Search results for "J60"
showing 10 items of 53 documents
Positive solutions for singular double phase problems
2021
Abstract We study the existence of positive solutions for a class of double phase Dirichlet equations which have the combined effects of a singular term and of a parametric superlinear term. The differential operator of the equation is the sum of a p-Laplacian and of a weighted q-Laplacian ( q p ) with discontinuous weight. Using the Nehari method, we show that for all small values of the parameter λ > 0 , the equation has at least two positive solutions.
Radial symmetry of p-harmonic minimizers
2017
"It is still not known if the radial cavitating minimizers obtained by Ball [J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Phil. Trans. R. Soc. Lond. A 306 (1982) 557--611] (and subsequently by many others) are global minimizers of any physically reasonable nonlinearly elastic energy". The quotation is from [J. Sivaloganathan and S. J. Spector, Necessary conditions for a minimum at a radial cavitating singularity in nonlinear elasticity, Ann. Inst. H. Poincare Anal. Non Lineaire 25 (2008), no. 1, 201--213] and seems to be still accurate. The model case of the $p$-harmonic energy is considered here. We prove that the planar radial minimizers are indee…
Nonlinear scalar field equations with general nonlinearity
2018
Consider the nonlinear scalar field equation \begin{equation} \label{a1} -\Delta{u}= f(u)\quad\text{in}~\mathbb{R}^N,\qquad u\in H^1(\mathbb{R}^N), \end{equation} where $N\geq3$ and $f$ satisfies the general Berestycki-Lions conditions. We are interested in the existence of positive ground states, of nonradial solutions and in the multiplicity of radial and nonradial solutions. Very recently Mederski [30] made a major advance in that direction through the development, in an abstract setting, of a new critical point theory for constrained functionals. In this paper we propose an alternative, more elementary approach, which permits to recover Mederski's results on the scalar field equation. T…
Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit
2011
In the context of self-stabilizing processes, that is processes attracted by their own law, living in a potential landscape, we investigate different properties of the invariant measures. The interaction between the process and its law leads to nonlinear stochastic differential equations. In [S. Herrmann and J. Tugaut. Electron. J. Probab. 15 (2010) 2087–2116], the authors proved that, for linear interaction and under suitable conditions, there exists a unique symmetric limit measure associated to the set of invariant measures in the small-noise limit. The aim of this study is essentially to point out that this statement leads to the existence, as the noise intensity is small, of one unique…
p-Laplacian type equations involving measures
2003
This is a survey on problems involving equations $-\operatorname{div}{\Cal A}(x,\nabla u)=\mu$, where $\mu$ is a Radon measure and ${\Cal A}:\bold {R}^n\times\bold R^n\to \bold R^n$ verifies Leray-Lions type conditions. We shall discuss a potential theoretic approach when the measure is nonnegative. Existence and uniqueness, and different concepts of solutions are discussed for general signed measures.
Local Gauge Conditions for Ellipticity in Conformal Geometry
2013
In this article we introduce local gauge conditions under which many curvature tensors appearing in conformal geometry, such as the Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors, become elliptic operators. The gauge conditions amount to fixing an $n$-harmonic coordinate system and normalizing the determinant of the metric. We also give corresponding elliptic regularity results and characterizations of local conformal flatness in low regularity settings.
Moduli spaces of rank two aCM bundles on the Segre product of three projective lines
2016
Let P^n be the projective space of dimension n on an algebraically closed field of characteristic 0 and F be the image of the Segre embedding of P^1xP^1xP^1 inside P^7. In the present paper we deal with the moduli spaces of locally free sheaves E on F of rank 2 with h^i(F,E(t))=0 for i=1,2 and each integer t.
Nonlinear diffusion in transparent media: the resolvent equation
2017
Abstract We consider the partial differential equation u - f = div ( u m ∇ u | ∇ u | ) u-f=\operatornamewithlimits{div}\biggl{(}u^{m}\frac{\nabla u}{|\nabla u|}% \biggr{)} with f nonnegative and bounded and m ∈ ℝ {m\in\mathbb{R}} . We prove existence and uniqueness of solutions for both the Dirichlet problem (with bounded and nonnegative boundary datum) and the homogeneous Neumann problem. Solutions, which a priori belong to a space of truncated bounded variation functions, are shown to have zero jump part with respect to the ℋ N - 1 {{\mathcal{H}}^{N-1}} -Hausdorff measure. Results and proofs extend to more general nonlinearities.
An unbounded family of log Calabi–Yau pairs
2016
We give an explicit example of log Calabi-Yau pairs that are log canonical and have a linearly decreasing Euler characteristic. This is constructed in terms of a degree two covering of a sequence of blow ups of three dimensional projective bundles over the Segre-Hirzebruch surfaces ${\mathbb F}_n$ for every positive integer $n$ big enough.
Exact simulation of diffusion first exit times: algorithm acceleration
2020
In order to describe or estimate different quantities related to a specific random variable, it is of prime interest to numerically generate such a variate. In specific situations, the exact generation of random variables might be either momentarily unavailable or too expensive in terms of computation time. It therefore needs to be replaced by an approximation procedure. As was previously the case, the ambitious exact simulation of exit times for diffusion processes was unreachable though it concerns many applications in different fields like mathematical finance, neuroscience or reliability. The usual way to describe exit times was to use discretization schemes, that are of course approxim…