12.2 Sobolev spaces and weak solutions of PDEs

Sobolev spaces are powerful tools in functional analysis, extending differentiability to function spaces. They're crucial for solving partial differential equations, providing a framework for weak solutions when classical derivatives don't exist.

These spaces, denoted as W^k,p(Ω), contain functions with weak derivatives up to order k in L^p(Ω). They're equipped with norms that make them Banach spaces, and for p=2, they become Hilbert spaces, ideal for many applications.

Sobolev Spaces

Definition of Sobolev spaces

• Generalize the notion of differentiability to function spaces denoted as $W^{k,p}(\Omega)$
  • $k$ represents the order of weak derivatives (highest order of derivatives considered)
  • $p$ represents the integrability of the function and its weak derivatives (Lebesgue space $L^p$)
  • $\Omega$ is the domain of the functions (open subset of $\mathbb{R}^n$)
• Contain functions with weak derivatives up to order $k$ that belong to $L^p(\Omega)$
• Equipped with the Sobolev norm $\|u\|_{W^{k,p}(\Omega)} = (\sum_{|\alpha| \leq k} \|D^\alpha u\|_{L^p(\Omega)}^p)^{1/p}$, making them Banach spaces
  • For $p=2$, Sobolev spaces are Hilbert spaces denoted as $H^k(\Omega)$
• Possess important properties such as density (smooth functions are dense), embedding (continuous embeddings between Sobolev spaces), and trace theorems (restrictions of functions to the boundary)

Concept of weak derivatives

• Extend the notion of classical derivatives to less regular functions
• A function $u \in L^1_{loc}(\Omega)$ has a weak derivative $v \in L^1_{loc}(\Omega)$ if $\int_\Omega u \frac{\partial \varphi}{\partial x_i} dx = -\int_\Omega v \varphi dx$ for all test functions $\varphi \in C_c^\infty(\Omega)$
  • Test functions are smooth and compactly supported
• Weak derivatives allow for the study of solutions to PDEs that may not have classical derivatives

Applications in differential equations

• Provide a suitable framework for solving partial differential equations (PDEs)
• Weak formulation of a PDE is posed in a Sobolev space setting
  • Solution space and test function space are chosen based on the regularity requirements of the PDE
• Examples of PDEs solved using Sobolev spaces:
  • Poisson equation: $-\Delta u = f$ in $\Omega$, $u = 0$ on $\partial \Omega$
    • Weak formulation: Find $u \in H_0^1(\Omega)$ such that $\int_\Omega \nabla u \cdot \nabla v dx = \int_\Omega fv dx$ for all $v \in H_0^1(\Omega)$
  • Elliptic equations: $-\sum_{i,j=1}^n \frac{\partial}{\partial x_i}(a_{ij}(x)\frac{\partial u}{\partial x_j}) + c(x)u = f(x)$ in $\Omega$, $u = 0$ on $\partial \Omega$
    • Weak formulation: Find $u \in H_0^1(\Omega)$ such that $\int_\Omega \sum_{i,j=1}^n a_{ij}(x)\frac{\partial u}{\partial x_i}\frac{\partial v}{\partial x_j} + c(x)uv dx = \int_\Omega fv dx$ for all $v \in H_0^1(\Omega)$

Weak solutions via Sobolev spaces

• Solutions to PDEs in the weak sense, obtained by multiplying the PDE by a test function and integrating by parts
• A function $u \in V$ (a suitable Sobolev space) is a weak solution to a PDE if it satisfies the weak formulation $a(u,v) = \langle f,v \rangle$ for all test functions $v \in V$
• Existence and uniqueness of weak solutions can be proven using functional analysis tools:
  • Lax-Milgram theorem: Guarantees existence and uniqueness for coercive ($a(u,u) \geq \alpha \|u\|_V^2$) and bounded ($|a(u,v)| \leq M \|u\|_V \|v\|_V$) bilinear forms
  • Riesz representation theorem: Establishes the existence of a unique solution $u \in V$ to $\langle f,v \rangle = (u,v)_V$ for all $v \in V$, where $f \in V'$ is a bounded linear functional
• These theorems are applied to the weak formulation of the PDE to prove the existence and uniqueness of weak solutions in the appropriate Sobolev space setting