1,721,144 research outputs found
A note on the construction of Hamiltonian trajectories along heteroclinic chains
No full textWe provide a short, simple proof of the existence of Hamiltonian trajectories arbitrarily close to a given chain of heteroclinic orbits connecting "codimension-one, KAM, whiskered tori"
Contemporary PDEs between theory and applications
No full textThis special issue of Discrete and Continuous Dynamical Systems is devoted to some recent developments in some important fields of partial differential equations.
The aim is to bring together several contributions in different fields that range from classical to modern topics with the intent to present new research perspectives, innovative methods and challenging applications.
Though it was of course impossible to take into account all the possible lines of research in PDEs, we tried to present a wide spectrum, hoping to capture the interest of both the general mathematical audience and the specialized mathematicians that work in differential equations and related fields.
We think that the Authors put a great effort to write their contributions in the clearest possible language. We are indeed grateful to all the Authors that contributed to this special issue, donating beautiful pieces of mathematics to the community and promoting further developments in the field.
We also thank the Managing Editor for his kind invitation to act as an editor of this special issue.
Also, we express our gratitude to all the Referees who kindly agreed to devote their time and efforts to read and check all the papers carefully, providing useful comments and recommendations. Indeed, each paper was submitted to the meticulous inspection of two independent and anonymous Experts, whose observations were fundamental to the final outcome of this special issue.
Finally, we would like to wish a `Happy reading!' to the Reader. This volume is for Her (or Him), after all
A Fractional Framework for Perimeters and Phase Transitions
No full textWe review some recent results on minimisers of a non-local perimeter functional, in connection with some phase coexistence models whose diffusion term is given by the fractional Laplacian
Pointwise estimates and monotonicity formulas without maximum principle
No full textWe study a second order elliptic partial differential equation for which a maximum principle is not available and whose nonlinearity is not C1. We discuss the role of a pointwise gradient bound and we derive a monotonicity estimate near flat points of the free boimdary
Geometric properties of Bernoulli-type minimizers
No full textWe consider a Bernoulli-type variational problem and we prove some geometric properties for minimizers, such as: gradient bounds, linear growth from the free boundary, density estimates, uniform convergence of level sets and the existence of plane-like minimizers in periodic media
Geometry of quasiminimal phase transitions
No full textWe consider the quasiminima of the energy functional ∫ Ω A(x, ∇ u) + F(x, u) dx, where A(x, ∇ u) ∼ |∇ u|p and F is a double-well potential. We show that the Lipschitz quasiminima, which satisfy an equipartition of energy condition, possess density estimates of Caffarelli-Cordoba-type, that is, roughly speaking, the complement of their interfaces occupies a positive density portion of balls of large radii. From this, it follows that the level sets of the rescaled quasiminima approach locally uniformly hypersurfaces of quasiminimal perimeter. If the quasiminimum is also a solution of the associated PDE, the limit hypersurface is shown to have zero mean curvature and a quantitative viscosity bound on the mean curvature of the level sets is given. In such a case, some Harnack-type inequalities for level sets are obtained and then, if the limit surface if flat, so are the level sets of the solution
A Brezis-Nirenberg result for non-local critical equations in low dimension
No full textThe present paper is devoted to the study of the following nonlocal fractional equation involving critical nonlinearities { (-δ) ∈u -u = u2-2u in ω u = 0 in Rn n ω where s 2 (0; 1) is fixed, (-δ)s is the fractional Laplace operator, is a positive parameter, 2 is the fractional critical Sobolev exponent and is an open bounded subset of Rn, n > 2s , with Lipschitz boundary. In the recent papers [14, 18, 19] we investigated the existence of non-trivial solutions for this problem when is an open bounded subset of Rn with n > 4s and, in this framework, we prove some existence results. Aim of this paper is to complete the investigation carried on in [14, 18, 19], by considering the case when 2s < n < 4s . In this context, we prove an existence theorem for our problem, which may be seen as a Brezis-Nirenberg type result in low dimension. In particular when s = 1 (and consequently n = 3) our result is the classical result obtained by Brezis and Nirenberg in the famous paper [4] . In this sense the present work may be considered as the extension of some classical results for the Laplacian to the case of non-local fractional operators
Gradient bounds for anisotropic partial differential equations
No full textWe consider solutions in the whole of the space of a partial differential equation driven by the anisotropic Laplacian. We prove a pointwise energy bound and we derive from that some rigidity results. © 2013 Springer-Verlag Berlin Heidelberg
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