Regularity of a free boundary problem

Let u be the Newtonian potential of a real analytic distribution in an open set Ω. In this paper we assume u is analytically continuable from the complement of Ω into some neighborhood of a point x₀ ∈ ∂Ω, and we study conditions under which the analytic continuation implies that ∂Ω is a real analytic hypersurface in some neighborhood of x₀.     

Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary.     

Regularity in a one-phase free boundary problem for the fractional Laplacian

For a one-phase free boundary problem involving a fractional Laplacian, we prove that “flat free boundaries” are C^1,α$C^{1,\alpha }$. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free boundary problem with the standard Laplacian.     

On some properties of the free boundary in a neighborhood of the points of contact with a given boundary

For the simplest elliptic obstacle problem, the behavior of the free boundary in the vicinity of the points where it meets the prescribed boundary of a domain is studied. The previous result of the author on the C¹ smoothness of the boundary ∂N of the noncoincidence set is improved. The Lipschitz condition on ∂N assumed earlier is shown to be superfluous. Bibliography: 4 titles. Dedicated to O. A. Ladyzhenskaya on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 249, 1997, pp. 303–312. Translated by I. Kostin.     

Asymptotics of Green's function for the Neumann problem in a neighborhood of a boundary point

The asymptotic behavior of Green's function of the exterior and interior Neumann problem is derived in the neighborhood of a pole lying on the boundary of the domain.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 131, pp. 142–147, 1983.     

Regularity of the free boundary for the one phase Hele-Shaw problem

In this paper we prove that, in a local neighborhood, Lipschitz continuous free boundary of a solution of the one-phase Hele-Shaw problem is indeed smooth if the solution is Lipschitz continuous and non-degenerate in the neighborhood.     

Motion of a vibrating string in the presence of a convex obstacle: A free boundary problem

Here we study the motion of a vibrating string in the presence of an arbitrary obstacle. We show that if the string always rebounds on the concave parts of the obstacle, it can either rebound or roll on the convex parts. The latter is the case if the velocity of the string is null at the contact point just before contact, or if the contact point propagates at a characteristic speed. Four examples are given. The three first correspond to the same obstacle, a sinusoidal arc, but with different initial conditions. In the first case, the string rebounds on the whole of the obstacle and the motion is explicitly determined when it is periodic. In the second case, the string rolls on the convex part of the obstacle up to the inflexion point and then rebounds on the concave part and unwinds on the convex part. In the third case, the string is initially at rest on the obstacle; then it instantaneously leaves the concave part while it unwinds progressively on the convex part. The fourth case is similar to the third but with a different obstacle; the motion, which is periodic, is determined explicitly.     

Regularity of solutions to a contact problem

We consider a variational inequality for the Lamé system which models an elastic body in contact with a rigid foundation. We give conditions on the domain and the contact set which allow us to prove regularity of solutions to the variational inequality. In particular, we show that the gradient of the solution is a square integrable function on the boundary.

     

Regularity of solutions to a free boundary problem modeling tumor growth by Stokes equation

In this paper we investigate regularity of solutions to a free boundary problem modeling tumor growth in fluid-like tissues. The model equations include a quasi-stationary diffusion equation for the nutrient concentration, and a Stokes equation with a source representing the proliferation density of the tumor cells, subject to a boundary condition with stress tensor effected by surface tension. This problem is a fully nonlinear problem involving nonlocal terms. Based on the employment of the functional analytic method and the theory of maximal regularity, we prove that the free boundary of this problem is real analytic in temporal and spatial variables for initial data of less regularity.     

The saddle point property for focusing selfsimilar solutions in a free boundary problem

We establish the saddle point property of the focusing selfsimilar solution of a free boundary problem for the heat equation with free boundary conditions given by and .

     

Regularity of a free boundary in parabolic potential theory

We study the regularity of the free boundary in a Stefan-type problem

with no sign assumptions on and the time derivative .

     

A free boundary problem for the stokes system with contact lines

In this paper we study an evolution free boundary problem for the two-dimensional Stoltes system in the case in which the free boundary intersects the solid boundary of a container. We prove existence and uniqueness of solutionsfor this problem in suitable classes of functions and under suitable smallness conditions for the initial data. For these solutions the contact point moves with an uniform velocity with respect to the container.     

Regularity of the obstacle problem for a fractional power of the laplace operator

Given a function φ and s ∈ (0, 1), we will study the solutions of the following obstacle problem:

• u ≥ φ in ?ⁿ,
• (??)^su ≥ 0 in ?ⁿ,
• (??)^su(x) = 0 for those x such that u(x) > φ(x),
• lim_{|x| → + ∞} u(x) = 0.

We show that when φ is C^{1, s} or smoother, the solution u is in the space C^{1, α} for every α < s. In the case where the contact set {u = φ} is convex, we prove the optimal regularity result u ∈ C^{1, s}. When φ is only C^{1, β} for a β < s, we prove that our solution u is C^{1, α} for every α < β. © 2006 Wiley Periodicals, Inc.