A singular perturbation problem with discontinuous data in a cuboid

We analyse the asymptotic behaviour of the solution of a 3Dsingularly perturbed convection–diffusion problem withdiscontinuous Dirichlet boundary data defined in a cuboid. Wewrite the solution in terms of a double series and we obtainan asymptotic approximation of the solution when the singularparameter 0. This approximation is given in terms of a finitecombination of products of error functions and characterizesthe effect of the discontinuities on the small -behaviour ofthe solution in the singular layers.     

Up-to boundary regularity for a singular perturbation problem of p-Laplacian type

In this paper we are interested in establishing up-to boundary uniform estimates for the one phase singular perturbation problem involving a nonlinear singular/degenerate elliptic operator. Our main result states: if Ω⊂Rn is a C1,α domain, for some 0<α<1 and uε verifies

     

On a class of sublinear singular elliptic problems with convection term

We establish several results related to existence, nonexistence or bifurcation of positive solutions for the boundary value problem −Δu+K(x)g(u)+a|∇u|=λf(x,u) in Ω, u=0 on ∂Ω, where Ω⊂RN(N?2) is a smooth bounded domain, 0<a?2, λ is a positive parameter, and f is smooth and has a sublinear growth. The main feature of this paper consists in the presence of the singular nonlinearity g combined with the convection term a|∇u|. Our approach takes into account both the sign of the potential K and the decay rate around the origin of the singular nonlinearity g. The proofs are based on various techniques related to the maximum principle for elliptic equations.     

Concentrating solutions in a two-dimensional elliptic problem with exponential Neumann data

We consider the elliptic equation -Δu+u=0 in a bounded, smooth domain Ω in R² subject to the nonlinear Neumann boundary condition . Here ?>0 is a small parameter. We prove that any family of solutions u_? for which ?∫_∂Ωe^u is bounded, develops up to subsequences a finite number m of peaks ξ_i∈∂Ω, in the sense that as ?→0. Reciprocally, we establish that at least two such families indeed exist for any given m?1.     

A Free Boundary Problem with Curvature

Abstract

In this paper we are interested in a free boundary problem with 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.     

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.     

Fourth order algorithms for a semilinear singular perturbation problem

Fourth order finite-difference algorithms for a semilinear singularly perturbed reaction–diffusion problem are discussed and compared both theoretically and numerically. One of them is the method of Sun and Stynes (1995) which uses a piecewise equidistant discretization mesh of Shishkin type. Another one is a simplified version of Vulanović's method (1993), based on a discretization mesh of Bakhvalov type. It is shown that the Bakhvalov mesh produces much better numerical results. This revised version was published online in June 2006 with corrections to the Cover Date.     

Long time regularity of solutions of the Hele–Shaw problem

In this paper we study the long-time behavior of solutions of the one phase Hele–Shaw problem without surface tension. We show that after a finite time solutions of the Hele–Shaw problem become starshaped and Lipschitz continuous in space. Based on this observation we then prove that the free boundary become smooth in space and time with nondegenerate free boundary speed.