Glossary

2.4 Variational methods

6 min read•august 20, 2024

Variational methods are powerful tools for solving partial differential equations. They involve finding functions that minimize or maximize certain quantities, called functionals, subject to specific constraints or boundary conditions. These methods provide a framework for analyzing the existence, uniqueness, and regularity of solutions to various types of PDEs.

The is a key component of variational methods, providing a necessary condition for a function to be a stationary point of a given . Other important concepts include the , calculus of variations, , and . These techniques are applied in numerous fields, from physics to image processing.

Variational principles

  • play a fundamental role in the study of partial differential equations (PDEs) and their solutions
  • These principles involve finding functions that minimize or maximize certain quantities, known as functionals, subject to specific constraints or boundary conditions
  • Variational methods provide a powerful framework for analyzing the existence, uniqueness, and regularity of solutions to various types of PDEs

Variational methods for PDEs

Euler-Lagrange equation

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  • The Euler-Lagrange equation is a necessary condition for a function to be a stationary point (minimizer or maximizer) of a given functional
  • It is derived by setting the first variation of the functional equal to zero
  • Solutions to the Euler-Lagrange equation are called extremals and often correspond to solutions of the associated PDE
  • The Euler-Lagrange equation takes the form ∂L∂u−ddx(∂L∂u′)=0\frac{\partial L}{\partial u} - \frac{d}{dx}\left(\frac{\partial L}{\partial u'}\right) = 0, where L(x,u,u′)L(x, u, u') is the

Existence and uniqueness of solutions

  • Variational methods can be used to establish the to certain classes of PDEs
  • The of calculus of variations involves showing that a minimizing sequence of functions converges to a function that minimizes the functional
  • Uniqueness of solutions can often be proved using the strict convexity of the functional or the monotonicity of the associated operator
  • The Lax-Milgram theorem provides conditions for the existence and uniqueness of weak solutions to linear elliptic PDEs

Dirichlet principle

Dirichlet energy functional

  • The measures the "smoothness" of a function and is defined as E[u]=12∫Ω∣∇u∣2dxE[u] = \frac{1}{2}\int_\Omega |\nabla u|^2 dx
  • Minimizing the Dirichlet energy functional subject to certain boundary conditions leads to
  • The Dirichlet energy functional plays a central role in the theory of harmonic functions and potential theory

Minimizers and harmonic functions

  • Functions that minimize the Dirichlet energy functional are called harmonic functions
  • Harmonic functions satisfy the Δu=0\Delta u = 0 and have important properties such as the mean value property and the maximum principle
  • The Dirichlet principle states that the solution to the Dirichlet problem (finding a harmonic function with prescribed boundary values) can be obtained by minimizing the Dirichlet energy functional

Calculus of variations

Functionals and variations

  • Functionals are mappings that assign a real number to each function in a certain function space
  • Examples of functionals include the Dirichlet energy functional, the area functional, and the length functional
  • The first variation of a functional J[u]J[u] is defined as δJ[u;v]=lim⁡ε→0J[u+εv]−J[u]ε\delta J[u; v] = \lim_{\varepsilon \to 0} \frac{J[u + \varepsilon v] - J[u]}{\varepsilon} and measures the rate of change of the functional with respect to small perturbations vv

Necessary conditions for extrema

  • The first variation of a functional must vanish at an extremal point (minimizer or maximizer)
  • The second variation of a functional can be used to determine the nature of an extremal point (minimum, maximum, or saddle point)
  • The Legendre condition and the Jacobi condition are necessary conditions for a function to be a minimizer of a functional
  • The is a sufficient condition for a function to be a minimizer of a functional

Variational inequalities

Obstacle problem

  • The is a variational inequality that arises in the study of elasticity, fluid mechanics, and other areas
  • It involves finding a function that minimizes a certain energy functional subject to the constraint that the function lies above a given obstacle function
  • The solution to the obstacle problem satisfies a variational inequality and has a free boundary, which is the set where the solution touches the obstacle

Free boundary problems

  • are a class of variational problems where the domain of the solution is not known a priori and must be determined as part of the solution
  • Examples of free boundary problems include the obstacle problem, the Stefan problem (melting or solidification), and the Hele-Shaw flow
  • The regularity and the structure of the free boundary are important questions in the study of free boundary problems

Gamma convergence

Definition and properties

  • Gamma convergence is a notion of convergence for functionals that is well-suited for studying the limit behavior of variational problems
  • A sequence of functionals FnF_n is said to Gamma converge to a functional FF if for every sequence unu_n converging to uu, we have lim inf⁡n→∞Fn[un]≥F[u]\liminf_{n \to \infty} F_n[u_n] \geq F[u] and there exists a sequence unu_n converging to uu such that lim sup⁡n→∞Fn[un]≤F[u]\limsup_{n \to \infty} F_n[u_n] \leq F[u]
  • Gamma convergence is stable under continuous perturbations and has compactness properties

Applications to variational problems

  • Gamma convergence can be used to study the asymptotic behavior of and minimum values of variational problems
  • It provides a framework for deriving effective models and homogenization results for materials with microstructure
  • Gamma convergence has been applied to problems in elasticity, phase transitions, image processing, and other areas

Numerical methods

Finite element method

  • The (FEM) is a numerical technique for solving PDEs by discretizing the domain into a mesh of elements (triangles, tetrahedra, etc.)
  • The solution is approximated by a linear combination of basis functions (usually piecewise polynomials) defined on the elements
  • The coefficients of the basis functions are determined by solving a system of linear equations obtained from the weak formulation of the PDE

Galerkin approximations

  • Galerkin methods are a class of numerical methods for solving variational problems and PDEs
  • The idea is to approximate the solution by a linear combination of basis functions and determine the coefficients by requiring that the residual is orthogonal to the space spanned by the basis functions
  • The finite element method is a particular case of the Galerkin method, where the basis functions are chosen to be piecewise polynomials on a mesh

Regularity theory

Hölder continuity

  • Hölder continuity is a stronger notion of continuity that measures the modulus of continuity of a function
  • A function uu is said to be Hölder continuous with exponent α∈(0,1]\alpha \in (0, 1] if there exists a constant CC such that ∣u(x)−u(y)∣≤C∣x−y∣α|u(x) - u(y)| \leq C|x - y|^\alpha for all x,yx, y in the domain
  • Solutions to elliptic PDEs often enjoy Hölder continuity, which can be proved using the De Giorgi-Nash-Moser theory

Higher regularity of solutions

  • Under suitable assumptions on the coefficients and the domain, solutions to elliptic PDEs can have higher regularity (smoothness)
  • The Schauder estimates provide bounds on the Hölder norms of derivatives of solutions in terms of the Hölder norms of the data
  • The Calderón-Zygmund theory gives LpL^p estimates for solutions to PDEs with discontinuous coefficients
  • Bootstrapping arguments can be used to derive higher regularity of solutions from lower regularity estimates

Nonlinear variational problems

Mountain pass theorem

  • The is a result in critical point theory that ensures the existence of a critical point (saddle point) of a functional under certain geometric conditions
  • It requires the functional to satisfy a compactness condition (the Palais-Smale condition) and to have a "mountain pass" geometry (two valleys separated by a mountain range)
  • The mountain pass theorem has been widely used to prove the existence of solutions to nonlinear PDEs, such as semilinear elliptic equations and systems

Existence via minimax principles

  • Minimax principles are a powerful tool for proving the existence of critical points (and hence solutions) of functionals
  • The idea is to construct a minimax level by taking the infimum of the functional over a certain class of sets and then showing that this level is a critical value
  • Examples of minimax principles include the mountain pass theorem, the saddle point theorem, and the linking theorem
  • These principles have been successfully applied to a wide range of nonlinear PDEs, including Hamiltonian systems, Schrödinger equations, and variational inequalities

Key Terms to Review (30)

Direct Method: The direct method is a technique in variational methods where one seeks to minimize a functional directly, often leading to the solution of a boundary value problem. This approach focuses on constructing minimizing sequences and demonstrating their convergence to a minimizer without needing to rely on a prior weak formulation or abstract framework, thereby providing a more straightforward route to finding solutions.
Dirichlet Boundary Conditions: Dirichlet boundary conditions are a type of constraint used in mathematical problems, particularly in partial differential equations, where the solution is required to take on specific values on the boundary of the domain. This concept is crucial for defining the behavior of physical systems and ensuring that solutions are well-posed. These conditions can be applied in various scenarios, including heat conduction, fluid flow, and electrostatics, establishing a foundation for solving boundary value problems effectively.
Dirichlet Energy Functional: The Dirichlet energy functional is a mathematical concept that quantifies the energy associated with a function defined on a domain, often used in variational methods to find functions that minimize this energy. It provides a way to analyze smoothness and variation of functions, often connecting physical concepts like potential energy to the minimization of energy states in mathematical models. This functional plays a crucial role in determining optimal solutions and understanding how variations affect the overall energy of a system.
Dirichlet Principle: The Dirichlet Principle states that if a function is harmonic within a certain domain and continuous on the boundary, then it achieves its minimum and maximum values on the boundary of that domain. This principle connects variational methods to potential theory by demonstrating how boundary conditions can influence the behavior of harmonic functions.
Euler-Lagrange Equation: The Euler-Lagrange equation is a fundamental equation in the calculus of variations that provides a necessary condition for a function to be an extremum of a functional. It connects the variation of a functional to the derivatives of the function, forming the backbone of variational methods used to find optimal solutions in physics and engineering.
Existence and Uniqueness of Solutions: Existence and uniqueness of solutions refers to the mathematical conditions under which a given problem, typically involving differential equations or variational methods, has a solution that not only exists but is also unique. This concept ensures that for a specified set of initial or boundary conditions, there is exactly one solution that satisfies the problem, which is crucial for both theoretical and practical applications in variational methods.
Finite Element Method: The finite element method (FEM) is a numerical technique used to find approximate solutions to boundary value problems for partial differential equations. It divides a large problem into smaller, simpler parts called finite elements, which are then analyzed to reconstruct the overall solution. This method is especially powerful for solving complex problems in various fields, including mechanics, heat transfer, and fluid dynamics.
Free boundary problems: Free boundary problems involve determining an unknown boundary or interface where a phase change occurs, such as the shape of a liquid droplet or the free surface of a fluid. These problems often arise in physics and engineering, requiring mathematical techniques to describe how the boundaries evolve over time in relation to governing equations.
Functional: In mathematics and physics, a functional is a special type of mapping that takes a function as input and produces a scalar output. It often plays a critical role in variational methods, which involve optimizing functionals to find functions that minimize or maximize certain criteria, leading to solutions of various problems in calculus of variations.
Galerkin Approximations: Galerkin approximations are a method used in numerical analysis and applied mathematics to convert a continuous problem, often a differential equation, into a discrete problem by projecting the continuous equations onto a finite-dimensional subspace. This technique involves selecting a set of basis functions and using them to approximate the solution, leading to a system of algebraic equations that can be solved more easily. By using Galerkin approximations, one can ensure that the residual error is minimized in the sense of an inner product, which makes it highly effective in variational methods.
Gamma convergence: Gamma convergence is a concept in the calculus of variations and functional analysis that describes the convergence of a sequence of functions in a specific way, focusing on their minimization properties. It connects closely to variational methods by providing a framework for understanding how minimizing sequences of functionals behave as they approach a limit, ensuring that certain properties are preserved under this convergence. This notion is fundamental in establishing conditions under which limits of minimization problems yield meaningful solutions.
Green's function: Green's function is a fundamental solution used to solve inhomogeneous linear differential equations subject to specific boundary conditions. It acts as a tool to express solutions to problems involving harmonic functions, allowing the transformation of boundary value problems into integral equations and simplifying the analysis of physical systems.
Harmonic functions: Harmonic functions are continuous functions that satisfy Laplace's equation, which states that the sum of the second partial derivatives of the function equals zero. These functions have important properties, such as being infinitely differentiable and exhibiting mean value behavior, making them crucial in various mathematical contexts, including boundary value problems and potential theory.
Joseph-Louis Lagrange: Joseph-Louis Lagrange was an 18th-century mathematician and astronomer known for his contributions to various fields, including mechanics, number theory, and calculus of variations. His work laid the groundwork for variational methods, which are essential in optimizing functionals and understanding the behavior of physical systems.
Lagrangian Function: The Lagrangian function is a mathematical formulation used to describe the dynamics of a system in classical mechanics, represented as the difference between kinetic and potential energy. It plays a crucial role in variational methods by allowing for the derivation of equations of motion through the principle of least action, leading to insights into the behavior of physical systems.
Laplace Equation: The Laplace Equation is a second-order partial differential equation given by $$ abla^2 u = 0$$, where $$u$$ is a scalar function and $$ abla^2$$ is the Laplacian operator. It describes the behavior of harmonic functions and is fundamental in various fields such as physics and engineering, particularly in potential theory. The solutions to the Laplace Equation provide critical insights into various physical phenomena, including gravitational and electrostatic fields, heat conduction, and fluid flow.
Laplace Operator: The Laplace operator, denoted as $$ abla^2$$, is a second-order differential operator that calculates the divergence of the gradient of a function. It plays a key role in various areas of mathematics and physics, especially in the study of harmonic functions and potential theory, where it helps to characterize properties of solutions to partial differential equations.
Maximization problem: A maximization problem is a type of optimization problem where the goal is to find the maximum value of a certain function, subject to given constraints. This concept is crucial in variational methods, as it often involves finding extremal points of functionals, which can represent physical systems or processes. Maximization problems are commonly encountered in various fields, such as economics, engineering, and physics, where they can be used to optimize resources or outcomes.
Minimization Problem: A minimization problem is an optimization problem that seeks to find the minimum value of a function, often subject to certain constraints. These problems are significant in various fields such as physics, engineering, and economics, and are closely tied to concepts like energy minimization and the stability of systems.
Minimizers: Minimizers are functions or configurations that yield the lowest possible value of a given functional, often arising in optimization problems where the goal is to find a state of least energy or cost. This concept plays a crucial role in variational methods, as these techniques often seek to identify minimizers that satisfy specific boundary conditions or constraints.
Mountain Pass Theorem: The Mountain Pass Theorem is a fundamental result in the calculus of variations and critical point theory that provides conditions under which a function attains a critical point in a constrained domain. It is often used to find minimizers or critical points of functionals, particularly in scenarios where traditional minimization techniques may fail due to the lack of lower bounds or specific topological constraints.
Necessary Conditions for Extrema: Necessary conditions for extrema refer to the criteria that must be satisfied for a function to achieve a local maximum or minimum at a certain point. These conditions help identify critical points where a function's derivative is either zero or undefined, serving as potential candidates for extremal values. Understanding these conditions is crucial in variational methods, as they provide insights into optimizing functions and finding the best possible solutions within given constraints.
Neumann Boundary Conditions: Neumann boundary conditions specify the values of the derivative of a function on the boundary of a domain, rather than the values of the function itself. These conditions are essential in variational methods, particularly when dealing with problems involving partial differential equations, as they allow for the modeling of physical situations where the flux or gradient is known at the boundary.
Obstacle Problem: The obstacle problem refers to a type of variational problem where the goal is to minimize a functional subject to certain constraints imposed by obstacles in the domain. This concept is crucial in understanding how functions behave when restricted by obstacles, as it explores the balance between minimizing energy and adhering to physical constraints. The solutions to this problem often arise in applications such as physics, materials science, and engineering, highlighting its significance in modeling real-world phenomena.
Richard Feynman: Richard Feynman was a renowned American theoretical physicist known for his work in quantum mechanics and quantum electrodynamics. His innovative approaches to teaching and problem-solving made complex ideas accessible, and his contributions to the field have had a lasting impact on physics, particularly in relation to variational methods which seek to find approximations to complex physical systems.
Strong convergence: Strong convergence refers to a type of convergence of sequences in a normed space where the sequence converges to a limit in such a way that the norms of the differences go to zero. This concept is crucial in mathematical analysis as it ensures that the convergence is robust, often leading to desirable properties in variational methods and weak solutions, making it essential for applications in optimization and partial differential equations.
Variational Inequalities: Variational inequalities are mathematical expressions that generalize the concept of inequalities, typically involving an unknown function and a differential operator. They arise in various applications, including optimization and control problems, where solutions must satisfy certain constraints. These inequalities provide a framework to study boundary value problems and optimal control within a variational approach.
Variational Principles: Variational principles are fundamental concepts in mathematics and physics that involve finding extrema (maximum or minimum values) of functionals, which are functions that map functions to real numbers. These principles provide powerful methods for solving problems in optimization, mechanics, and other fields by translating them into finding critical points of a functional, often leading to differential equations that describe the system's behavior.
Weak convergence: Weak convergence refers to a type of convergence for a sequence of functions or measures, where the sequence converges in the sense that it preserves certain integrals against a fixed test function, even if the functions themselves do not converge pointwise. This concept is important because it allows for a more generalized notion of convergence, especially in functional analysis and probability, facilitating the analysis of equilibrium measures, variational methods, and weak solutions.
Weierstrass Condition: The Weierstrass Condition is a fundamental criterion used in calculus of variations to determine whether a given functional has an extremum. It essentially states that for a functional to reach its extremum, it must satisfy certain regularity and boundary conditions. This condition plays a critical role in ensuring that the extremal functions are well-defined and lead to meaningful physical interpretations.
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