Uniform Bounds for Strongly Competing Systems: The Optimal Lipschitz Case

By Nicola Soave, Alessandro Zilio. 2014

68 citations recorded by the source corpus.

AbstractFor a class of systems of semi-linear elliptic equations, including $$-\Delta u_i=f_i(x,u_i) - \beta u_i\sum_{j\neq i}a_{ij}u_j^p,\quad i=1,\dots,k,$$-Δui=fi(x,ui)-βui∑j≠iaijujp,i=1,⋯,k,for p = 2 (variational-type interaction) or p = 1 (symmetric-type interaction), we prove that uniform $${L^\infty}$$L∞ boundedness of the solutions implies uniform boundedness of their Lipschitz norm as $${\beta \to +\infty}$$β→+∞, that is, in the limit of strong competition. This extends known quasi-optimal regularity results and covers the optimal case for this class of problems. The proofs rest on monotonicity formulae of Alt–Caffarelli–Friedman and Almgren type in the variational setting, and on the Caffarelli–Jerison–Kenig almost monotonicity formula in the symmetric one.

Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/61e199f031ec7f75b4433cf71086beba0b0d15fa

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Control and Systems Engineering

1. Thomas Bartsch
2. Nicola Soave
3. Wenming Zou
4. Houwang Li
5. Louis Jeanjean

In Mathematical Physics

1. Thomas Bartsch
2. Nicola Soave
3. Louis Jeanjean
4. Wenming Zou
5. Houwang Li
6. Xinfu Li
7. Gianmaria Verzini
8. R. Molle

In Statistical and Nonlinear Physics

1. Thomas Bartsch
2. Nicola Soave
3. Louis Jeanjean
4. Xinfu Li
5. Wenming Zou
6. Houwang Li

Citations in the corpus

• A natural constraint approach to normalized solutions of nonlinear Schr\"odinger equations and systems (2016, background): “At this point, the rest of the proof follows directly by the general theory developed in [34, 43, 44, 49].”