1,721,083 research outputs found
(Quasi)periodic solutions in (in)finite dimensional Hamiltonian systems with applications to celestial mechanics and wave equation
No full textFractional Laplacian equations with critical Sobolev exponent
No full textIn this paper we complete the study of a critical elliptic equation, driven by a general non-local integrodifferential operator and depending on a real parameter, started by Servadei and Valdinoci (Commun Pure Appl Anal 12( 6): 24452464, 2013). Under suitable growth condition on the nonlinearity, we show that this problem admits non-trivial solutions for any positive parameter. This existence theorem extends some results obtained in previous papers by the same authors. In the model case, that is when the non-local operator is the fractional Laplacian, the existence result proved along the paper may be read as the non-local fractional counterpart of the one obtained in [12] (see also [9]) in the framework of the classical Laplace equation with critical nonlinearities
The Brezis-Nirenberg result for the fractional Laplacian
Full text linkThe aim of this paper is to deal with the non-local fractional counterpart of the Laplace equation involving critical non-linearities studied in the famous paper of Brezis and Nirenberg (1983). In this paper we first study the problem in a general framework; indeed we consider an equation driven by a general non-local integrodifferential operator and in presence of a lower order perturbation of the critical power. In this setting we prove an existence result through variational techniques. Then, as a concrete example, we derive a Brezis-Nirenberg type result for an equation driven by the fractional Laplacian (-Delta)(s); that is, we show that if lambda(1,s) is the first eigenvalue of the non-local operator (-Delta)(s) with homogeneous Dirichlet boundary datum, then for any lambda is an element of (0, lambda(1,s)) there exists a non-trivial solution of the above model equation, provided n >= 4s. In this sense the present work may be seen as the extension of the classical Brezis-Nirenberg result to the case of non-local fractional operators
Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators
No full textThe purpose of this paper is to derive some Lewy-Stampacchia estimates in some cases of interest, such as the ones driven by non-local operators. Since we will perform an abstract approach to the problem, this will provide, as a byproduct, Lewy-Stampacchia
estimates in more classical cases as well. In particular, we can recover the known estimates for the standard Laplacian, the -Laplacian, and the Laplacian in the Heisenberg group. In the non-local framework we prove a Lewy-Stampacchia estimate for a general integrodifferential operator and, as a particular case, for the fractional Laplacian. As far as we know, the abstract framework and the results in the non-local setting are new
Multibump solutions and asymptotic expansions for mesoscopic Allen-Cahn type equations
No full textWe consider a mesoscopic model for phase transitions in a periodic medium and we construct multibump solutions. The rational perturbative case is dealt with by explicit asymptotics
The geometry of mesoscopic phase transition interfaces
No full textWe consider a mesoscopic model of phase transitions and investigate the geometric properties of the interfaces of the associated minimal solutions. We provide density estimates for level sets and, in the periodic setting, we construct minimal interfaces at a universal distance from any given hyperplane
On the spectrum of two different fractional operators
No full textIn this paper we deal with two non-local operators that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. The aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences
A notion of nonlocal curvature
No full textWe consider a nonlocal (or fractional) curvature and we investigate similarities and differences with respect to the classical local case. In particular, we show that the nonlocal mean curvature can be seen as an average of suitable nonlocal directional curvatures and there is a natural asymptotic convergence to the classical case. Nevertheless, different from the classical cases, minimal and maximal nonlocal directional curvatures are not in general attained at perpendicular directions and, in fact, one can arbitrarily prescribe the set of extremal directions for nonlocal directional curvatures. Also the classical directional curvatures naturally enjoy some linear properties that are lost in the nonlocal framework. In this sense, nonlocal directional curvatures are somewhat intrinsically nonlinear. © 2014 Copyright Taylor & Francis Group, LLC
- …
