Harmonic functions | Mathematical principles | Partial differential equations

Maximum principle

In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. In the simplest case, consider a function of two variables u(x,y) such that The weak maximum principle, in this setting, says that for any open precompact subset M of the domain of u, the maximum of u on the closure of M is achieved on the boundary of M. The strong maximum principle says that, unless u is a constant function, the maximum cannot also be achieved anywhere on M itself. Such statements give a striking qualitative picture of solutions of the given differential equation. Such a qualitative picture can be extended to many kinds of differential equations. In many situations, one can also use such maximum principles to draw precise quantitative conclusions about solutions of differential equations, such as control over the size of their gradient. There is no single or most general maximum principle which applies to all situations at once. In the field of convex optimization, there is an analogous statement which asserts that the maximum of a convex function on a compact convex set is attained on the boundary. (Wikipedia).

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Maximum principle for PDE

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From playlist Partial differential equations

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Maximum Principle

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From playlist Partial Differential Equations

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Maximum and Minimum

Maximum and Minimum of a set In this video, I define the maximum and minimum of a set, and show that they don't always exist. Enjoy! Check out my Real Numbers Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCZggpJZvUXnUzaw7fHCtoh

From playlist Real Numbers

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Maximum modulus principle

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From playlist Complex Analysis

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Calculus: Absolute Maximum and Minimum Values

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From playlist Calculus

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From playlist 241Fall13Ex3

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From playlist Calculus Pt 1: Limits and Derivatives

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From playlist Limits

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From playlist Applications of Differentiation

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From playlist Mechanical Engineering

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Differential Equations 1 - Maximum Principle: Oxford Mathematics 2nd Year Student Lecture

This is the final lecture from Melanie Rupflin's Second Year Differential Equations Course. In this lecture we prove the parabolic maximum principle and discuss how it can be used also for nonlinear versions of the heat equation to control the behaviour of solutions in terms of the initial

From playlist Oxford Mathematics Student Lectures - Differential Equations

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From playlist Mechanical Engineering

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Alejandro Morales: "Asymptotics of principal evaluations of Schubert polynomials"

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From playlist Asymptotic Algebraic Combinatorics 2020

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ch11 7. Heat equation, CFL stability condition for explicit forward Euler method. Wen Shen

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From playlist CMPSC/MATH 451 Videos. Wen Shen, Penn State University

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Complex analysis: Maximum modulus principle

This lecture is part of an online undergraduate course on complex analysis. We prove the maximum modulus principle, and use to to prove the fundamental theorem of algebra and to find the symmetries of the unit disk. For the other lectures in the course see https://www.youtube.com/playli

From playlist Complex analysis

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From playlist Mathematics

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From playlist Course 6: Introduction to Analysis (Fall 2017)

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Convex function | Convex optimization | Elliptic partial differential equation | Geometric analysis | Parabolic partial differential equation | Spectral theorem | Pontryagin's maximum principle | Hopf maximum principle | Gradient | Boundary (topology) | Maximum modulus principle | Convex set | Olga Ladyzhenskaya