Existence of lattice solutions to semilinear elliptic systems with periodic potential
Under the assumption that the potential W is invariant under a general discrete reflection group $G'=TG$ acting on $mathbb{R}^n$, we establish existence of G'-equivariant solutions to $Delta u - W_u(u) = 0$, and find an estimate. By taking the size of the cell of the lattice in space d...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Texas State University
2012-01-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2012/15/abstr.html |
| _version_ | 1828948092604383232 |
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| author | Nicholas D. Alikakos Panayotis Smyrnelis |
| author_facet | Nicholas D. Alikakos Panayotis Smyrnelis |
| author_sort | Nicholas D. Alikakos |
| collection | DOAJ |
| description | Under the assumption that the potential W is invariant under a general discrete reflection group $G'=TG$ acting on $mathbb{R}^n$, we establish existence of G'-equivariant solutions to $Delta u - W_u(u) = 0$, and find an estimate. By taking the size of the cell of the lattice in space domain to infinity, we obtain that these solutions converge to G-equivariant solutions connecting the minima of the potential W along certain directions at infinity. When particularized to the nonlinear harmonic oscillator $u''+alpha sin u=0$, $alpha>0$, the solutions correspond to those in the phase plane above and below the heteroclinic connections, while the G-equivariant solutions captured in the limit correspond to the heteroclinic connections themselves. Our main tool is the G'-positivity of the parabolic semigroup associated with the elliptic system which requires only the hypothesis of symmetry for W. The constructed solutions are positive in the sense that as maps from $mathbb{R}^n$ into itself leave the closure of the fundamental alcove (region) invariant. |
| first_indexed | 2024-12-14T05:41:32Z |
| format | Article |
| id | doaj.art-22f507a61d0049549da744a0c34239d4 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-14T05:41:32Z |
| publishDate | 2012-01-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-22f507a61d0049549da744a0c34239d42022-12-21T23:15:01ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912012-01-01201215,115Existence of lattice solutions to semilinear elliptic systems with periodic potentialNicholas D. AlikakosPanayotis SmyrnelisUnder the assumption that the potential W is invariant under a general discrete reflection group $G'=TG$ acting on $mathbb{R}^n$, we establish existence of G'-equivariant solutions to $Delta u - W_u(u) = 0$, and find an estimate. By taking the size of the cell of the lattice in space domain to infinity, we obtain that these solutions converge to G-equivariant solutions connecting the minima of the potential W along certain directions at infinity. When particularized to the nonlinear harmonic oscillator $u''+alpha sin u=0$, $alpha>0$, the solutions correspond to those in the phase plane above and below the heteroclinic connections, while the G-equivariant solutions captured in the limit correspond to the heteroclinic connections themselves. Our main tool is the G'-positivity of the parabolic semigroup associated with the elliptic system which requires only the hypothesis of symmetry for W. The constructed solutions are positive in the sense that as maps from $mathbb{R}^n$ into itself leave the closure of the fundamental alcove (region) invariant.http://ejde.math.txstate.edu/Volumes/2012/15/abstr.htmlLattice solutioninvariantperiodic Potentialelliptic systemreflectiondiscrete groupvariational calculus |
| spellingShingle | Nicholas D. Alikakos Panayotis Smyrnelis Existence of lattice solutions to semilinear elliptic systems with periodic potential Electronic Journal of Differential Equations Lattice solution invariant periodic Potential elliptic system reflection discrete group variational calculus |
| title | Existence of lattice solutions to semilinear elliptic systems with periodic potential |
| title_full | Existence of lattice solutions to semilinear elliptic systems with periodic potential |
| title_fullStr | Existence of lattice solutions to semilinear elliptic systems with periodic potential |
| title_full_unstemmed | Existence of lattice solutions to semilinear elliptic systems with periodic potential |
| title_short | Existence of lattice solutions to semilinear elliptic systems with periodic potential |
| title_sort | existence of lattice solutions to semilinear elliptic systems with periodic potential |
| topic | Lattice solution invariant periodic Potential elliptic system reflection discrete group variational calculus |
| url | http://ejde.math.txstate.edu/Volumes/2012/15/abstr.html |
| work_keys_str_mv | AT nicholasdalikakos existenceoflatticesolutionstosemilinearellipticsystemswithperiodicpotential AT panayotissmyrnelis existenceoflatticesolutionstosemilinearellipticsystemswithperiodicpotential |