Piotr Rybka

Global weak solutions to a sixth order Cahn-Hilliard type equation

We study a sixth order Cahn--Hilliard type equation that arises as a model for the faceting of a growing surface. We show existence of global-in-time weak solutions in two space dimensions, assuming periodic boundary conditions. We also establish exponential-in-time a priori estimates on the

A crystalline motion: uniqueness and geometric properties

H^3

On a Higher Order Convective Cahn--Hilliard-Type Equation

norm of solutions. These bounds enable us to prove the uniqueness of weak solutions. We also show the regularizing effect of the equation on the data.

A NEW LOOK AT EQUILIBRIA IN STEFAN-TYPE PROBLEMS IN THE PLANE

The author studies a model of crystalline motion in the plane. Existence and uniqueness of local in time solutions are shown. Geometric properties of the flow are studied, assuming that the Wulff shape is a regular N-sided polygon. The author shows that a small, convex grain of ice immersed in the cold melt shrinks, provided that the flow does not overly deform the initial interface. The author is able to verify the last condition only if the initial interface is a scaled Wulff shape. In this case, the free boundary will remain a scaled Wulff shape at later times. This is shown by using an isoperimetric inequality.

Existence of Energy Minimizers for Magnetostrictive Materials

A higher order convective Cahn--Hilliard-type equation that describes the faceting of a growing surface is considered with periodic boundary conditions. By using a Galerkin approach, the existence ...

On convergence of solutions to the equation of viscoelasticity with capillarity

We study steady states of Stefan-type problems in the plane with the Gibbs–Thomson correction involving a general anisotropic energy density function. By a local analysis we prove the global result showing that the solution is the Wulff shape. The key element is a stability result which enables us to approximate singular models by regular ones.

A comparison principle for Hamilton-Jacobi equations with discontinuous Hamiltonians

The existence of a deformation and magnetization minimizing the magnetostrictive free energy is given. Mathematical challenges are presented by a free energy that includes elastic contributions defined in the reference configuration and magnetic contributions defined in the spatial frame. The one-to-one a.e. and orientation-preserving property of the deformation is demonstrated, and the satisfaction of the nonconvex saturation constraint for the magnetization is proven.

The crystalline version of the modified Stefan problem in the plane and its properties

aug~nented with appropriate boundary and initial conditions. In the above cq~iation p, v and 6 are positive numbers, u : R C IR2 + IR and a ( F ) = DEW(F) . We are interested in particular in W having more than one local niiuilnum. Let us mention that several authors including Slemrod, Truskinovsky, and later Abeyaratne and Knowles justified (1.1) as a proper model for dynamics of phase transitions in van der Waals fluids ([28], [32]) and interface motion in solids ([2], 131). Most of the work on (1.1) was performed on the case of one spatial variable, see e.g. [19], [la] for analytical treatment or [9] on numerical approximation of (1.1) and passing with v, 6 to zero. We al~grnent (1.1) with initial and boundary conditions. However, the particular form of boundary conditions is not important. What we require is that they mike the bi-harmonic operator the square of a sectorial one. For instance we may consider

A caricature of a singular curvature flow in the plane

We show a comparison principle for viscosity super- and subsolutions to HamiltonJacobi equations with discontinuous Hamiltonians. The key point is that the Hamiltonian depends upon u and it has a special structure. The supersolution must enjoy some additional regularity.

Existence of self-similar evolution of crystals grown from supersaturated vapor

We study the modified Stefan problem in the plane for polygonal interfacial curves. Uniqueness of local in time solutions is shown while existence of local in time solutions has been proved in an earlier work of the author [P. Rybka, Advances in Differential Equations, 3 (1998), pp. 687--713]. Geometric properties of the flow are studied if the Wulff shape is a regular N-sided polygon and the initial interface has sufficiently small perimeter. Namely, if the isoperimetric quotient of the initial interface does not differ much from the isoperimetric quotient of the Wulff shape, then the interface shrinks to a point in finite time and the isoperimetric quotient decreases.