Search results for "LIOUVILLE"
showing 10 items of 32 documents
Multiplicity results for asymptotically linear equations, using the rotation number approach
2007
By using a topological approach and the relation between rotation numbers and weighted eigenvalues, we give some multiplicity results for the boundary value problem u′′ + f(t, u) = 0, u(0) = u(T) = 0, under suitable assumptions on f(t, x)/x at zero and infinity. Solutions are characterized by their nodal properties.
Infinitely many solutions to boundary value problem for fractional differential equations
2018
Variational methods and critical point theorems are used to discuss existence of infinitely many solutions to boundary value problem for fractional order differential equations where Riemann-Liouville fractional derivatives and Caputo fractional derivatives are used. An example is given to illustrate our result.
Integrability of the one dimensional Schrödinger equation
2018
We present a definition of integrability for the one dimensional Schroedinger equation, which encompasses all known integrable systems, i.e. systems for which the spectrum can be explicitly computed. For this, we introduce the class of rigid functions, built as Liouvillian functions, but containing all solutions of rigid differential operators in the sense of Katz, and a notion of natural boundary conditions. We then make a complete classification of rational integrable potentials. Many new integrable cases are found, some of them physically interesting.
Classification and non-existence results for weak solutions to quasilinear elliptic equations with Neumann or Robin boundary conditions
2021
Abstract We classify positive solutions to a class of quasilinear equations with Neumann or Robin boundary conditions in convex domains. Our main tool is an integral formula involving the trace of some relevant quantities for the problem. Under a suitable condition on the nonlinearity, a relevant consequence of our results is that we can extend to weak solutions a celebrated result obtained for stable solutions by Casten and Holland and by Matano.
EXISTENCE OF THREE SOLUTIONS FOR A MIXED BOUNDARY VALUE PROBLEM WITH THE STURM-LIOUVILLE EQUATION
2012
Abstract. The aim of this paper is to establish the existence of threesolutions for a Sturm-Liouville mixed boundary value problem. The ap-proach is based on multiple critical points theorems. 1. IntroductionThe aim of this paper is to establish, under a suitable set of assumptions, theexistence of at least three solutions for the following Sturm-Liouville problemwith mixed boundary conditions(RS λ )ˆ−(pu ′ ) ′ +qu = λf(t,u) in I =]a,b[u(a) = u ′ (b) = 0,where λ is a positive parameter and p, q, f are regular functions. To be precise,if f : [a,b] × R→ Ris a L 2 -Carath´eodory function and p,q ∈ L ∞ ([a,b]) suchthatp 0 := essinf t∈[a,b] p(t) > 0, q 0 := essinf t∈[a,b] q(t) ≥ 0,then we prove …
Uncertainty quantification analysis of the biological Gompertz model subject to random fluctuations in all its parameters
2020
[EN] In spite of its simple formulation via a nonlinear differential equation, the Gompertz model has been widely applied to describe the dynamics of biological and biophysical parts of complex systems (growth of living organisms, number of bacteria, volume of infected cells, etc.). Its parameters or coefficients and the initial condition represent biological quantities (usually, rates and number of individual/particles, respectively) whose nature is random rather than deterministic. In this paper, we present a complete uncertainty quantification analysis of the randomized Gomperz model via the computation of an explicit expression to the first probability density function of its solution s…
Heisenberg Uncertainty Relation in Quantum Liouville Equation
2009
We consider the quantum Liouville equation and give a characterization of the solutions which satisfy the Heisenberg uncertainty relation. We analyze three cases. Initially we consider a particular solution of the quantum Liouville equation: the Wigner transformf(x,v,t) of a generic solutionψ(x;t) of the Schrödinger equation. We give a representation ofψ(x,t) by the Hermite functions. We show that the values of the variances ofxandvcalculated by using the Wigner functionf(x,v,t) coincide, respectively, with the variances of position operatorX^and conjugate momentum operatorP^obtained using the wave functionψ(x,t). Then we consider the Fourier transform of the density matrixρ(z,y,t) =ψ∗(z,t)…
Conformality and $Q$-harmonicity in sub-Riemannian manifolds
2016
We prove the equivalence of several natural notions of conformal maps between sub-Riemannian manifolds. Our main contribution is in the setting of those manifolds that support a suitable regularity theory for subelliptic $p$-Laplacian operators. For such manifolds we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth. In particular, we prove that contact manifolds support the suitable regularity. The main new technical tools are a sub-Riemannian version of p-harmonic coordinates and a technique of propagation of regularity from horizontal layers.
Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths
2018
Many open quantum systems encountered in both natural and synthetic situations are embedded in classical-like baths. Often, the bath degrees of freedom may be represented in terms of canonically conjugate coordinates, but in some cases they may require a non-canonical or non-Hamiltonian representation. Herein, we review an approach to the dynamics and statistical mechanics of quantum subsystems embedded in either non-canonical or non-Hamiltonian classical-like baths which is based on operator-valued quasi-probability functions. These functions typically evolve through the action of quasi-Lie brackets and their associated Quantum-Classical Liouville Equations, or through quasi-Lie brackets a…
Optimal control and Clairaut-Liouville metrics with applications
2014
The work of this thesis is about the study of the conjugate and cut loci of 2D riemannian or almost-riemannian metrics. We take the point of view of optimal control to apply the Pontryagin Maximum Principle in the purpose of characterize the extremals of the problem considered.We use geometric, numerical and integrability methods to study some Liouville and Clairaut-Liouville metrics on the sphere. In the degenerate case of revolution, the study of the ellipsoid uses geometric methods to fix the cut locus and the nature of the conjugate locus in the oblate and prolate cases. In the general case, extremals will have two distinct type of comportment which correspond to those observed in the r…