int ||\nabla f||^p \right)^{1/p} \\ &\space= || \nabla f ||_{L^p(U)} \end{aligned}$$ - $W_0^{1,p}(U) := \overline{C_c^\infty(U)}$ functions have [[Trace Operator]] equal to zero and encode boundary and initial conditions of a PDE - $H^k(U) := W^{k,2}(U)$ is a Hilbert Space with inner product $$\left\langle f, g \right\rangle_{H^k(U)} := \int_U fg \space + \space \sum_{|\alpha| \leq k} \int_U D^\alpha f \space D^\alpha g$$ with $H_0^1(U) \subset H^1(U) \subset L^2(U) \subset H^{-1}(U)$ - For $n=1$ and $f \in W^{1,p}\left((a,b)\right)$: $$f = g \space\space \text{a.e.}$$where g is an [[Absolutely Continuous Function]] with $g' \in L^p\left((a,b)\right)$ (3a of [[PDE HW2]]) - For $n > 1$ Sobolev functions can be discontinuous or unbounded For [[Bochner Integrable]] functions Sobolev spaces are defined as $$W^{1,p}([0,T]; X) := \left\{ u \in \space L^p([0, T]; X) \space \Big{|} \space \left(\int_0^T \big{|}\big{|} u(t) \big{|}\big{|}_X^p + \big{|}\big{|} \frac{d}{dt}u(t) \big{|}\big{|}_X^p\space dt\right)^{1/p} < \infty \right\}$$ with the following properties - **Regularity**: $$u \in C([0,T]; X)$$ - **Fundamental Theorem of Calculus**: $$u(t) - u(s) = \int_s^t \frac{d}{dt} u(\tau) \space d\tau$$ - **Estimates**: $$\max_{t \in [0,T]} \big{|} \big{|} u(t) \big{|} \big{|}_X \leq C_T \big{|}\big{|} u\big{|}\big{|}_{W^{1,p}([0,T]; X)}$$