2024/11/03 by Margherita Solci, Solci, Margherita · 1 citation
Engineering · #35B25 #35R11 #49J45 #82B26 #Analysis of PDEs (math.AP) #FOS: Mathematics #Heat Transfer and Mathematical Modeling #Material Science and Thermodynamics #Thermoelastic and Magnetoelastic Phenomena
paper · pdf · doi:10.48550/arxiv.2411.01586
openalex publication_date 2024/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the asymptotic behaviour of double-well energies perturbed by a higher-order fractional term, which, in the one-dimensional case, take the form (1)/(ε)∫I W(u(x))dx+ε2(k+s)-1\fracs(1-s)21-s∫I× I \frac|u(k)(x)-u(k)(y)|2|x-y|1+2s dx dy defined on the higher-order fractional Sobolev space Hk+s(I), where W is a double-well potential, k∈ \mathbb N and s∈(0,1) with k+s>\frac12. We show that these functionals Γ-converge as ε→ 0 to a sharp-interface functional with domain BV(I;\-1,1\) of the form mk+s#(S(u)), with mk+s given by the optimal-profile problem mk+s =inf\∫\mathbb R W(v)dx+\fracs(1-s)21-s∫\mathbb R2\frac|v(k)(x)-v(k)(y)|2|x-y|1+2s dx dy : v∈ Hk+s\rm loc(\mathbb R), limx→±∞v(x)=±1\. The normalization coefficient \fracs(1-s)21-s is such that mk+s interpolates continuously the corresponding mk defined on standard higher-order Sobolev space Hk(I), obtained by Modica and Mortola in the case k=1, Fonseca and Mantegazza in the case k=2 and Brusca, Donati and Solci for k≥ 3. The results also extends previous works by Alberti, Bouchitté and Seppecher, Savin and Valdinoci, and Palatucci and Vincini, in the case k=0 and s∈(\frac12,1).