arXiv subjects

Alessandro Zilio

Publications and source records attributed to Alessandro Zilio.

22 records · Page 2Linked to original sources

arXiv2013

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 $$(-Δ)^s u_i = f_i(x,u_i) - βu_i^p \sum_{j\neq i} a_{ij} u_j^p,$$ where $i = i,\dots, k$, $s\in(0,1)$, $p>0$, $a_{ij}>0$ and $β>0$. When $k=2$ we develop a quasi-optimal regularity theory in $C^{0,α}$, uniformly w.r.t. $β$, for every $α< α_{\mathrm opt}=min(1,2s)$; moreover we show that the traces of the limiting profiles as $β\to+\infty$ are Lipschitz continuous and segregated. Such results are extended to the case of $k\geq3$ densities, with some restrictions on $s$, $p$ and $a_{ij}$.

math.AP↗

arXiv2013

Uniform Hölder regularity with small exponent in competition-fractional diffusion systems

For a class of competition-diffusion nonlinear systems involving fractional powers of the Laplacian, including as a special case the fractional Gross-Pitaevskii system, we prove that uniform boundedness implies Hölder boundedness for sufficiently small positive exponents, uniformly as the interspecific competition parameter diverges. This implies strong convergence of the family of solutions as the segregation of their support occurs.

math.AP↗

arXiv2013

Entire solutions with exponential growth for an elliptic system modeling phase-separation

We prove the existence of entire solutions with exponential growth for the semilinear elliptic system [\begin{cases} -Δu = -u v^2 & \text{in $\R^N$} -Δv= -u^2 v & \text{in $\R^N$} u,v>0, \end{cases}] for every $N \ge 2$. Our construction is based on an approximation procedure, whose convergence is ensured by suitable Almgren-type monotonicity formulae. The construction of \emph{some} solutions is extended to systems with $k$ components, for every $k > 2$.

math.AP↗

arXiv2012

Uniform Holder 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.

math.AP↗