Gradient estimates for elliptic systems with measurable coefficients in nonsmooth domains

2010 ◽  
Vol 133 (1-2) ◽  
pp. 225-245 ◽  
Author(s):  
Sun-Sig Byun ◽  
Seungjin Ryu ◽  
Lihe Wang
Keyword(s):  
Elliptic Systems ◽  
Gradient Estimates ◽  
Measurable Coefficients ◽  
Nonsmooth Domains

scholarly journals Gradient estimates for semilinear elliptic systems and other related results

2015 ◽  
Vol 145 (6) ◽  
pp. 1313-1330 ◽  
Author(s):  
Panayotis Smyrnelis
Keyword(s):  
Liouville Theorem ◽  
Elliptic Systems ◽  
Gradient Estimates ◽  
Alternative Form ◽  
Strong Monotonicity ◽  
Energy Tensor ◽  
Planar Domains ◽  
Stress Energy ◽  
General Phase ◽  
Double Well Potential

A periodic connection is constructed for a double well potential defined in the plane. This solution violates Modica's estimate as well as the corresponding Liouville theorem for general phase transition potentials. Gradient estimates are also established for several kinds of elliptic systems. They allow us to prove the Liouville theorem in some particular cases. Finally, we give an alternative form of the stress–energy tensor for solutions defined in planar domains. As an application, we deduce a (strong) monotonicity formula.

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