The interplay between analysis and topology in some nonlinear PDE problems
By Haim Brezis. 2003
47 citations recorded by the source corpus.
The “balayage” method introduced by H. Poincaré in his proof of Theorem 1 relies heavily on tools of Potential theory: maximum principle, Harnack’s inequality, explicit representation formulas for the Dirichlet problem in a ball (Poisson integral), etc. In 1900, D. Hilbert [39], in a celebrated address, followed by a (slightly) more detailed paper in 1904, announced that he had solved the Dirichlet problem (0.1) (0.2) via the Dirichlet principle which had been discovered by G. Green in 1833, with later contributions by C. F. Gauss (1837), W. Thomson (=Lord Kelvin) (1847) and G. Riemann (1853). Dirichlet’s principle asserts that any solution u of (0.1) (0.2) is a minimizer of the Dirichlet integral
Subfields: Computational Theory and Mathematics, Mathematical Physics, Modeling and Simulation
View Paper: https://www.semanticscholar.org/paper/d45f5f7a8796a01a368386acba25e1b3b206a4a0
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Citations in the corpus
- Characterization of Sobolev and BV Spaces (2011, background): “We finally remark that there is a large body of work on related non-local functionals, including papers addressing compactness (see [6], [33]), applications to problems and further questions (see [10], [11]), extended looks at non-radial mollifiers (see [32], [33]), Γ-convergence of non-local functi”
- Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond (2008, background): “The Oseen-Frank theory has been extensively studied in the literature, see the review [5], and there are rigorous results on the existence, regularity and singularities of Oseen-Frank minimizers.”
- Some Historical Remarks on the Positivity of Boundary Integral Operators (2007, background): “is, however, a quote from a recent paper[1] by one of the specialists in the calculus of variations:”