1,721,243 research outputs found
Overdetermined problems for the fractional Laplacian in exterior and annular sets
Full text linkWe consider a fractional elliptic equation in an unbounded set with both Dirichlet and fractional normal derivative datum prescribed. We prove that the domain and the solution are necessarily radially symmetric. We also study the extension of the result in bounded non-convex regions, as well as the radial symmetry of the solution when the set is assumed a priori to be rotationally symmetric
(Dis)connectedness of nonlocal minimal surfaces in a cylinder and a stickiness property
No full textWe consider nonlocal minimal surfaces in a cylinder with prescribed datum given by the complement of a slab. We show that when the width of tthe slab is large the minimizers are disconnected and when the width of the slab is small the minimizers are connected. This feature is in agreement with the classical case of the minimal surfaces. Nevertheless, we show that when the width of the slab is large the minimizers are not fflat discs, as it happens in the classical setting, and, in particular, in dimension 2 we provide a quantitative bound on the stickiness property exhibited by the minimizers. Moreover, differently from the classical case, we show that when the width of the slab is small then the minimizers completely adhere to the side of the cylinder, thus providing a further example of stickiness phenomenon
Quantitative flatness results and -estimates for stable nonlocal minimal surfaces
Full text linkWe establish quantitative properties of minimizers and stable sets for nonlocal interaction functionals, including the s-fractional perimeter as a particular case.On the one hand, we establish universal BV-estimates in every dimension n >= 2 for stable sets. Namely, we prove that any stable set in B-1 has finite classical perimeter in B-1/2, with a universal bound. This nonlocal result is new even in the case of s-perimeters and its local counterpart (for classical stable minimal surfaces) was known only for simply connected two-dimensional surfaces immersed in R-3.On the other hand, we prove quantitative flatness estimates for minimizers and stable sets in low dimensions n = 2, 3. More precisely, we show that a stable set in B-R, with R large, is very close in measure to being a half space in B-1 - with a quantitative estimate on the measure of the symmetric difference. As a byproduct, we obtain new classification results for stable sets in the whole plane
Prescribed conditions at infinity for fractional parabolic and elliptic equations with unbounded coefficients
Full text linkWe investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed point-wise conditions at infinity (in space), which can be time-dependent. Moreover, we study the asymptotic behavior of such solutions. We also consider solutions of elliptic equations satisfying appropriate conditions at infinity
The Phillip island penguin parade (A mathematical treatment)
Full text linkPenguins are flightless, so they are forced to walk while on land. In particular, they show rather specific behaviours in their homecoming, which are interesting to observe and to describe analytically. We observed that penguins have the tendency to waddle back and forth on the shore to create a sufficiently large group, and then walk home compactly together. The mathematical framework that we introduce describes this phenomenon, by taking into account "natural parameters", such as the eyesight of the penguins and their cruising speed. The model that we propose favours the formation of conglomerates of penguins that gather together, but, on the other hand, it also allows the possibility of isolated and exposed individuals.
The model that we propose is based on a set of ordinary differential equations. Due to the discontinuous behaviour of the speed of the penguins, the mathematical treatment (to get existence and uniqueness of the solution) is based on a "stop-and-go" procedure. We use this setting to provide rigorous examples in which at least some penguins manage to safely return home (there are also cases in which some penguins remain isolated). To facilitate the intuition of the model, we also present some simple numerical simulations that can be compared with the actual movement of the penguin parade
Lipschitz regularity of almost minimizers in one-phase problems driven by the p-Laplace operator
Full text linkWe prove that, given~p>\max\left\{\frac{2n}{n+2},1\right\}, the nonnegative
almost minimizers of the nonlinear free boundary functional
J_p(u,\Omega):=\int_{\Omega}\Big( |\nabla u(x)|^p+\chi_{\{u>0\}}(x)\Big)\,dx
are Lipschitz continuous
Uniqueness in weighted Lebesgue spaces for a class of fractional parabolic and elliptic equations
No full textWe investigate uniqueness, in suitable weighted Lebesgue spaces, of solutions to a class of fractional parabolic and elliptic equations
Singularity Formation in Fractional Burgers’ Equations
No full textThe formation of singularities in finite time in nonlocal Burgers’ equations, with time-fractional derivative, is studied in detail. The occurrence of finite-time singularity is proved, revealing the underlying mechanism, and precise estimates on the blowup time are provided. The employment of the present equation to model a problem arising in job market is also analyzed
A New Lotka-Volterra Model of Competition With Strategic Aggression
No full textThis monograph introduces a new mathematical model in population dynamics that describes two species sharing the same environmental resources in a situation of open hostility. Its main feature is the expansion of the family of Lotka-Volterra systems by introducing a new term that defines aggression. Because the model is flexible, it can be applied to various scenarios in the context of human populations, such as strategy games, competition in the marketplace, and civil wars.
Drawing from a variety of methodologies within dynamical systems, ODEs, and mathematical biology, the authors' approach focuses on the dynamical properties of the system. This is accomplished by detecting and describing all possible equilibria, and analyzing the strategies that may lead to the victory of the aggressive population. Techniques typical of two-dimensional dynamical systems are used, such as asymptotic behaviors regulated by the Poincaré–Bendixson Theorem
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