An inverse problem in population dynamics

The evolution of population densities of two interacting species in presence of diffusion phenomena is governed by a system of semilinear Volterra integrodifferential parabolic equations. In this system there are time convolution integrals, accounting for past history effects, which are essentially characterized by kernels depending on time only. These delay kernels can be viewed as entries of a 2x2 matrix K. The inverse problem of determining K via suitable population measurements is analyzed.     

Existence of minimizers for a class of quasi-convex functionals with non-standard growth

We study the lower semicontinuity properties of non-autonomous variational integrals whose energy densities satisfy general growth conditions. We apply these results to solve Dirichlet’s boundary value problems for such functionals. Received: June 14, 2000; in final form: November 25, 2000 Published online: December 19, 2001     

Semicontinuity of a functional depending on curvature

In this paper we prove a lower semicontinuity result for a functional , defined on a class of bounded subsets of with a piecewise boundary, with respect to the -convergence of the sets. The functional depends on the curvature of in a linear way and contains a penalizing term which prevents the appearance of thin sets in the symmetric difference , where in an -approximating sequence of . Received: 3 January 2001 / Accepted: 11 May 2001 / Published online: 19 October 2001     

Counterexample to lower semicontinuity in Calculus of Variations

An example is shown of a functional which is not lower semicontinuous with respect to -convergence. The function f is lower semicontinuous, convex in the second variable and linearly coercive. Application to nonexistence of minimizers in BV-setting is also given. Received: 2 May 2000 / Published online: 4 May 2001     

Higher integrability of the gradient and dimension of the singular set for minimisers of the Mumford–Shah functional

The paper is concerned with the higher regularity properties of the minimizers of the Mumford–Shah functional. It is shown that, near to singular points where the scaled Dirichlet integral tends to 0, the discontinuity set is close to an Almgren area minimizing set. As a byproduct, the set of singular points of this type has Hausdorff dimension at most N-2, N being the dimension of the ambient space. Assuming higher integrability of the gradient this leads to an optimal estimate of the Hausdorff dimension of the full singular set. Received: 5 July 2001 / Accepted: 29 November 2001 / Published online: 23 May 2002     

On lower semicontinuity in BH and 2-quasiconvexification

It is proved that if , with p > 1, if is bounded in , , and if in then provided is 2-quasiconvex and satisfies some appropriate growth and continuity condition. Characterizations of the 2-quasiconvex envelope when admissible test functions belong to BH^p are provided. Received: 10 October 2001 / Accepted: 8 May 2002 / Published online: 17 December 2002     

Minimizers of non convex scalar functionals and viscosity solutions of Hamilton-Jacobi equations

We consider a class of non convex scalar functionals of the form

On the equivalence of two variational problems

We consider the following problems where is a convex function, is an open bounded subset of is a closed convex subset of such that and and are suitable obstacles. We give conditions on the function {\it g} under which the two problems are equivalent. Received March 24, 1999/ Accepted January 14, 2000 / Published online June 28, 2000     

A discretization theory for a class of semi-coercive unilateral problems

Summary. In this paper, we present a convergence analysis applicable to the approximation of a large class of semi-coercive variational inequalities. The approach we propose is based on a recession analysis of some regularized Galerkin schema. Finite-element approximations of semi-coercive unilateral problems in mechanics are discussed. In particular, a Signorini-Fichera unilateral contact model and some obstacle problem with frictions are studied. The theoretical conditions proved are in good agreement with the numerical ones. Received January 14, 1999 / Revised version received June 24, 1999 / Published online July 12, 2000     

Relaxation in magnetostriction

For a family of nonlocal variational problems, a relaxation in terms of Young measures associated with minimizing sequences is discussed and explicitly computed. The nonlocality character is the main new feature. These computations generalize the same sort of ideas previously used in the analysis of micromagnetics to the case of magnetostriction in which interactions between elastic and magnetic properties are considered. This situation, however, is analyzed under important simplifying assumptions in dimension two. Received June 25, 1998 / Accepted February 26, 1999     

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.     

Non-local approximation of the Mumford-Shah functional

The Mumford-Shah functional, introduced to study image segmentation problems, is approximated in the sense of -convergence by a sequence of non-local integral functionals. Received June 6, 1996 / Accepted July 11, 1996     

Error estimates for the approximation of semicoercive variational inequalities

Summary. An abstract error estimate for the approximation of semicoercive variational inequalities is obtained provided a certain condition holds for the exact solution. This condition turns out to be necessary as is demonstrated analytically and numerically. The results are applied to the finite element approximation of Poisson's equation with Signorini boundary conditions and to the obstacle problem for the beam with no fixed boundary conditions. For second order variational inequalities the condition is always satisfied, whereas for the beam problem the condition holds if the center of forces belongs to the interior of the convex hull of the contact set. Applying the error estimate yields optimal order of convergence in terms of the mesh size . The numerical convergence rates observed are in good agreement with the predicted ones. Received August 16, 1993 / Revised version received March 21, 1994     

Characterization of optimal shapes and masses through Monge-Kantorovich equation

We study some problems of optimal distribution of masses, and we show that they can be characterized by a suitable Monge-Kantorovich equation. In the case of scalar state functions, we show the equivalence with a mass transport problem, emphasizing its geometrical approach through geodesics. The case of elasticity, where the state function is vector valued, is also considered. In both cases some examples are presented. Received February 10, 2000 / final version received July 21, 2000?Published online November 8, 2000     

Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian

Summary. In this paper, we derive quasi-norm a priori and a posteriori error estimates for the Crouzeix-Raviart type finite element approximation of the p-Laplacian. Sharper a priori upper error bounds are obtained. For instance, for sufficiently regular solutions we prove optimal a priori error bounds on the discretization error in an energy norm when . We also show that the new a posteriori error estimates provide improved upper and lower bounds on the discretization error. For sufficiently regular solutions, the a posteriori error estimates are further shown to be equivalent on the discretization error in a quasi-norm. Received January 25, 1999 / Revised version received June 5, 2000 Published online March 20, 2001     

On theoretical and numerical aspects of the bang-bang-principle

Summary In the first part of this paper we are dealing with theoretical statements and conditions which finally lead to bang-bang-principles. A careful analysis of these theorems is used for the development of a numerical method. This method consists of two stages: During the first iterations the number and approximate location of the switching points of the optimal control are determined. In the second phase a rapidly convergent algorithm determines the exact location. We apply this method successfully to a parabolic boundary control problem and give an extensive discussion of numerical results.The work of the second author on this paper was partially done during his stay at North Carolina State University, Graduate Program in Operations Research and Department of Mathematics, Raleigh, USA     

On two-dimensional parametric variational problems

Let be a two-dimensional parametric variational integral the Lagrangian F(x,z) of which is positive definite and elliptic, and suppose that is a closed rectifiable Jordan curve in . We then prove that there is a conformally parametrized minimizer of in the class of surfaces of the type of the disk B which are bounded by . An immediate consequence of this theorem is that the Dirichlet integral and the area functional have the same infima, a result whose proof usually requires a Lichtenstein-type mapping theorem or else Morrey's lemma on -conformal mappings. In addition we show that the minimizer of is H?lder continuous in B, and even in if satisfies a chord-arc condition. In Section 1 it is described how our results are related to classical investigations, in particular to the work of Morrey. Without difficulty our approach can be carried over to two-dimensional surfaces of codimension greater than one. Received July 20, 1998 / Accepted October 23, 1998     

Numerical approximation of non-homogeneous, non-convex vector variational problems

Summary. In this paper we study a numerical scheme for non-convex vector variational problems allowing for microstructure, based on the approximation of gradient Young measures. We present a convergence result and some numerical experiments. Received March 26, 2000 / Revised version received November 13, 2000 / Published online March 20, 2001     

Partial regularity for variational integrals with (s,\mu ,q)–Growth

We introduce integrands of –type, which are, roughly speaking, of lower (upper) growth rate ) satisfying in addition for some . Then, if , we prove partial –regularity of local minimizers by the way including integrands f being controlled by some N–function and also integrands of anisotropic power growth. Moreover, we extend the known results up to a certain limit and present examples which are not covered by the standard theory. Received: 17 February 2000 / Accepted: 23 January 2001 / Published online: 4 May 2001