arXiv subjects

Panayotis Smyrnelis

Publications and source records attributed to Panayotis Smyrnelis.

21 records · Page 2Linked to original sources

arXiv2016

Theory of light-matter interaction in nematic liquid crystals and the second Painlevé equation

We study global minimizers of an energy functional arising as a thin sample limit in the theory of light-matter interaction in nematic liquid crystals. We show that depending on the parameters various defects are predicted by the model. In particular we show existence of a new type of topological defect which we call the {\it shadow kink}. Its local profile is described by the second Painlevé equation. As part of our analysis we find new solutions to this equation thus generalizing the well known result of Hastings and McLeod.

math.AP↗

arXiv2014

Gradient estimates for semilinear elliptic systems and other related results

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 in some particular cases the Liouville Theorem. 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.

math.AP↗

arXiv2014

Multyphase solutions to the vector Allen-Cahn equation: Crystalline and other complex symmetric structures

We present a systematic study of entire symmetric solutions $u:R^n\rightarrow R^m$ of the vector Allen-Cahn equation $Δu-W_u(u)=0, x \in R^n$, where $W:R^m\rightarrow R$ is smooth, symmetric, nonnegative with a finite number of zeros and $W_u=(\frac{\partial W}{\partial u_1},\ldots,\frac{\partial W}{\partial u_m})^\top$. We introduce a general notion of equivariance with respect to a homomorphism $f:G\rightarrowΓ$ ($G,Γ$ reflection groups) and prove two abstract results, concerning the cases of $G$ finite and $G$ discrete, for the existence of equivariant solutions. Our approach is variational and based on a mapping property of the parabolic vector Allen-Cahn equation and on a pointwise estimate for vector minimizers.

math.AP↗