Double Phase Variational Problems in Nonlinear Analysis

Double phase variational problems arise from energy functionals whose integrands switch between two distinct growth regimes. Typically one considers integrals of the form ∫Ω(|∇u|^p + a(x)|∇u|^q) dx, where 1

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Technical terms

Double phase functional: An integral energy characterised by two power-law growth regimes, switching between p- and q-powers of the gradient according to a spatial weight function.

Nehari manifold: A constraint set in variational calculus defined by the vanishing of the directional derivative of the energy along rays, used to locate nontrivial critical points with prescribed sign or nodal structure.

Sobolev–Orlicz space: A generalised Sobolev space built upon Orlicz spaces, allowing for energy densities with nonstandard growth conditions beyond polynomial exponents, and equipped with modular topology.

References

1. On double phase Kirchhoff problems with singular nonlinearity. Advances in Nonlinear Analysis (2023).
2. Regularity for minimizers for functionals of double phase with variable exponents. Advances in Nonlinear Analysis (2019).
3. New embedding results for double phase problems with variable exponents and a priori bounds for corresponding generalized double phase problems. Calculus of Variations and Partial Differential Equations (2023).