Top Tweets for #boundaryvalueproblem

over 1 year ago

HARMONIC FUNCTIONS, BOUNDARY VALUE PROBLEMS No calculations are required to solve the exercise below — can you figure out why? #Mathematics #partialdifferentialequations #boundaryvalueproblem #laplacian #maximumprinciple Hint. You don’t need to compute the solutions of the two problems. The maximum principle for harmonic functions is one of my favorite theorems (together with the complex version for holomorphic functions). It states that if a function is twice differentiable on an open, bounded, and connected set D, continuous on the closure of D, and harmonic, then the maximum and minimum of the function on the closure of D (which exists by Weierstrass theorem) occur at the boundary. This theorem proves useful in many situations. See for example the exercise below. The maximum of u1 on the boundary is 1, this is also the maximum of u1 on the whole disk. Similarly, the minimum of u2 on the boundary is 4, and this is also the minimum of u2 on disk. So we proved that u1(x,y) <= 1 < 4 <= u2(x,y) for all (x,y) in D.

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De Gruyter Journals➡️ @degruyterbrill.bsky.social

@DGJournals

about 9 years ago

Reduction of Schwartz-type #BoundaryValueProblem for Biharmonic Monogenic Functions to #Fredholm #IntegralEquations https://t.co/6B6Ht7sG30

DGJournals's tweet photo. Reduction of Schwartz-type #BoundaryValueProblem for Biharmonic Monogenic Functions to #Fredholm #IntegralEquations
https://t.co/6B6Ht7sG30 https://t.co/8tqr5JCfkI

Math Stack Exchange @StackMath

over 9 years ago

Bounty offered: Sturm-Liouville-Problem with confusing result https://t.co/n5GSGKRv3A #boundaryvalueproblem

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