Singular perturbations models in phase transitions for anisotropic higher-order materials
Giuseppe Cosma Brusca, Davide Donati, Chiara Trifone · Calculus of Variations and Partial Differential Equations · 2025
Abstract We discuss a model for phase transitions in which a double-well potential is singularly perturbed by possibly several terms involving different, arbitrarily high orders of derivation. We study by $$\Gamma $$ Γ -convergence the asymptotic behaviour as $$\varepsilon \rightarrow 0$$ ε → 0 of the functionals $$\begin{aligned} F_\varepsilon (u):=\int _\Omega \Bigl [\frac{1}{\varepsilon }W(u)+\sum _{\ell =1}^{k}q_\ell \varepsilon ^{2\ell -1}| abla ^{(\ell )}u|_\ell ^2\Bigr ]\,dx, \qquad u\in H^k(\Omega ), \end{aligned}$$ F ε ( u ) : = ∫ Ω [ 1 ε W ( u ) + ∑ ℓ = 1 k q ℓ ε 2 ℓ - 1 | ∇ ( ℓ ) u | ℓ 2 ] d x , u ∈ H k ( Ω ) , for fixed $$k>1$$ k > 1 integer, addressing also the case in which the coefficients $$q_1,...,q_{k-1}$$ q 1 , . . . , q k - 1 are negative and $$|\cdot |_\ell $$ | · | ℓ is any norm on the space of symmetric $$\ell $$