arXiv subjects
Publications and source records attributed to Piotr Rybka.
We combine the total variation flow suitable for crystal modeling and image analysis with the dynamic boundary conditions. We analyze the behavior of facets at the parts of the boundary where these conditions are imposed. We devote particular attention to the radially symmetric data. We observe that the boundary layer detachment actually can happen at concave parts of the boundary
We study a basic linear elliptic equation on a lower dimensional rectifiable set $S$ in $\mathbb{R}^N$ with the Neumann boundary data. Set $S$ is a support of a finite Borel measure $μ$. We will use the measure theoretic tools to interpret the equation and the Neumann boundary condition. For this purpose we recall the Sobolev-type space dependent on the measure $μ$. We establish existence and uniqueness of weak solutions provided that an appropriate source term is given.
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.
We study the two dimensional least gradient problem in a convex, but not necessary strictly convex region. We look for solutions in the space of $BV$ functions satisfying the boundary data $f$ in trace sense. We assume that $f$ is in $BV$ too. We state admissibility conditions on the trace and on the domain that are sufficient for existence of solutions.
We consider a gradient flow of the total variation in a negative Sobolev space $H^{-s}$ $(0\leq s \leq 1)$ under the periodic boundary condition. If $s=0$, the flow is nothing but the classical total variation flow. If $s=1$, this is the fourth order total variation flow. We consider a convex variational problem which gives an implicit-time discrete scheme for the flow. By a duality based method, we give a simple numerical scheme to calculate this minimizing problem numerically and discuss convergence of a forward-backward splitting scheme. Several numerical experiments are given.
We study integrability of the derivative of solutions to a singular one-dimensional parabolic equation with initial data in $W^{1,1}$. In order to avoid additional difficulties we consider only the periodic boundary conditions. The problem we study is a gradient flow of a convex, linear growth variational functional. We also prove a similar result for the elliptic companion problem, i.e. the time semidiscretization.
We study a time fractional heat equation in a noncylindrical domain. The problem is one-dimensional. We prove existence of properly defined weak solutions by means of the Galerkin approximation.
We study two special cases of the planar least gradient problem. In the first one, the boundary conditions are imposed on a part of the strictly convex domain. In the second case, we impose the Dirichlet data on the boundary of a rectangle, an example of convex but not strictly convex domain. We show the existence of solutions and study their properties for particular cases of data.
We study one-dimensional very singular parabolic equations with periodic boundary conditions and initial data in $BV$, which is the energy space. We show existence of solutions in this energy space and then we prove that they are viscosity solutions in the sense of Giga-Giga.
We study Berg's effect on special domains. This effect is understood as monotonicity of a harmonic function (with respect to the distance from the center of a flat part of the boundary) restricted to the boundary. The harmonic function must satisfy piecewise constant Neumann boundary conditions. We show that Berg's effect is a rare and fragile phenomenon.
We study a singular one-dimensional parabolic problem with initial data in the $BV$ space, the energy space, for various boundary data. We pay special attention to Dirichlet conditions, which need not satisfied in a pointwise manner. We study the facet creation process and the extinction of solutions caused by the evolution of facets. Our major tool is the comparison principle provided by the theory of viscosity solutions developed in \cite{miyory}.
We present here a fine singularity analysis of solutions to the Laplace equation in special polygonal domains in the plane. We assume piecewise constant Neumann on one component of the boundary. Our motivation is to find the rigorous proof the so-called Berg's effect.
We are interested in an anisotropic singular diffusion equation in the plane and in its regularization. We establish existence, uniqueness and basic regularity of solutions to both equations. We construct explicit solutions showing the creation of facets, i.e. flat regions of solutions. By using the formula for solutions, we rigorously prove that both equations create ruled surfaces out of convex initial conditions as well as do not admit point (local) extrema. We present numerical experiments suggesting that the two flows seem not differ much. Possible applications to image reconstruction is pointed out, too.
Our goal is to establish existence with suitable initial data of solutions to general parabolic equation in one dimension, $u_t = L(u_x)_x$, where $L$ is merely a monotone function. We also expose the basic properties of solutions, concentrating on maximal possible regularity. Analysis of solutions with convex initial data explains why we may call them {\it almost classical}. Some qualitative aspects of solutions, like facets -- flat regions of solutions, are studied too.
The paper examines one-dimensional total variation flow equation with Dirichlet boundary conditions. Thanks to a new concept of "almost classical" solutions we are able to determine evolution of facets -- flat regions of solutions. A key element of our approach is the natural regularity determined by nonlinear elliptic operator, for which $x^2$ is an irregular function. Such a point of view allows us to construct solutions. We apply this idea to implement our approach to numerical simulations for typical initial data. Due to the nature of Dirichlet data any monotone function is an equilibrium. We prove that each solution reaches such steady state in a finite time.
To separate stars and galaxies in the far infrared AKARI All-Sky Survey data, we have selected a sample with the complete color information available in the low extinction regions of the sky and constructed color-color plots for these data. We looked for the method to separate stars and galaxies using the color information. We performed an extensive search for the counterparts of these selected All-Sky Survey sources in the NED and SIMBAD databases. Among 5176 objects, we found 4272 galaxies, 382 other extragalactic objects, 349 Milky Way stars, 50 other Galactic objects, and 101 sources detected before in various wavelengths but of an unknown origin. 22 sources were left unidentified. Then, we checked colors of stars and galaxies in the far-infrared flux-color and color-color plots. In the resulting diagrams, stars form two clearly separated clouds. One of them is easy to be distinguished from galaxies and allows for a simple method of excluding a large part of stars using the far-infrared data. The other smaller branch, overplotting galaxies, consists of stars known to have an infrared excess, like Vega and some fainter stars discovered by IRAS or 2MASS. The color properties of these objects in any case make them very difficult to distinguish from galaxies. We conclude that the FIR color-color diagrams allow for a high-quality star-galaxy separation. With the proposed simple method we can select more that 95 % of galaxies rejecting at least 80 % of stars.
We study a singular parabolic equation of the total variation type in one dimension. The problem is a simplification of the singular curvature flow. We show existence and uniqueness of weak solutions. We also prove existence of weak solutions to the semi-discretization of the problem as well as convergence of the approximating sequences. The semi-discretization shows that facets must form. For a class of initial data we are able to study in details the facet formation and interactions and their asymptotic behavior. We notice that our qualitative results may be interpreted with the help of a special composition of multivalued operators.