arXiv subjects

Michal Kowalczyk

Publications and source records attributed to Michal Kowalczyk.

16 recordsLinked to original sources

arXiv2020

Doubling construction for $O(m)\times O(n)$-invariant solutions to the Allen-Cahn equation

We construct new families of two-ended $O(m)\times O(n)$-invariant solutions to the Allen- Cahn equation Δu+u-u3=0 in $\mathbb{R}^{N+1}$, with $N\ge 7$, whose zero level sets diverge logarithmically from the Lawson cone at infinity. The construction is based on a careful study of the Jacobi-Toda system on a given $O(m)\times O(n)$-invariant manifold, which is asymptotic to the Lawson cone at infinity.

math.AP↗

arXiv2020

Exotic states of matter in an oscillatory driven liquid crystal cell

Matter under different equilibrium conditions of pressure and temperature exhibits different states such as solid, liquid, gas, and plasma. Exotic states of matter, such as Bose- Einstein condensates, superfluidity, chiral magnets, superconductivity, and liquid crystalline blue phases are observed in thermodynamic equilibrium. Rather than being a result of an aggregation of matter, their emergence is due to a change of a topological state of the system. Here we investigate topological states of matter in a system with injection and dissipation of energy. In an experiment involving a liquid crystal cell under the influence of a low-frequency oscillatory electric field, we observe a transition from non-vortex state to a state in which vortices persist. Depending on the period and the type of the forcing, the vortices self-organise forming square lattices, glassy states, and disordered vortex structures. Based on a stochastic amplitude equation, we recognise the origin of the transition as the balance between stochastic creation and deterministic annihilation of vortices. Our results show that the matter maintained out of equilibrium by means of the temporal modulation of parameters can exhibit exotic states.

cond-mat.soft↗

arXiv2019

Soliton dynamics for the 1D NLKG equation with symmetry and in the absence of internal modes

We consider the dynamics of even solutions of the one-dimensional nonlinear Klein-Gordon equation $\partial_t^2 ϕ- \partial_x^2 ϕ+ ϕ- |ϕ|^{2α} ϕ=0$ for $α>1$, in the vicinity of the unstable soliton $Q$. Our main result is that stability in the energy space $H^1(\mathbb R)\times L^2(\mathbb R)$ implies asymptotic stability in a local energy norm. In particular, there exists a Lipschitz graph of initial data leading to stable and asymptotically stable trajectories. The condition $α>1$ corresponds to cases where the linearized operator around $Q$ has no resonance and no internal mode. Recall that the case $α>2$ is treated in Krieger-Nakanishi-Schlag using Strichartz and other local dispersive estimates. Since these tools are not available for low power nonlinearities, our approach is based on virial type estimates and the particular structure of the linearized operator observed in Chang-Gustafson-Nakanishi-Tsai.

math.AP↗

arXiv2018

Multiple Delaunay ends solutions of the Cahn-Hilliard equation

Let $Σ$ be a surface of constant mean curvature in ${\mathbb R}^3$ with multiple Delaunay ends. Assuming that $Σ$ is non degenerate in this paper we construct new solutions to the Cahn-Hilliard equation $\varepsilonΔu+\varepsilon^{-1}u(1-u^2)=\ell_\varepsilon$ in ${\mathbb R}^3$ such that as $\varepsilon\to 0$ the zero level set of $u_\varepsilon$ approaches $Σ$. Moreover, on compacts of the connected components of ${\mathbb R}^3\setminus Σ$ we have $1-|u_\varepsilon|\to 0$ uniformly.

math.AP↗

arXiv2018

Gradient theory of domain walls in thin, nematic liquid crystals films

In this paper we describe domain walls appearing in a thin, nematic liquid crystal sample subject to an external field with intensity close to the Fréedericksz transition threshold. Using the gradient theory of the phase transition adopted to this situation, we show that depending on the parameters of the system, domain walls occur in the bistable region or at the border between the bistable and the monostable region.

math.AP↗

arXiv2018

Free boundary problems arising in the theory of maximal solutions of equations with exponential nonlinearities

We consider equations of the form $Δu +λ^2 V(x)e^{\,u}=ρ$ in various two dimensional settings. We assume that $V>0$ is a given function, $λ>0$ is a small parameter and $ρ=\mathcal O(1)$ or $ρ\to +\infty$ as $λ\to 0$. In a recent paper we prove the existence of the maximal solutions for a particular choice $V\equiv 1$, $ρ=0$ when the problem is posed in doubly connected domains under Dirichlet boundary conditions. We related the maximal solutions with a novel free boundary problem. The purpose of this note is to derive the corresponding free boundary problems in other settings. Solvability of such problems is, viewed formally, the necessary condition for the existence of the maximal solution.

math.AP↗

arXiv2018

Maximal solution of the Liouville equation in doubly connected domains

In this paper we consider the Liouville equation $Δu +λ^2 e^{\,u}=0$ with Dirichlet boundary conditions in a two dimensional, doubly connected domain $Ω$. We show that there exists a simple, closed curve $γ\subset Ω$ such that for a sequence $λ_n\to 0$ and a sequence of solutions $u_{n}$ it holds $\frac{u_{n}}{\log\frac{1}{λ_n}}\to H$, where $H$ is a harmonic function in $Ω\setminusγ$ and $\frac{λ_n^2}{\log\frac{1}{λ_n}}\int_Ωe^{\,u_n}\,dx\to 8πc_Ω$, where $c_Ω$ is a constant depending on the conformal class of $Ω$ only.

math.AP↗

arXiv2017

Nondegeneracy and the Jacobi fields of rotationally symmetric solutions to the Cahn-Hillard equation

In this paper we study rotationally symmetric solutions of the Cahn-Hilliard equation in $\mathbb R^3$ constructed by the authors. These solutions form a one parameter family analog to the family of Delaunay surfaces and in fact the zero level sets of their blowdowns approach these surfaces. Presently we go a step further and show that their stability properties are inherited from the stability properties of the Delaunay surfaces. Our main result states that the rotationally symmetric solutions are non degenerate and that they have exactly $6$ Jacobi fields of temperate growth coming from the natural invariances of the problem (3 translations and 2 rotations) and the variation of the Delaunay parameter.

math.AP↗

arXiv2016

Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities and spectral estimates

This paper is devoted to the Lin-Ni conjecture for a semi-linear elliptic equation with a super-linear, sub-critical nonlinearity and homogeneous Neumann boundary conditions. We establish a new rigidity result, that is, we prove that the unique positive solution is a constant if the parameter of the problem is below an explicit bound that we relate with an optimal constant for a Gagliardo-Nirenberg-Sobolev interpolation inequality and also with an optimal Keller-Lieb-Thirring inequality. Our results are valid in a sub-linear regime as well. The rigidity bound is obtained by nonlinear flow methods inspired by recent results on compact manifolds, which unify nonlinear elliptic techniques and the carr{é} du champ method in semi-group theory. Our method requires the convexity of the domain. It relies on integral quantities, takes into account spectral estimates and provides improved functional inequalities.

math.AP↗

arXiv2015

Rotationally symmetric solutions to the Cahn-Hilliard equation

This paper is devoted to construction of new solutions to the Cahn-Hilliard equation in $\mathbb R^d$. Staring from a Delaunay unduloid $D_τ$ with parameter $τ\in (0,τ^*)$ we find for each sufficiently small $\varepsilon$ a solution $u$ of this equation which is periodic in the direction of the $x_d$ axis and rotationally symmetric with respect to rotations about this axis. The zero level set of $u$ approaches as $\varepsilon\to 0$ the surface $D_τ$. We use a refined version of the Lyapunov-Schmidt reduction method which simplifies very technical aspects of previous constructions for similar problems.

math.AP↗

arXiv2014

A hybrid variational principle for the Keller-Segel system in $\mathbb R^2$

We construct weak global in time solutions to the classical Keller-Segel system cell movement by chemotaxis in two dimensions when the total mass is below the well-known critical value. Our construction takes advantage of the fact that the Keller-Segel system can be realized as a gradient flow in a suitable functional product space. This allows us to employ a hybrid variational principle which is a generalisation of the minimising implicit scheme for Wasserstein distances introduced by Jordan, Kinderlehrer and Otto (1998).

math.AP↗

arXiv2014

Improved interpolation inequalities on the sphere

This paper contains a review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents. Pushing further these techniques we also establish some new results, clarify the range of applicability of the various existing methods and state several explicit estimates.

math.AP↗

arXiv2012

Sharp interpolation inequalities on the sphere : new methods and consequences

These notes are devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension two and higher interpolate between Poincaré, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. We emphasize the connexion between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere. We shall address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.

math.AP↗

arXiv2011

The space of 4-ended solutions to the Allen-Cahn equation on the plane

An entire solution of the Allen-Cahn equation $Δu=F'(u)$, where $F$ is an even, bistable function, is called a $2k$-end solution if its nodal set is asymptotic to $2k$ half lines, and if along each of these half lines the function $u$ looks like the one dimensional, heteroclinic solution. In this paper we initiate a program to classify the four-end solutions of the Allen-Cahn equation in $\R^2$. We show that there exists a one parameter family of solutions containing the saddle solution, for which the angle between the nodal lines is $\fracπ{2}$, as well as solutions for which the angle between the asymptotic half lines is any $θ\in (0, \fracπ{2})$. This justifies the definition of the angle map for a four-end solution $u$, which is the angle $θ=θ(u)\in (0, \fracπ{2})$ between the asymptote to the nodal line in the first quadrant and the x axis. Then we show that on any connected component in the moduli space of four-end solutions the angle map is surjective onto $(0,\fracπ{2})$.

math.AP↗

arXiv2011

The classification of four end solutions of the Allen-Cahn equation on the plane

An entire solution of the Allen-Cahn equation $Δu=f(u)$, where $f$ has exactly three zeros at $\pm 1$ and 0, is balanced and odd, e.g. $f(u)=u(u^2-1)$, is called a $2k$-ended solution if its nodal set is asymptotic to $2k$ half lines, and if along each of these half lines the function $u$ looks like the one dimensional, heteroclinic solution. In this paper we consider the family of four ended solutions whose ends are almost parallel at $\infty$. We show that this family can be parametrized by the family of solutions of the two component Toda system. As a result we obtain the uniqueness of four ended solutions with almost parallel ends. Combining this result with the classification of connected components in the moduli space of the four ended solutions we can classify all such solutions. Thus we show that four end solutions form, up to rigid motions, a one parameter family. This family contains the saddle solution, for which the angle between the nodal lines is $\fracπ{2}$ as well as solutions for which the angle between the asymptotic half lines of the nodal set is arbitrary small (almost parallel nodal sets).

math.AP↗

arXiv2010

Interface Foliation Near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature

Let $(\MM ,{\tilde g})$ be an $N$-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen-Cahn equation $$ ε^2Δ_{ {\tilde g}} {u}\,+\, (1 - {u}^2)u \,=\,0\quad \mbox{in } \MM, $$ where $ε$ is a small parameter. Let $\KK\subset \MM$ be an $(N-1)$-dimensional smooth minimal submanifold that separates $\MM$ into two disjoint components. Assume that $\KK$ is non-degenerate in the sense that it does not support non-trivial Jacobi fields, and that $|A_{\KK}|^2+\mbox{Ric}_{\tilde g}(ν_{\KK}, ν_{\KK})$ is positive along $\KK$. Then for each integer $m\geq 2$, we establish the existence of a sequence $ε= ε_j\to 0$, and solutions $u_ε$ with $m$-transition layers near $\KK$, with mutual distance $O(ε|\ln ε|)$.

math.AP↗