Wellposed convex integral functionals

Sufficient conditions are obtained for wellposedness of convex minimum problems of the calculus of variations for multiple integrals under strong or weak perturbations of the boundary data. Problems with a unique minimizer as well as problems with several solutions are treated. Wellposedness under weak convergence of the boundary data in W^{1 p}

ω is proved if p

>2 and a counterexample is exhibited if p

=2.     

Der fàltungsalgorithmus für dreidimensionale radon–probleme †

Filtered backprojection is an efficient numerical method for inversion of the Radon and x-ray transform. It can be adapted to the three dimensional case ([10], [11]). In this paper we present a method to calculate convolution kernels for filtered backprojection in three dimensions.     

Uniqueness of Dirichlet forms associated with systems of infinitely many Brownian balls in ℝ d

Summary. Dirichlet forms associated with systems of infinitely many Brownian balls in ℝ ^d are studied. Introducing a linear operator L ₀ defined on a space of smooth local functions, we show the uniqueness of Dirichlet forms associated with self adjoint Markovian extensions of L ₀. We also discuss the ergodicity of the reversible process associated with the Dirichlet form. Received: 18 July 1996/In revised form: 13 February 1997     

A fractional version of the peano-sard theorem

Sard's classical generalization of the Peano kernel theorem provides an extremely useful method for expressing and calculating sharp bounds for approximation errors. The error is expressed in terms of a derivative of the underlying function. However, we can apply the theorem only if the approximation is exact on a certain set of polynomials.

In this paper, we extend the Peano-Sard theorem to the case that the approximation is exact for a class of generalized polynomials (with non-integer exponents). As a result, we obtain an expression for the remainder in terms of a fractional derivative of the function under consideration. This expression permits us to give sharp error bounds as in the classical situation. An application of our results to the classical functional (vanishing on polynomials) gives error bounds of a new type involving weighted Sobolev-type spaces. In this way, we may state estimates for functions with weaker smoothness properties than usual.

The standard version of the Peano-Sard theory is contained in our results as a special case.     

Global bounds of solutions for a strongly coupled system of reaction-diffusion equations

Boundedness results for a strongly coupled system of reaction-diffusion equations on spatially bounded region are proved.     

Isometrical identities and inverse formulas in the one-dimensional heat equation

By a general method using the theory of reproducing kernels, some isometrical identities and inverse formulas for the analytical solutions of the heat equation are examined     

Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model

Summary. A positivity-preserving numerical scheme for a strongly coupled cross-diffusion model for two competing species is presented, based on a semi-discretization in time. The variables are the population densities of the species. Existence of strictly positive weak solutions to the semidiscrete problem is proved. Moreover, it is shown that the semidiscrete solutions converge to a non-negative solution of the continuous system in one space dimension. The proofs are based on a symmetrization of the problem via an exponential transformation of variables and the use of an entropy functional. Received September 10, 2001 / Revised version received February 25, 2002 / Published online June 17, 2002     

Branching random walks and contact processes on homogeneous trees

Summary. Branching random walks and contact processes on the homogeneous tree in which each site has d+1 neighbors have three possible types of behavior (for d≧ 2): local survival, local extinction with global survival, and global extinction. For branching random walks, we show that if there is local extinction, then the probability that an individual ever has a descendent at a site n units away from that individual’s location is at most d ^{− n/2} , while if there is global extinction, this probability is at most d ⁻ⁿ . Next, we consider the structure of the set of invariant measures with finite intensity for the system, and see how this structure depends on whether or not there is local and/or global survival. These results suggest some problems and conjectures for contact processes on trees. We prove some and leave others open. In particular, we prove that for some values of the infection parameter λ, there are nontrivial invariant measures which have a density tending to zero in all directions, and hence are different from those constructed by Durrett and Schinazi in a recent paper. Received: 26 April 1996/In revised form: 20 June 1996     

On the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity

This paper is concerned with a Cauchy problem where and is a nonnegative radially symmetric function in with compact support. Denote the solution of (P) by . Let if and $p^{\ast} = 1+6/(N-10) N \geq 11 p_{\ast} < p < p^{\ast} \lambda_{\varphi} > 0 $ such that: (i) If $ \lambda < \lambda_{\varphi} u_{\lambda} $ exists globally in time in the classical sense and converges to zero locally uniformly in as . (ii) If , then $ u_{\lambda} $ blows upincompletely in finite time. (iii) If , then blows upcompletely in finite time. Received: 20 December 1999; in final form: 26 May 2000 / Published online: 4 May 2001     

Recurrence and ergodicity of interacting particle systems

Many interacting particle systems with short range interactions are not ergodic, but converge weakly towards a mixture of their ergodic invariant measures. The question arises whether a.s.the process eventually stays close to one of these ergodic states, or if it changes between the attainable ergodic states infinitely often (“recurrence”). Under the assumption that there exists a convergence–determining class of distributions that is (strongly) preserved under the dynamics, we show that the system is in fact recurrent in the above sense. We apply our method to several interacting particle systems, obtaining new or improved recurrence results. In addition, we answer a question raised by Ed Perkins concerning the change of the locally predominant type in a model of mutually catalytic branching. Received: 22 January 1999 / Revised version: 24 May 1999     

Stability of infinite clusters in supercritical percolation

. A recent theorem by Häggström and Peres concerning independent percolation is extended to all the quasi-transitive graphs. This theorem states that if 0<p ₁<p ₂≤1 and percolation occurs at level p ₁, then every infinite cluster at level p ₂ contains some infinite cluster at level p ₁. Consequences are the continuity of the percolation probability above the percolation threshold and the monotonicity of the uniqueness of the infinite cluster, i.e., if at level p ₁ there is a unique infinite cluster then the same holds at level p ₂. These results are further generalized to graphs with a “uniform percolation” property. The threshold for uniqueness of the infinite cluster is characterized in terms of connectivities between large balls.     

Second-order subelliptic operators on Lie groups III: Hölder continuous coefficients

Let G be a connected Lie group with Lie algebra and an algebraic basis of . Further let denote the generators of left translations, acting on the -spaces formed with left Haar measure dg, in the directions . We consider second-order operators corresponding to a quadratic form with complex coefficients , , , . The principal coefficients are assumed to be H?lder continuous and the matrix is assumed to satisfy the (sub)ellipticity condition uniformly over G. We discuss the hierarchy relating smoothness properties of the coefficients of H with smoothness of the kernel. Moreover, we establish Gaussian type bounds for the kernel and its derivatives. Similar theorems are proved for operators in nondivergence form for which the principal coefficients are at least once differentiable. Received January 24, 1997 / Accepted June 5, 1998     

A method for the numerical solution of the one-dimensional inverse Stefan problem

Summary In this paper we suggest the use of complete families of solutions of the heat equation for the numerical solution of the inverse Stefan problem. Our approach leads to linear optimization problems which can be established and solved easily. Convergence results are proved. In a final section the method is applied to some examples.     

Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously

. Consider site or bond percolation with retention parameter p on an infinite Cayley graph. In response to questions raised by Grimmett and Newman (1990) and Benjamini and Schramm (1996), we show that the property of having (almost surely) a unique infinite open cluster is increasing in p. Moreover, in the standard coupling of the percolation models for all parameters, a.s. for all p ₂>p ₁>p _c , each infinite p ₂-cluster contains an infinite p ₁-cluster; this yields an extension of Alexander's (1995) “simultaneous uniqueness” theorem. As a corollary, we obtain that the probability θ _v (p) that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p>p _c . All our results extend to quasi-transitive infinite graphs with a unimodular automorphism group. Received: 22 December 1997 / Revised version: 1 July 1998     

Dependent percolation in two dimensions

For a natural number k, define an oriented site percolation on ℤ² as follows. Let x _i , y _j be independent random variables with values uniformly distributed in {1, …, k}. Declare a site (i, j) ∈ℤ² closed if x _i = y _j , and open otherwise. Peter Winkler conjectured some years ago that if k≥ 4 then with positive probability there is an infinite oriented path starting at the origin, all of whose sites are open. I.e., there is an infinite path P = (i ₀, j ₀)(i ₁, j ₁) · · · such that 0 = i ₀≤i ₁≤· · ·, 0 = j ₀≤j ₁≤· · ·, and each site (i _n , j _n ) is open. Rather surprisingly, this conjecture is still open: in fact, it is not known whether the conjecture holds for any value of k. In this note, we shall prove the weaker result that the corresponding assertion holds in the unoriented case: if k≤ 4 then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive. Furthermore, we shall show that our method can be applied to a wide variety of distributions of (x _i ) and (y _j ). Independently, Peter Winkler [14] has recently proved a variety of similar assertions by different methods. Received: 4 March 1999 / Revised version: 27 September 1999 / Published online: 21 June 2000     

Exponential decay rate of survival probability in a disastrous random environment

Summary. We study the exponential decay rate of the survival probability up to time t>0 of a random walker moving in Zopf; ^d in a temporally and spatially fluctuating random environment. When the random walker has a speed parameter κ>0, we investigate the influence of κ on the exponential decay rate λ(d,κ). In particular we prove that for any fixed d≥1, λ(d,κ) behaves like as logκ as κ↘0. Received: 21 May 1996 / In revised form: 2 February 1997     

Some remarks on the level set flow by anisotropic curvature

We show that almost all level sets of the unique viscosity solution for general anisotropic mean curvature flow satisfy a weak form of the flow equation. This generalizes the case of isotropic mean curvature flow studied by Evans and Spruck, in which a relation between the viscosity solution and Brakke's varifold mean curvature flow is established. Received June 4, 1998 / Accepted February 26, 1999     

Backward differentiation type formulas for Volterra integral equations of the second kind

Summary Numerical integration formulas are discussed which are obtained by differentiation of the Volterra integral equation and by applying backward differentiation formulas to the resulting integro-differential equation. In particular, the stability of the method is investigated for a class of convolution kernels. The accuracy and stability behaviour of the method proposed in this paper is compared with that of (i) a block-implicit Runge-Kutta scheme, and (ii) the scheme obtained by applying directly a quadrature rule which is reducible to the backward differentiation formulas. The present method is particularly advantageous in the case of stiff Volterra integral equations.     

A fully discrete numerical scheme for weighted mean curvature flow

Summary. We analyze a fully discrete numerical scheme approximating the evolution of n–dimensional graphs under anisotropic mean curvature. The highly nonlinear problem is discretized by piecewise linear finite elements in space and semi–implicitly in time. The scheme is unconditionally stable und we obtain optimal error estimates in natural norms. We also present numerical examples which confirm our theoretical results. Received October 2, 2000 / Published online July 25, 2001     

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