Search results for "uniqueness"

A singularly perturbed Kirchhoff problem revisited

2020

Abstract In this paper, we revisit the singularly perturbation problem (0.1) − ( ϵ 2 a + ϵ b ∫ R 3 | ∇ u | 2 ) Δ u + V ( x ) u = | u | p − 1 u in  R 3 , where a , b , ϵ > 0 , 1 p 5 are constants and V is a potential function. First we establish the uniqueness and nondegeneracy of positive solutions to the limiting Kirchhoff problem − ( a + b ∫ R 3 | ∇ u | 2 ) Δ u + u = | u | p − 1 u in  R 3 . Then, combining this nondegeneracy result and Lyapunov-Schmidt reduction method, we derive the existence of solutions to (0.1) for ϵ > 0 sufficiently small. Finally, we establish a local uniqueness result for such derived solutions using this nondegeneracy result and a type of local Pohozaev identity.

Global Existence for Nonlinear Parabolic Problems With Measure Data– Applications to Non-uniqueness for Parabolic Problems With Critical Gradient ter…

2011

Abstract In the present article we study global existence for a nonlinear parabolic equation having a reaction term and a Radon measure datum: where 1 < p < N, Ω is a bounded open subset of ℝN (N ≥ 2), Δpu = div(|∇u|p−2∇u) is the so called p-Laplacian operator, sign s ., ϕ(ν0) ∈ L1(Ω), μ is a finite Radon measure and f ∈ L∞(Ω×(0, T)) for every T > 0. Then we apply this existence result to show wild nonuniqueness for a connected nonlinear parabolic problem having a gradient term with natural growth.

L∞-variational problems associated to measurable Finsler structures

2016

Abstract We study L ∞ -variational problems associated to measurable Finsler structures in Euclidean spaces. We obtain existence and uniqueness results for the absolute minimizers.

The Uniqueness of Planktonic Ecosystems in the Mediterranean Sea: The Response to Orbital- and Suborbital-Climatic Forcing over the Last 130,000 Years

2016

AbstractThe Mediterranean Sea is an ideal location to test the response of organisms to hydrological transformations driven by climate change. Here we review studies carried out on planktonic foraminifera and coccolithophores during the late Quaternary and attempt the comparison of data scattered in time and space. We highlight the prompt response of surface water ecosystems to both orbital- and suborbital-climatic variations.A markedly different spatial response was observed in calcareous plankton assemblages, possibly due to the influence of the North Atlantic climatic system in the western, central and northern areas and of the monsoon system in the easternmost and southern sites. Orbita…

Sliding solutions of second-order differential equations with discontinuous right-hand side

2017

We consider second-order ordinary differential equations with discontinuous right-hand side. We analyze the concept of solution of this kind of equations and determine analytical conditions that are satisfied by typical solutions. Moreover, the existence and uniqueness of solutions and sliding solutions are studied. Copyright © 2017 John Wiley & Sons, Ltd.

A singular elliptic equation and a related functional

2021

We study a class of Dirichlet boundary value problems whose prototype is [see formula in PDF] where 0 < p < 1 and f belongs to a suitable Lebesgue space. The main features of this problem are the presence of a singular term |u|p−2u and a datum f which possibly changes its sign. We introduce a notion of solution in this singular setting and we prove an existence result for such a solution. The motivation of our notion of solution to problem above is due to a minimization problem for a non–differentiable functional on [see formula in PDF] whose formal Euler–Lagrange equation is an equation of that type. For nonnegative solutions a uniqueness result is obtained.

Non-autonomous rough semilinear PDEs and the multiplicative Sewing Lemma

2021

We investigate existence, uniqueness and regularity for local solutions of rough parabolic equations with subcritical noise of the form $du_t- L_tu_tdt= N(u_t)dt + \sum_{i = 1}^dF_i(u_t)d\mathbf X^i_t$ where $(L_t)_{t\in[0,T]}$ is a time-dependent family of unbounded operators acting on some scale of Banach spaces, while $\mathbf X\equiv(X,\mathbb X)$ is a two-step (non-necessarily geometric) rough path of H\"older regularity $\gamma >1/3.$ Besides dealing with non-autonomous evolution equations, our results also allow for unbounded operations in the noise term (up to some critical loss of regularity depending on that of the rough path $\mathbf X$). As a technical tool, we introduce a versi…

Existence of dynamical low-rank approximations to parabolic problems

2021

The existence and uniqueness of weak solutions to dynamical low-rank evolution problems for parabolic partial differential equations in two spatial dimensions is shown, covering also non-diagonal diffusion in the elliptic part. The proof is based on a variational time-stepping scheme on the low-rank manifold. Moreover, this scheme is shown to be closely related to practical methods for computing such low-rank evolutions.

Quantum Groups, Star Products and Cyclic Cohomology

1993

After some historical remarks, we start with a rapid overview of the star-product theory (deformation of algebras of functions on phase space) and its applications to deformation-quantization. We then concentrate on Poisson-Lie groups and their “quantization”, give a star-product realization of quantum groups and discuss uniqueness and the rigidity as bialgebra of a universal model for the quantum SL(2) groups. In the last part we develop the notion of closed star-product (for which a trace can be defined on the algebra), show that it is classified by cyclic cohomology, permits to define a character and that there always exists one; finally we show that the pseudodifferential calculus on a …

2020

Abstract This paper shows global uniqueness in two inverse problems for a fractional conductivity equation: an unknown conductivity in a bounded domain is uniquely determined by measurements of solutions taken in arbitrary open, possibly disjoint subsets of the exterior. Both the cases of infinitely many measurements and a single measurement are addressed. The results are based on a reduction from the fractional conductivity equation to the fractional Schrodinger equation, and as such represent extensions of previous works. Moreover, a simple application is shown in which the fractional conductivity equation is put into relation with a long jump random walk with weights.