scholarly journals Correction to: Weighted Energy-Dissipation approach to doubly nonlinear problems on the half line

2021 ◽  
Author(s):  
Goro Akagi ◽  
Stefano Melchionna ◽  
Ulisse Stefanelli
Keyword(s):  
Energy Dissipation ◽  
Nonlinear Problems ◽  
Half Line ◽  
Weighted Energy ◽  
Doubly Nonlinear

scholarly journals Weighted Energy-Dissipation approach to doubly nonlinear problems on the half line

2017 ◽  
Vol 18 (1) ◽  
pp. 49-74 ◽  
Author(s):  
Goro Akagi ◽  
Stefano Melchionna ◽  
Ulisse Stefanelli
Keyword(s):  
Energy Dissipation ◽  
Nonlinear Problems ◽  
Half Line ◽  
Weighted Energy ◽  
Doubly Nonlinear

The weighted energy-dissipation principle and evolutionary Γ-convergence for doubly nonlinear problems

2019 ◽  
Vol 25 ◽  
pp. 36 ◽  
Author(s):  
Matthias Liero ◽  
Stefano Melchionna
Keyword(s):  
Energy Dissipation ◽  
Evolution Equations ◽  
Regularization Parameter ◽  
Nonlinear Problems ◽  
Nonlinear Evolution Equations ◽  
Energy Functionals ◽  
Weighted Energy ◽  
Doubly Nonlinear ◽  
Γ Convergence ◽  
External Forces

We consider a family of doubly nonlinear evolution equations that is given by families of convex dissipation potentials, nonconvex energy functionals, and external forces parametrized by a small parameter ε. For each of these problems, we introduce the so-called weighted energy-dissipation (WED) functional, whose minimizers correspond to solutions of an elliptic-in-time regularization of the target problems with regularization parameter δ. We investigate the relation between the Γ-convergence of the WED functionals and evolutionary Γ-convergence of the associated systems. More precisely, we deal with the limits δ → 0, ε → 0, as well as δ + ε → 0 either in the sense of Γ-convergence of functionals or in the sense of evolutionary Γ-convergence of functional-driven evolution problems, or both. Additionally, we provide some quantitative estimates on the rate of convergence for the limit ε → 0, in the case of quadratic dissipation potentials and uniformly λ-convex energy functionals. Finally, we discuss a homogenization and a dimension reduction problem as examples of application.

scholarly journals Weighted Energy-Dissipation principle for gradient flows in metric spaces

2019 ◽  
Vol 127 ◽  
pp. 1-66 ◽  
Author(s):  
Riccarda Rossi ◽  
Giuseppe Savaré ◽  
Antonio Segatti ◽  
Ulisse Stefanelli
Keyword(s):  
Energy Dissipation ◽  
Metric Spaces ◽  
Gradient Flows ◽  
Weighted Energy

scholarly journals Eigenvalue bifurcation in doubly nonlinear problems with an application to surface plasmon polaritons

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Tomáš Dohnal ◽  
Giulio Romani
Keyword(s):  
Surface Plasmon ◽  
Surface Plasmon Polaritons ◽  
Eigenvalue Problems ◽  
Nonlinear Problems ◽  
Localized Solutions ◽  
Eigenvalue Parameter ◽  
Doubly Nonlinear ◽  
Isolated Eigenvalues ◽  
Plasmon Polaritons ◽  
Nonlinear Eigenvalue

AbstractWe consider a class of generally non-self-adjoint eigenvalue problems which are nonlinear in the solution as well as in the eigenvalue parameter (“doubly” nonlinear). We prove a bifurcation result from simple isolated eigenvalues of the linear problem using a Lyapunov–Schmidt reduction and provide an expansion of both the nonlinear eigenvalue and the solution. We further prove that if the linear eigenvalue is real and the nonlinear problem $${\mathcal {PT}}$$ PT -symmetric, then the bifurcating nonlinear eigenvalue remains real. These general results are then applied in the context of surface plasmon polaritons (SPPs), i.e. localized solutions for the nonlinear Maxwell’s equations in the presence of one or more interfaces between dielectric and metal layers. We obtain the existence of transverse electric SPPs in certain $${\mathcal {PT}}$$ PT -symmetric configurations.

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