Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems
Alberto Farina
Discrete & Continuous Dynamical Systems - A 2015, 35(12): 5869-5877 doi: 10.3934/dcds.2015.35.5869
We prove the symmetry of components and some Liouville-type theorems for, possibly sign changing, entire distributional solutions to a family of nonlinear elliptic systems encompassing models arising in Bose-Einstein condensation and in nonlinear optics. For these models we also provide precise classification results for non-negative solutions. The sharpness of our results is also discussed.
keywords: rigidity and symmetry results. Nonlinear systems
A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature
Alberto Farina Enrico Valdinoci
Discrete & Continuous Dynamical Systems - A 2011, 30(4): 1139-1144 doi: 10.3934/dcds.2011.30.1139
keywords: a-priori estimates. Geometric analysis of PDEs
Some symmetry results for entire solutions of an elliptic system arising in phase separation
Alberto Farina
Discrete & Continuous Dynamical Systems - A 2014, 34(6): 2505-2511 doi: 10.3934/dcds.2014.34.2505
We study the one dimensional symmetry of entire solutions to an elliptic system arising in phase separation for Bose-Einstein condensates with multiple states. We prove that any monotone solution, with arbitrary algebraic growth at infinity, must be one dimensional in the case of two spatial variables. We also prove the one dimensional symmetry for half-monotone solutions, i.e., for solutions having only one monotone component.
keywords: Almgrens monotonicity formulae. Symmetry results for elliptic systems entire solutions
A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature
Diego Castellaneta Alberto Farina Enrico Valdinoci
Communications on Pure & Applied Analysis 2012, 11(5): 1983-2003 doi: 10.3934/cpaa.2012.11.1983
We consider a singular or degenerate elliptic problem in a proper domain and we prove a gradient bound and some symmetry results.
keywords: a priori estimates. Singular and degenerate elliptic pde's

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