Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries

By Luis Caffarelli, F. Lin. 2008

166 citations recorded by the source corpus.

Here we study the asymptotic limits of solutions of some singularly perturbed elliptic systems. The limiting problems involve multiple valued harmonic functions or, in general, harmonic maps to singular spaces and free interfaces between supports of various components of the maps. The main results of the paper are the uniform Lipschitz regularity of solutions as well as the regularity of free interfaces.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/b21ed801f80734e6bc595f210b1a9e3020e61b32

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case (2015, background): “For other kinds of elliptic systems with strong competition, phase separation has also been well studied, we refer to [10,11,17] and references therein.”
Some new results in competing systems with many species (2010, background): “This result is valid for locally energy minimizing maps too, because it also satisfy the same conditions such as monotonicity of the frequency function (see [3]).”
Uniform Bounds for Strongly Competing Systems: The Optimal Lipschitz Case (2014, background): “Concerning the regularity of the limit configurations and of their free-boundary, we mainly refer to [7] for the variational setting, to [6] for the symmetric one, and in particular to [27], which provides a unified approach investigating the regularity of a wide class of segregated vector valued fu”
A minimization problem with free boundary related to a cooperative system (2016, background): “nce supui= supuj for some of these i,j.2 Those gi that are still 1Competitive systems, which gives rise to disjoint support of limiting solutions, have been much in focus in the last decade (see e.g. [10], [11]). Competitive system of more than two equations usually give rise to the so-called juncti”
De Giorgi type results for elliptic systems (2012, background): “olds: |∇u|2 ≤ 2H(u). (53) Note that this is an extension of the pointwise estimate that Modica [13] proved in the case of a single equation. It is still not known for systems, though Caﬀarelli-Lin in [9] and later, Alikakos in [2] have shown, in the case where H≥ 0, the following weaker monotonicity”
Positive Least Energy Solutions and Phase Separation for Coupled Schrödinger Equations with Critical Exponent (2012, background): “For other kinds of elliptic systems with strong competition, phase separation has also been well studied, we refer to [10, 11, 14] and references therein.”