A free boundary problem with curvature
نویسنده
چکیده
In this paper we are interested in a free boundary problem whith a motion law involving the mean curvature term of the free boundary. Viscosity solutions are introduced as a notion of global-time solutions past singularities. We show the comparison principle for viscosity solutions, which yields the existence of minimal and maximal solutions for given initial data. We also prove uniqueness of the solution for several classes of initial data and discuss the possibility of nonunique solutions. 0 Introduction In this paper we introduce a method to study existence and uniqueness properties of viscosity solutions for the free boundary problems of the following form:
منابع مشابه
Free and constrained equilibrium states in a variational problem on a surface
We study the equilibrium states for an energy functional with a parametric force field on a region of a surface. Consideration of free equilibrium states is based on Lyusternik - Schnirelman's and Skrypnik's variational methods. Consideration of equilibrium states under a constraint of geometrical character is based on an analog of Skrypnik's method, described in [P. Vyridis, {it Bifurcation in...
متن کاملTHE STEFAN PROBLEM WITH KINETIC FUNCTIONS AT THE FREE BOUNDARY
This paper considers a class of one-dimensional solidification problem in which kinetic undercooling is incorporated into the temperature condition at the interface. A model problem with nonlinear kinetic law is considered. The main result is an existence theorem. The mathematical effects of the kinetic term are discussed
متن کاملBifurcation in a variational problem on a surface with a constraint
We describe a variational problem on a surface under a constraintof geometrical character. Necessary and sufficient conditions for the existence ofbifurcation points are provided. In local coordinates the problem corresponds toa quasilinear elliptic boundary value problem. The problem can be consideredas a physical model for several applications referring to continuum medium andmembranes.
متن کاملA two-phase free boundary problem for a semilinear elliptic equation
In this paper we study a two-phase free boundary problem for a semilinear elliptic equation on a bounded domain $Dsubset mathbb{R}^{n}$ with smooth boundary. We give some results on the growth of solutions and characterize the free boundary points in terms of homogeneous harmonic polynomials using a fundamental result of Caffarelli and Friedman regarding the representation of functions whose ...
متن کاملNumerical Solution of a Free Boundary Problem from Heat Transfer by the Second Kind Chebyshev Wavelets
In this paper we reduce a free boundary problem from heat transfer to a weakly Singular Volterra integral equation of the first kind. Since the first kind integral equation is ill posed, and an appropriate method for such ill posed problems is based on wavelets, then we apply the Chebyshev wavelets to solve the integral equation. Numerical implementation of the method is illustrated by two ben...
متن کاملConvexity Properties of the Free Boundary and Gradient Estimates for Quasi-linear Elliptic Equations
This paper is devoted to a proof of the convexity of a free boundary for a quasi-linear problem deened on a convex domain in IR 2 and to the obtention of L 1-bounds on the gradient of a solution. The free boundary can be seen as the boundary of the coincidence set of an obstacle problem. The bound on the gradient explicitely depends on the curvature of the boundary of the domain. The main tool ...
متن کامل