Singular perturbation of domains and semilinear elliptic equations III

We provide a general approach to the classification results of stable solutions of (possibly nonlinear) elliptic problems with Robin conditions. The method is based on a geometric formula of Poincaré type, which is inspired by a classical work of Sternberg and Zumbrun and which gives an accurate description of the curvatures of the level sets of the stable solutions. From this, we show that the stable solutions of a quasilinear problem with Neumann data are necessarily constant.As a byproduct of this, we obtain an alternative proof of a celebrated result of Casten and Holland, and Matano.In addition, we will obtain as a consequence a new proof of a result recently established by Bandle, Mastrolia, Monticelli and Punzo.

show abstract

“…This gives a contradiction with (30), thus proving (29). Hence, the proof of Theorem 1.5 is completed.…”

Section: Proof Of Theorem 15mentioning

confidence: 75%

“…This and (28) give that ∇u(x) = 0, in contradiction with (26), which proves (25). Now, we let c be the value attained by u along ∂Ω, as given by (25), and we complete the proof of Theorem 1.5, by showing that (29) u is constant in Ω.…”

Section: Proof Of Theorem 15mentioning

confidence: 83%

Rigidity Results for Elliptic Boundary Value Problems: Stable Solutions for Quasilinear Equations with Neumann or Robin Boundary Conditions

Dipierro

Pinamonti

Valdinoci

2018

International Mathematics Research Notices

View full text Add to dashboard Cite

show abstract

“…Convergence problem in the case of tabular region is well studied. Here we refer to [19], [31]. We focus only vertex region.…”

Section: Stationary Nlse On a Fat Graphmentioning

confidence: 99%

“…First we show that ϕ 1 = ϕ 2 = ϕ 3 . According to [31], [19] and maximum principle [32] estimate (see, e.g. [19], [32]) v C 2 (Ω1) C 3 .…”

Section: Stationary Nlse On a Fat Graphmentioning

confidence: 99%

Nonlinear standing waves on planar branched systems: shrinking into metric graph

Sobirov¹,

Babajanov²,

Matrasulov³

2017

Nanosystems: Phys. Chem. Math.

View full text Add to dashboard Cite

We treat the stationary nonlinear Schrödinger equation on two-dimensional branched domains, so-called fat graphs. The shrinking limit when the domain becomes one-dimensional metric graph is studied by using analytical estimate of the convergence of fat graph boundary conditions into those for metric graph. Detailed analysis of such convergence on the basis of numerical solution of stationary nonlinear Schrödinger equation on a fat graph is provided. The possibility for reproducing different metric graph boundary conditions studied in earlier works is shown. Practical applications of the proposed model for such problems as Bose-Einstein condensation in networks, branched optical media, DNA, conducting polymers and wave dynamics in branched capillary networks are discussed.

show abstract

“…The most studied one was the dumbbell, see Fig. 1 ([8], [9], [10], [11], [12], [13]- [14], [15]). Other shapes have been considered in [16]- [17], [13]- [14], [15].…”

Section: Introductionmentioning

confidence: 99%

Simulating Phase Transition Dynamics on Non-trivial Domains

Bolikowski

Gokieli

2014

Parallel Processing and Applied Mathematics

View full text Add to dashboard Cite

Our goal is to investigate the influence of the geometry and topology of the domain Ω on the solutions of the phase transition and other diffusion-driven phenomena in Ω, modeled e.g. by the Allen-Cahn, Cahn-Hilliard, reaction-diffusion equations. We present FEM numerical schemes for the Allen-Cahn and Cahn-Hilliard equation based on the Eyre's algorithm and present some numerical results on split and dumbbell domains.

show abstract

Singular perturbation of domains and semilinear elliptic equations III

Cited by 9 publications

References 0 publications

Rigidity Results for Elliptic Boundary Value Problems: Stable Solutions for Quasilinear Equations with Neumann or Robin Boundary Conditions

Rigidity Results for Elliptic Boundary Value Problems: Stable Solutions for Quasilinear Equations with Neumann or Robin Boundary Conditions

Nonlinear standing waves on planar branched systems: shrinking into metric graph

Simulating Phase Transition Dynamics on Non-trivial Domains

Contact Info

Product

Resources

About