Glossary

12.2 Sobolev spaces and weak solutions of PDEs

3 min read•july 22, 2024

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 (Ω). They're equipped with norms that make them , and for p=2, they become , ideal for many applications.

Sobolev Spaces

Definition of Sobolev spaces

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  • Generalize the notion of differentiability to function spaces denoted as [Wk,p](https://www.fiveableKeyTerm:wk,p)(Ω)[W^{k,p}](https://www.fiveableKeyTerm:w^{k,p})(\Omega)
    • kk represents the order of weak derivatives (highest order of derivatives considered)
    • pp represents the integrability of the function and its weak derivatives (Lebesgue space LpL^p)
    • Ω\Omega is the domain of the functions (open subset of Rn\mathbb{R}^n)
  • Contain functions with weak derivatives up to order kk that belong to Lp(Ω)L^p(\Omega)
  • Equipped with the ∥u∥Wk,p(Ω)=(∑∣α∣≤k∥Dαu∥Lp(Ω)p)1/p\|u\|_{W^{k,p}(\Omega)} = (\sum_{|\alpha| \leq k} \|D^\alpha u\|_{L^p(\Omega)}^p)^{1/p}, making them Banach spaces
    • For p=2p=2, Sobolev spaces are Hilbert spaces denoted as [Hk](https://www.fiveableKeyTerm:hk)(Ω)[H^k](https://www.fiveableKeyTerm:h^k)(\Omega)
  • Possess important properties such as (smooth functions are dense), embedding (continuous embeddings between Sobolev spaces), and (restrictions of functions to the boundary)

Concept of weak derivatives

  • Extend the notion of classical derivatives to less regular functions
  • A function u∈[Lloc1](https://www.fiveableKeyTerm:lloc1)(Ω)u \in [L^1_{loc}](https://www.fiveableKeyTerm:l^1_{loc})(\Omega) has a v∈Lloc1(Ω)v \in L^1_{loc}(\Omega) if ∫Ωu∂φ∂xidx=−∫Ωvφdx\int_\Omega u \frac{\partial \varphi}{\partial x_i} dx = -\int_\Omega v \varphi dx for all test functions φ∈Cc∞(Ω)\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)
  • of a PDE is posed in a Sobolev space setting
    • Solution space and space are chosen based on the regularity requirements of the PDE
  • Examples of PDEs solved using Sobolev spaces:
    • Poisson equation: −Δu=f-\Delta u = f in Ω\Omega, u=0u = 0 on ∂Ω\partial \Omega
      • Weak formulation: Find u∈H01(Ω)u \in H_0^1(\Omega) such that ∫Ω∇u⋅∇vdx=∫Ωfvdx\int_\Omega \nabla u \cdot \nabla v dx = \int_\Omega fv dx for all v∈H01(Ω)v \in H_0^1(\Omega)
    • Elliptic equations: −∑i,j=1n∂∂xi(aij(x)∂u∂xj)+c(x)u=f(x)-\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 Ω\Omega, u=0u = 0 on ∂Ω\partial \Omega
      • Weak formulation: Find u∈H01(Ω)u \in H_0^1(\Omega) such that ∫Ω∑i,j=1naij(x)∂u∂xi∂v∂xj+c(x)uvdx=∫Ωfvdx\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∈H01(Ω)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∈Vu \in V (a suitable Sobolev space) is a to a PDE if it satisfies the weak formulation a(u,v)=⟨f,v⟩a(u,v) = \langle f,v \rangle for all test functions v∈Vv \in V
  • Existence and uniqueness of weak solutions can be proven using functional analysis tools:
    • : Guarantees existence and uniqueness for coercive (a(u,u)≥α∥u∥V2a(u,u) \geq \alpha \|u\|_V^2) and bounded (∣a(u,v)∣≤M∥u∥V∥v∥V|a(u,v)| \leq M \|u\|_V \|v\|_V) bilinear forms
    • : Establishes the existence of a unique solution u∈Vu \in V to ⟨f,v⟩=(u,v)V\langle f,v \rangle = (u,v)_V for all v∈Vv \in V, where f∈V′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

Key Terms to Review (36)

Banach spaces: A Banach space is a complete normed vector space, which means it is a vector space equipped with a norm that allows for the measurement of distances, and every Cauchy sequence in the space converges to an element within that space. This concept is essential in functional analysis as it provides a framework for discussing convergence, continuity, and the structure of various mathematical functions. The completeness and the properties of the norm facilitate the analysis of linear operators and play a significant role in solving equations, especially in relation to weak solutions and Sobolev spaces.
Brezis: Brezis refers to the foundational contributions of Haim Brezis in the field of functional analysis, particularly regarding Sobolev spaces and weak solutions of partial differential equations (PDEs). His work helped establish rigorous mathematical frameworks for understanding the behavior of functions that are not necessarily smooth, enabling the analysis of solutions to PDEs in a broader context than traditional methods allowed.
C_c^{\infty}: The space $c_c^{\infty}$ consists of all smooth functions that have compact support, meaning they are infinitely differentiable and vanish outside of a compact set. This space is crucial in the context of distribution theory and is used extensively in the study of partial differential equations, as it provides a suitable setting for the definition of distributions and weak derivatives.
Density: In the context of Sobolev spaces and weak solutions of PDEs, density refers to the property that a subset of a space is dense if every point in the space can be approximated arbitrarily closely by points from that subset. This concept is crucial when dealing with functions in Sobolev spaces, where certain approximations or weak solutions can be approached through smoother or more regular functions, enabling analysis and existence proofs for PDEs.
Dirichlet Boundary Condition: A Dirichlet boundary condition specifies the value of a function on a boundary of the domain, providing a fixed constraint. This type of boundary condition is crucial in various mathematical contexts, particularly in eigenvalue problems where it helps define the solution space of differential equations. It also plays a significant role in weak formulations of partial differential equations, allowing for the specification of solution behavior at the boundaries.
Dirichlet problem: The Dirichlet problem is a type of boundary value problem where one seeks to find a function that solves a specified partial differential equation (PDE) within a domain, subject to given values on the boundary of that domain. It is particularly important in the context of harmonic functions and relates closely to Sobolev spaces, which are essential for understanding weak solutions of PDEs.
Embedding Theorem: The Embedding Theorem refers to a result in functional analysis that provides conditions under which a Sobolev space can be continuously embedded into another function space, typically a space of continuous functions. This theorem is crucial for understanding how weak solutions of partial differential equations behave and ensuring that these solutions possess additional regularity properties.
Embedding Theorems: Embedding theorems are mathematical results that establish conditions under which one functional space can be continuously and densely embedded into another. These theorems are significant in understanding how different Sobolev spaces relate to one another, especially when analyzing weak solutions of partial differential equations (PDEs). They help characterize the behavior of functions in different spaces, allowing for the transfer of properties like compactness and continuity between them.
Existence Theorems: Existence theorems are mathematical statements that assert the existence of solutions to a given problem, particularly in the context of differential equations and functional analysis. These theorems often provide conditions under which solutions can be guaranteed, offering a foundation for understanding weak solutions and Sobolev spaces, which are crucial in solving partial differential equations (PDEs). By establishing the criteria for existence, these theorems help mathematicians understand the behavior of solutions even when explicit forms are difficult to obtain.
H^k: The term h^k refers to a specific Sobolev space, denoted as W^{k,p}(Ω), which includes functions that possess weak derivatives up to order k that are in L^p(Ω). This concept is crucial in analyzing weak solutions to partial differential equations (PDEs), as it allows us to study the regularity and existence of solutions by incorporating the behavior of functions under certain norms.
Hilbert Spaces: Hilbert spaces are complete inner product spaces that generalize the concept of Euclidean space to infinite dimensions. They provide a fundamental framework for studying various mathematical concepts, including functional analysis, quantum mechanics, and partial differential equations. The properties of Hilbert spaces allow for a rigorous treatment of convergence and orthogonality, making them essential in the understanding of weak solutions and the Closed Graph Theorem.
L^1_{loc}: The space l^1_{loc} consists of locally integrable functions, which means that a function belongs to this space if it is integrable over every compact subset of its domain. This concept is crucial when dealing with Sobolev spaces and weak solutions of partial differential equations, as many functions that arise in these contexts may not be globally integrable but are still manageable on smaller scales.
L^p: The term l^p refers to a family of spaces consisting of all sequences whose p-th power is summable. More specifically, for a sequence (x_n), it belongs to the space l^p if the series $$\sum_{n=1}^{\infty} |x_n|^p$$ converges. This concept is crucial in functional analysis, particularly when dealing with Sobolev spaces and weak solutions of partial differential equations (PDEs), as it helps characterize functions in terms of their integrability and behavior under various norms.
Lax-Milgram Theorem: The Lax-Milgram Theorem provides a powerful framework for establishing the existence and uniqueness of solutions to certain types of linear operator equations, particularly those involving unbounded operators in Hilbert spaces. It essentially states that if a bilinear form is continuous and coercive, then there exists a unique solution to the associated linear problem. This theorem is crucial for understanding how weak formulations arise, especially in the context of differential equations and Sobolev spaces.
Neumann Boundary Condition: The Neumann boundary condition specifies the value of the derivative of a function on the boundary of its domain, often representing the flux or gradient at that boundary. This type of condition is crucial in various mathematical problems, especially in relation to differential equations and variational methods, as it helps define how solutions behave at the edges of the domain. It plays an important role in applications such as heat transfer, fluid dynamics, and potential theory.
Neumann Problem: The Neumann Problem is a type of boundary value problem in partial differential equations where the solution is sought along with its normal derivative on the boundary of the domain. This problem is essential in understanding how to solve certain physical phenomena, particularly when dealing with heat conduction or fluid flow, where the rate of change at the boundary is as crucial as the value itself.
Poincaré Inequality: The Poincaré inequality is a fundamental result in functional analysis that establishes a relationship between the integral of a function and the integral of its gradient over a given domain. It essentially states that the average deviation of a function from its mean can be controlled by the average of its gradient, providing crucial estimates for functions in Sobolev spaces, particularly in the context of weak solutions of partial differential equations.
Regularity Theory: Regularity theory is a branch of mathematical analysis that investigates the smoothness properties of weak solutions to partial differential equations (PDEs). It connects the spaces of weak solutions, like Sobolev spaces, to classical solutions, exploring how regularity can be achieved under certain conditions. This theory is essential in understanding how solutions behave and can provide insights into their continuity and differentiability.
Riesz Representation Theorem: The Riesz Representation Theorem states that every continuous linear functional on a Hilbert space can be represented as an inner product with a fixed element from that space. This theorem connects linear functionals to geometry and analysis, showing how functional behavior can be understood in terms of vectors and inner products.
Sobolev: Sobolev refers to a class of function spaces that extend the concept of classical derivatives to weak derivatives, allowing for the analysis of functions that may not be differentiable in the traditional sense. These spaces are fundamental in studying partial differential equations (PDEs), as they enable the formulation of weak solutions, which are essential when classical solutions do not exist.
Sobolev Embedding Theorem: The Sobolev Embedding Theorem establishes a relationship between Sobolev spaces and continuous function spaces, showing that functions in certain Sobolev spaces can be embedded into L^p spaces for various p. This theorem is essential in understanding the regularity of weak solutions to partial differential equations (PDEs) and plays a critical role in the analysis of distributions and generalized functions.
Sobolev norm: The Sobolev norm is a mathematical tool used to measure the size of functions in Sobolev spaces, which are spaces that consider not only the functions themselves but also their derivatives up to a certain order. This norm allows for the analysis of weak solutions of partial differential equations (PDEs) by capturing both the function's behavior and its regularity properties. Understanding Sobolev norms is essential for studying the existence, uniqueness, and stability of solutions to PDEs.
Sobolev Norm: The Sobolev norm is a mathematical tool used to measure the size or smoothness of functions in Sobolev spaces, which are spaces of functions equipped with a norm that accounts for both the function itself and its derivatives. This norm is crucial when analyzing weak solutions of partial differential equations (PDEs) because it allows us to handle functions that may not be classically differentiable but still possess some regularity. The Sobolev norm plays a pivotal role in the study of functional analysis and PDEs by providing a framework to ensure the existence and uniqueness of solutions under weaker conditions than traditional methods would require.
Strong Convergence: Strong convergence refers to a type of convergence in a normed space where a sequence of elements converges to a limit in the sense that the norm of the difference between the sequence and the limit approaches zero. This concept is crucial when dealing with bounded linear operators, as it ensures stability and continuity in various mathematical settings, including Banach spaces and Hilbert spaces.
Test function: A test function is a smooth function that has compact support, used in various areas of analysis, particularly in the context of distributions and Sobolev spaces. These functions play a critical role in defining and understanding weak derivatives, enabling the formulation of weak solutions to partial differential equations (PDEs) through integration by parts. Test functions are essential because they allow us to extend the notion of differentiation to functions that may not be classically differentiable.
Trace Theorem: The Trace Theorem states that if a function is in a Sobolev space, its restriction to a lower-dimensional subset retains certain properties of continuity and integrability. This theorem connects Sobolev spaces, which are crucial in the study of weak solutions to partial differential equations, to boundary value problems by allowing functions to be evaluated at lower dimensions without losing their essential characteristics.
Trace Theorems: Trace theorems provide essential tools in functional analysis, particularly concerning Sobolev spaces, by establishing how certain functionals can be evaluated on lower-dimensional subsets of their domain. They offer a way to connect the properties of functions defined in higher-dimensional spaces to their behavior on boundaries or lower-dimensional manifolds, which is critical when dealing with partial differential equations and weak solutions. This is especially important in analyzing how solutions behave at the edges of their domains.
Variational formulation: Variational formulation is a mathematical approach that translates a problem, often a partial differential equation (PDE), into a minimization or maximization problem involving functionals. This method connects to weak solutions and Sobolev spaces by allowing the use of less regular functions, facilitating the study of PDEs in settings where traditional solutions may not exist. It provides a framework for finding approximate solutions using techniques from calculus of variations.
W^{k,p}: The term w^{k,p} refers to a specific type of Sobolev space that consists of functions whose weak derivatives up to order k are in the L^p space. This means that these functions not only possess certain regularity properties but also satisfy integrability conditions. The importance of w^{k,p} lies in its application to the study of weak solutions of partial differential equations, as it allows for the inclusion of functions that may not be classically differentiable but still exhibit desirable analytical features.
Weak Convergence: Weak convergence refers to a type of convergence in a topological vector space where a sequence converges to a limit if it converges with respect to every continuous linear functional. This concept is crucial for understanding the behavior of sequences in various mathematical structures, particularly in the context of functional analysis and applications in areas like differential equations and optimization.
Weak derivative: A weak derivative is a generalization of the classical derivative that allows for the differentiation of functions that may not be differentiable in the traditional sense. Instead of requiring pointwise differentiability, weak derivatives are defined through integration by parts, making them suitable for Sobolev spaces, where functions can have weak derivatives even if they are not smooth. This concept is crucial for understanding weak solutions of partial differential equations, as it provides a framework for working with functions that arise in various applications without strict differentiability requirements.
Weak formulation: Weak formulation refers to a way of expressing the solution of a partial differential equation (PDE) that relaxes the traditional requirements for differentiability. Instead of requiring solutions to be classically differentiable, weak formulations allow for functions that may not be smooth but still satisfy the PDE in an 'averaged' sense when integrated against test functions, often leading to solutions in Sobolev spaces.
Weak solution: A weak solution is a function that satisfies a partial differential equation (PDE) not in the classical sense, but rather in an integral form. This concept is crucial in the context of Sobolev spaces, where functions may not possess traditional derivatives, allowing for broader applications and solutions to equations that might be otherwise unsolvable using classical methods. Weak solutions often arise in situations where regularity conditions of the solution may be relaxed, leading to better existence and uniqueness results for various types of PDEs.
Weak solution: A weak solution is a generalized solution to a differential equation that may not be continuously differentiable but satisfies the equation in an 'averaged' sense when integrated against test functions. This concept allows for the inclusion of solutions that exhibit irregular behavior or are less regular than classical solutions, which is particularly important in the study of partial differential equations (PDEs) and Sobolev spaces.
Weak solution to the heat equation: A weak solution to the heat equation is a function that satisfies the equation in an integral sense rather than pointwise, allowing for solutions that may not be continuously differentiable. This approach extends the concept of a solution by incorporating Sobolev spaces, which provide a framework for analyzing functions and their derivatives in a generalized manner. The significance of weak solutions lies in their ability to exist even when classical solutions may not be found, particularly in cases with irregular or non-smooth data.
Weak solution to the Laplace equation: A weak solution to the Laplace equation is a function that satisfies the equation in a weak sense, meaning it fulfills the equation's requirements when integrated against test functions, rather than requiring pointwise differentiation. This approach allows for solutions that may not be classically differentiable but still exhibit relevant properties, connecting to Sobolev spaces, which provide a framework for handling such functions and their behavior under various conditions.
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