Alessandro Zilio

Uniform Hölder bounds for strongly competing systems involving the square root of the laplacian

For a class of competition-diffusion nonlinear systems involving the square root of the Laplacian, including the fractional Gross-Pitaevskii system, we prove that uniform boundedness implies Holder boundedness for every exponent less than 1/2, uniformly as the interspecific competition parameter diverges. Moreover we prove that the limiting profile is Holder continuous of exponent 1/2. This system arises, for instance, in the relativistic Hartree-Fock approximation theory for mixtures of Bose-Einstein condensates in different hyperfine states.

Hierarchical Model Reduction: Three Different Approaches

We present three different approaches to model, in a computationally cheap way, problems characterized by strong horizontal dynamics, even though in the presence of transverse heterogeneities. The three approaches are based on the hierarchical model reduction setting introduced in Ern et al. (Hierarchical model reduction for advection-diffusion-reaction problems. In: Kunisch K, Of G, Steinbach O (eds) Numerical mathematics and advanced applications. Springer (2008), pp 703–710) and Perotto et al. (Multiscale Model Simul 8(4):1102–1127, 2010).

Strong Competition versus Fractional Diffusion: The Case of Lotka-Volterra Interaction

We consider a system of differential equations with nonlinear Steklov boundary conditions, related to the fractional problem where u = (u 1,…, u k ), s ∈ (0, 1), p > 0, a ij > 0 and β > 0. When k = 2 we develop a quasi-optimal regularity theory in 𝒞0, α, uniformly w.r.t. β, for every α < αopt = min (1, 2s); moreover we show that the traces of the limiting profiles as β → + ∞ are Lipschitz continuous and segregated. Such results are extended to the case of k ≥ 3 densities, with some restrictions on s, p and a ij . Since for competition of variational type the optimal regularity is known to be , these results mark a substantial difference with the case of standard diffusion s = 1, where the two competitions cannot be distinguished from each other in the limit.

Space-time adaptive hierarchical model reduction for parabolic equations.

BackgroundSurrogate solutions and surrogate models for complex problems in many fields of science and engineering represent an important recent research line towards the construction of the best trade-off between modeling reliability and computational efficiency. Among surrogate models, hierarchical model (HiMod) reduction provides an effective approach for phenomena characterized by a dominant direction in their dynamics. HiMod approach obtains 1D models naturally enhanced by the inclusion of the effect of the transverse dynamics. MethodsHiMod reduction couples a finite element approximation along the mainstream with a locally tunable modal representation of the transverse dynamics. In particular, we focus on the pointwise HiMod reduction strategy, where the modal tuning is performed on each finite element node. We formalize the pointwise HiMod approach in an unsteady setting, by resorting to a model discontinuous in time, continuous and hierarchically reduced in space (c[M(

Optimal Regularity Results Related to a Partition Problem Involving the Half-Laplacian

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Variational Problems with Long-Range Interaction

M)G(s)]-dG(q) approximation). The selection of the modal distribution and of the space–time discretization is automatically performed via an adaptive procedure based on an a posteriori analysis of the global error. The final outcome of this procedure is a table, named HiMod lookup diagram, that sets the time partition and, for each time interval, the corresponding 1D finite element mesh together with the associated modal distribution.Results The results of the numerical verification confirm the robustness of the proposed adaptive procedure in terms of accuracy, sensitivity with respect to the goal quantity and the boundary conditions, and the computational saving. Finally, the validation results in the groundwater experimental setting are promising.Conclusion The extension of the HiMod reduction to an unsteady framework represents a crucial step with a view to practical engineering applications. Moreover, the results of the validation phase confirm that HiMod approximation is a viable approach.

A nonlinear Steklov problem arising in corrosion modeling

For the system of semilinear elliptic equations \[ \Delta V_i = V_i \sum_{j \neq i} V_j^2, \qquad V_i > 0 \qquad \text{in

Uniform Bounds for Strongly Competing Systems: The Optimal Lipschitz Case

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