We obtain some Poincaré type formulas, that we use, together with the level set analysis, to detect the one-dimensional symmetry of monotone and stable solutions of possibly degenerate elliptic systems of the form div (a (jruj)ru) = F1(u; v); div (b (jrvj)rv) = F2(u; v); where F 2 C1;1 loc (R2). Our setting is very general, and it comprises, as a particular case, a conjec- ture of De Giorgi for phase separations in R2.
Geometric inequalities and symmetry results for elliptic systems / S. Dipierro. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. - ISSN 1078-0947. - 33:8(2013), pp. 3473-3496.
Geometric inequalities and symmetry results for elliptic systems
S. Dipierro
2013
Abstract
We obtain some Poincaré type formulas, that we use, together with the level set analysis, to detect the one-dimensional symmetry of monotone and stable solutions of possibly degenerate elliptic systems of the form div (a (jruj)ru) = F1(u; v); div (b (jrvj)rv) = F2(u; v); where F 2 C1;1 loc (R2). Our setting is very general, and it comprises, as a particular case, a conjec- ture of De Giorgi for phase separations in R2.
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