What are some common techniques and tools for proving weak lower semicontinuity of functionals?

Weak lower semicontinuity (WLC) is an important property of functionals that arise in variational problems, such as optimization, calculus of variations, and partial differential equations. A functional is WLC if its value at the limit of a sequence of functions is less than or equal to the limit of its values at the sequence. In other words, the functional does not increase when the functions converge weakly. In this article, we will discuss some common techniques and tools for proving WLC of functionals in the context of functional analysis.

1 Weak convergence and weak topology

One of the key concepts for WLC is weak convergence, which is a weaker notion of convergence than the usual norm convergence. A sequence of functions converges weakly to a function if it converges to the function in the sense of the dual pairing, that is, for every linear and continuous functional, the value of the functional at the sequence converges to the value of the functional at the limit. Weak convergence implies that the sequence is bounded in norm, but not vice versa. Weak convergence is related to the weak topology, which is the coarsest topology that makes all linear and continuous functionals continuous. The weak topology is often useful for proving compactness and continuity results.

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Abhiram Sripat

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Proving weak lower semicontinuity of functionals is a fundamental task in the calculus of variations and functional analysis. Common techniques include the direct methods, which involve using the definition of weak lower semicontinuity and working directly with sequences converging weakly. Another approach is leveraging convexity; proving that a functional is convex often leads to establishing its weak lower semicontinuity since convex functionals possess this property. Coercivity is another useful technique, where demonstrating that the functional is coercive can aid in proving lower semicontinuity...

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Abhiram Sripat

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..Sequential characterization is also employed, which uses the principle that a functional is weakly lower semicontinuous if it satisfies a sequential lower semicontinuity condition. The tools essential for these techniques include properties of Sobolev spaces, particularly the Rellich-Kondrachov compactness theorem, which are frequently utilized. A solid understanding of weak topology and weak convergence principles is crucial. Compactness arguments, such as those provided by the Banach-Alaoglu theorem, are also leveraged to handle weak convergence scenarios. Mazur's Lemma is another critical tool, used to approximate weakly convergent sequences with strongly convergent convex combinations.

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2 Lower semicontinuity and coercivity

Another key concept for WLC is lower semicontinuity, which is a generalization of continuity to functions that may not be defined everywhere. A function is lower semicontinuous if for every point in its domain and every sequence that converges to that point, the value of the function at the limit is less than or equal to the limit inferior of the values of the function at the sequence. Lower semicontinuity is a weaker property than continuity, but it still preserves some important features, such as the existence of minima and the subadditivity of convex functions. A special case of lower semicontinuity is coercivity, which means that the function tends to infinity as the norm of the argument tends to infinity. Coercivity is often a sufficient condition for WLC.

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Abhiram Sripat

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Proving weak lower semicontinuity of functionals is essential in variational calculus. Common techniques include leveraging convexity, as convex functionals are inherently weakly lower semicontinuous, and using the direct method in the calculus of variations, which involves showing that a minimizing sequence is precompact. Sequential characterization verifies that the functional value at any weak limit point is less than or equal to the limit infimum. Coercivity ensures the functional grows sufficiently at infinity, aiding in the precompactness of sequences. Utilizing Sobolev spaces and compactness theorems, as well as Mazur's Lemma, further supports these proofs.

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3 Direct methods and relaxation

One of the common techniques for proving WLC is the direct method, which is based on the principle of the minimum energy. The idea is to show that the functional is lower semicontinuous and coercive on a convex and weakly closed set, and then use the Banach-Alaoglu theorem to prove the existence of a minimizer. The Banach-Alaoglu theorem states that every bounded set in a normed space is weakly compact, that is, every sequence in the set has a weakly convergent subsequence. The direct method can be applied to a wide range of functionals, such as the Dirichlet energy, the Sobolev norm, and the p-Laplacian. Sometimes, the functional may not be lower semicontinuous or coercive on the original set, but it can be modified by a process called relaxation, which involves extending the functional to a larger set and minimizing its lower semicontinuous envelope.

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Abhiram Sripat

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The direct method involves showing that a sequence of functions converges weakly to a limit function, ensuring the functional is lower semicontinuous at this limit. Relaxation involves approximating a non-lower semicontinuous functional with a lower semicontinuous one, typically by using convexification techniques. Historically, these methods have roots in the calculus of variations and the study of partial differential equations. They relate to broader mathematics by providing foundational tools for analyzing optimization problems, stability in physical systems, and solutions to differential equations in various applied fields.

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4 Semicontinuity theorems and compactness

Another common technique for proving WLC is to use semicontinuity theorems, which are results that link the lower semicontinuity of a functional to some compactness or convergence properties of its arguments. For example, one of the simplest semicontinuity theorems is the sequential lower semicontinuity theorem, which states that a functional is WLC if and only if it is sequentially lower semicontinuous, that is, for every weakly convergent sequence in its domain, the value of the functional at the limit is less than or equal to the limit inferior of the values of the functional at the sequence. Another example is the Mazur lemma, which states that every weakly convergent sequence can be approximated by a norm convergent sequence with the same limit. This lemma can be used to prove WLC of continuous functionals. There are many other semicontinuity theorems that involve different types of compactness and convergence, such as weak* convergence, strong convergence, compact embeddings, and compact operators.

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Abhiram Sripat

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These are essential concepts in mathematical analysis, particularly in the calculus of variations and functional analysis. Semicontinuity theorems help determine the behavior of functionals under limit processes, ensuring lower semicontinuity, which is crucial for proving the existence of minimizers in optimization problems. Compactness, often achieved through the use of compact embedding theorems, ensures that bounded sequences have convergent subsequences. This interplay between semicontinuity and compactness facilitates the application of the direct method in the calculus of variations, allowing for the effective analysis of minimization problems, stability in physical systems, and solutions to differential equations

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5 Convexity and subdifferential

A final common technique for proving WLC is to use convexity and subdifferential, which are tools that describe the shape and behavior of convex functions. A function is convex if for every pair of points in its domain and every number between zero and one, the value of the function at the convex combination of the points is less than or equal to the convex combination of the values of the function at the points. Convexity implies lower semicontinuity and subadditivity, and also allows for various optimization methods and duality principles. The subdifferential of a convex function at a point is the set of all linear and continuous functionals that are less than or equal to the function at every point in its domain. The subdifferential characterizes the direction and rate of increase of the function at the point, and also provides necessary and sufficient conditions for optimality. The subdifferential can be used to prove WLC of convex functionals by showing that the subdifferential at the limit contains the limit of a sequence of subdifferentials at the sequence.

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Abhiram Sripat

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Convexity and subdifferentials are key concepts in optimization and functional analysis. Convexity ensures that a functional's value at any point on the line segment between two points is less than or equal to the weighted average of its values at those points. This property simplifies the analysis and solution of optimization problems. The subdifferential extends the concept of derivatives to convex functionals, representing a set of slopes that define supporting hyperplanes at a given point. These concepts are crucial for understanding the structure of optimization problems, allowing for the development of efficient algorithms and providing insights into the stability and sensitivity of solutions.

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6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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Abhiram Sripat

Can Artify anything | Founder & CEO at Florence Quantum Labs | Quantum Biology | Quantum Biosensing | Photonics | Nanotechnology | Quantum Machine Learning
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Exploring the interplay between convexity and subdifferentials can deepen understanding of modern optimization techniques. Further readings include "Convex Analysis" by R. Tyrrell Rockafellar and "Convex Optimization" by Stephen Boyd and Lieven Vandenberghe, which provide comprehensive insights into these concepts. Recent research focuses on extending these ideas to non-convex settings, stochastic optimization, and machine learning applications. Investigating the latest papers on arXiv.org and journals like the "Journal of Optimization Theory and Applications" can offer updates on cutting-edge developments and practical applications

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What are some common techniques and tools for proving weak lower semicontinuity of functionals?

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1 Weak convergence and weak topology

2 Lower semicontinuity and coercivity

3 Direct methods and relaxation

4 Semicontinuity theorems and compactness

5 Convexity and subdifferential

6 Here’s what else to consider

Functional Analysis

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