caa a22 4500 605516065 CHVBK 20210128100712.0 cr unu---uuuuu 210128e20151101xx s 000 0 eng 10.1007/s00205-015-0867-9 doi (NATIONALLICENCE)springer-10.1007/s00205-015-0867-9 Uniform Bounds for Strongly Competing Systems: The Optimal Lipschitz Case [Elektronische Daten] [Nicola Soave, Alessandro Zilio] For a class of systems of semi-linear elliptic equations, including $$-\Delta u_i=f_i(x,u_i) - \beta u_i\sum_{j\neq i}a_{ij}u_j^p,\quad i=1,\dots,k,$$ - Δ u i = f i ( x , u i ) - β u i ∑ j ≠ i a i j u j p , 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. Springer-Verlag Berlin Heidelberg, 2015 Soave Nicola Mathematisches Institut, Justus Liebig Universität Giessen, Arndtstrasse 2, 35392, Giessen, Germany aut Zilio Alessandro Centre d'analyse et de mathématique sociales, École des Hautes Études en Sciences Sociales, 190-198 Avenue de France, 75244, Paris Cedex 13, France aut Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 218/2(2015-11-01), 647-697 0003-9527 218:2<647 2015 218 205 https://doi.org/10.1007/s00205-015-0867-9 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00205-015-0867-9 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Soave Nicola Mathematisches Institut, Justus Liebig Universität Giessen, Arndtstrasse 2, 35392, Giessen, Germany aut NATIONALLICENCE 700 1- Zilio Alessandro Centre d'analyse et de mathématique sociales, École des Hautes Études en Sciences Sociales, 190-198 Avenue de France, 75244, Paris Cedex 13, France aut NATIONALLICENCE 773 0- Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 218/2(2015-11-01), 647-697 0003-9527 218:2<647 2015 218 205