We study the spectral theory of mixed local and nonlocal operators with lower-order terms in the right-hand side of the equation. This kind of problems is motivated by the analysis of superposition operators of mixed order and with the "wrong sign" of the lower-order terms with respect to the classical elliptic theory. Our results include: -convergence to classical cases when the right-hand side of the eigenvalye equations "localizes", recovering the simplicity and sign-definiteness of eigenfunctions in the limit; -a detailed analysis of disconnected domains, showing that, unlike the classical case, any eigenfunction associated with the first eigenvalue must change sign, and that the first eigenvalue of a union of disconnected domains is strictly smaller than that of its individual components; -examples in which the first eigenvalue is either simple or non-simple in disconnected domains; -a regularity theory that underpins these results.
We analyze fractional phase-transition energies with periodic heterogeneities at the critical H1/2 scaling. We prove that the Γ-limit is a sharp-interface functional whose surface energy density combines homogenization and averaging effects. The resulting coefficient is a weighted combination of the minimum and the mean of the oscillatory parameter, reflecting the coexistence of multiple interaction scales. This behavior is specific to the critical regime and differs from the case s > 1/2.
In this paper we prove some new Strichartz estimates related to the Cauchy problem for the Bessel operator on the half-line and we establish a fractal version of the Tomas-Stein restriction theorem for the Hankel transform. Then we use the proved Strichartz estimates to show global in time well-posedness for a class of nonlinear La2-critical problems, and local in time well-posedness in the sub-critical case.
A photovoltaic (PV)-based multi-generation system applied to a hotel building is considered, and a comfortaware design and assessment framework is proposed, in which occupant thermal and visual comfort requirements are explicitly incorporated into the evaluation of energy, economic, techno-economic, and environmental indicators. Moving beyond conventional frameworks in which occupant comfort is either neglected, treated as a secondary assessment criterion, or simplified through static indoor-condition assumptions, thermal comfort (represented by the Predicted Mean Vote (PMV)) and visual comfort (expressed through indoor illuminance level (Lux)) are introduced as active operational variables. The integrated system supplies electricity, hydrogen, and methanol as a green fuel while utilizing captured carbon dioxide, thereby establishing a coupled energy-fuel-carbon pathway. The adopted methodology integrates one-dimensional parametric analysis with two-dimensional heatmap visualization to reveal both the separate and interactive effects of thermal and visual comfort targets on system sizing and performance. The results show that comfort requirements substantially affect the system size, with progressively larger capacities observed at higher Lux and lower PMV levels. For the considered case-study, which is a 100-unit hotel building complex in Honolulu, Hawai'i, USA, the variation ranges are from 1130 kW to 1515 kW for PV capacity, from 987 kW to 1324 kW for proton exchange membrane electrolyzer (PEMEC) capacity, from 161 kW to 215 kW for proton exchange membrane fuel cell (PEMFC) capacity, and from 6842 m3 to 9202 m3 for the hydrogen storage tank volume. The captured CO2 also changes between 96.0 and 128.7 ton/year. Moreover, from the economic perspective, an optimum is identified at PMVset = 0 and Luxset = 300 lx, where the system achieves a maximum benefit-to-cost ratio (BCR) of 2.54 and a minimum payback period (PBP) of 6.04 years. Taken together, the investigation demonstrates that visual comfort has a greater influence on electricity demand, component sizing, and captured CO2, whereas thermal comfort more strongly governs economic performance.
