Uniqueness of harmonic maps with small energy

Harmonic maps from B ₁ (0, ℝ³) to a smooth compact target manifold N with uniformly small scaled energy (see assumption (2) below) are shown to be unique for their boundary values. Received: 12 May 1997     

Partial regularity of weak solutions for the curve shortening flow

We study the regularity of certain weak solutions for the curve shortening flow in arbitrary codimension. These solutions arise as limits of a regularization process which is related to an approach suggested by Calabi. We prove that the set of times for which such a weak solution is not smooth has Hausdorff dimension at most ?. Received: 23 May 1998 / Revised version: 7 September 1998     

Four positive solutions for the semilinear elliptic equation: -Delta u+u=a(x)u^p+f(x) in {mathbb R}^N

We consider the existence of positive solutions of the following semilinear elliptic problem in : where , , , and . Under the conditions: 1° for all , 2° as , 3° there exist and such that 4°, we show that (*) has at least four positive solutions for sufficiently small but . Received December 11, 1998 / Accepted July 16, 1999 / Published online April 6, 2000     

Multiple solutions of hemivariational inequalities with area-type term

Hemivariational inequalities containing both an area-type and a non-locally Lipschitz term are considered. Multiplicity results are obtained by means of techniques of nonsmooth critical point theory. Received: December 24, 1998 / Accepted: October 1, 1999     

Zero sets of solutions to semilinear elliptic systems of first order

Consider a nontrivial smooth solution to a semilinear elliptic system of first order with smooth coefficients defined over an n-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution is contained in a countable union of smooth (n−2)-dimensional submanifolds. Hence it is countably (n−2)-rectifiable and its Hausdorff dimension is at most n−2. Moreover, it has locally finite (n−2)-dimensional Hausdorff measure. We show by example that every real number between 0 and n−2 actually occurs as the Hausdorff dimension (for a suitable choice of operator). We also derive results for scalar elliptic equations of second order. Oblatum 22-V-1998 & 26-III-1999 / Published online: 10 June 1999     

Convergence of fluctuation-splitting schemes for two dimensional scalar conservation laws with a kinetic solver

Summary. The “fluctuation-splitting schemes” (FSS in short) have been introduced by Roe and Sildikover to solve advection equations on rectangular grids and then extended to triangular grids by Roe, Deconinck, Struij... For a two dimensional nonlinear scalar conservation law, we consider the case of a triangular grid and of a kinetic approach to reduce the discretization of the nonlinear equation to a linear equation and apply a particular FSS called N-scheme. We show that the resulting scheme converges strongly in in a finite volume sense. Received February 25, 1997 / Revised version received November 8, 1999 / Published online August 24, 2000     

Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory

We study a perturbed semilinear problem with Neumann boundary condition where is a bounded smooth domain of , , , if or if and is the unit outward normal at the boundary of . We show that for any fixed positive integer K any “suitable” critical point of the function generates a family of multiple interior spike solutions, whose local maximum points tend to as tends to zero. Received March 7, 1999 / Accepted October 1, 1999 / Published online April 6, 2000     

An operator splitting method for nonlinear convection-diffusion equations

Summary. We present a semi-discrete method for constructing approximate solutions to the initial value problem for the -dimensional convection-diffusion equation . The method is based on the use of operator splitting to isolate the convection part and the diffusion part of the equation. In the case , dimensional splitting is used to reduce the -dimensional convection problem to a series of one-dimensional problems. We show that the method produces a compact sequence of approximate solutions which converges to the exact solution. Finally, a fully discrete method is analyzed, and demonstrated in the case of one and two space dimensions. ReceivedFebruary 1, 1996 / Revised version received June 24, 1996     

Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations

Summary. We prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically stable and unstable periodic solutions of the linear delay differential equation by a (periodic) boundary value approach. This equation arises, e.g., in the study of the numerical stability of collocation methods for computing periodic solutions of nonlinear delay equations. We obtain convergence results for the standard collocation algorithm and for two variants. In particular, estimates of the difference between the collocation solution and the true solution are derived. For the standard collocation scheme the convergence results are “unconditional”, that is, they do not require mesh-ratio restrictions. Numerical results that support the theoretical findings are also given. Received June 9, 2000 / Revised version received December 14, 2000 / Published online October 17, 2001     

Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg group

Denoting Δ_? the Laplacian operator on the (2N+1)-dimensional Heisenberg group ? ^N , we prove some nonexistence results for solutions of inequalities of the three types

Uniform boundary controllability of a semi-discrete 1-D wave equation

Summary. A numerical scheme for the controlled semi-discrete 1-D wave equation is considered. We analyze the convergence of the boundary controls of the semi-discrete equations to a control of the continuous wave equation when the mesh size tends to zero. We prove that, if the high modes of the discrete initial data have been filtered out, there exists a sequence of uniformly bounded controls and any weak limit of this sequence is a control for the continuous problem. The number of the eliminated frequencies depends on the mesh size and the regularity of the continuous initial data. The case of the HUM controls is also discussed. Received March 3, 2001 / Published online October 17, 2001