We will show optimal regularity for minimizers of the Signorini problem for the Lame system. In particular if (Formula presented.) minimizes (Formula presented.)in the convex set (Formula presented.)where (Formula presented.) say. Then (Formula presented.). Moreover the free boundary, given by (Formula presented.)will be a (Formula presented.) graph close to points where (Formula presented.) is not degenerate. Similar results have been know before for scalar partial differential equations (see for instance [5, 6]). The novelty of this approach is that it does not rely on maximum principle methods and is therefore applicable to systems of equations.

In this paper we are concerned with singular points of solutions to the unstable free boundary problem Delta u = -chi({u>0}) in B-1. The problem arises in applications such as solid combustion, composite membranes, climatology and fluid dynamics. It is known that solutions to the above problem may exhibit singularities-that is points at which the second derivatives of the solution are unbounded-as well as degenerate points. This causes breakdown of by-now classical techniques. Here we introduce new ideas based on Fourier expansion of the non-linearity chi({u>0}). The method turns out to have enough momentum to accomplish a complete description of the structure of the singular set in R-3. A surprising fact in R-3 is that although u(rx)/sup(B1) vertical bar u(rx)vertical bar can converge at singularities to each of the harmonic polynomials xy, x(2) + y(2)/2 - z(2) and z(2) - x(2) + y(2)/2, it may not converge to any of the non-axially-symmetric harmonic polynomials alpha((1 + delta)x(2) + (1 - delta)y(2) - 2z(2)) with delta not equal 1/2. We also prove the existence of stable singularities in R-3.

We consider families of nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure of , as a function of on the set of Collet-Eckmann (CE) parameters: Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set of CE parameters , and, for each , a set of polynomially recurrent parameters containing as a Lebesgue density point, and constants , , so that, for every -Holder function , In addition, for all , the renormalisation period of satisfies , and there are uniform bounds on the rates of mixing of for all with . If , the set contains almost all CE parameters. Lower bounds: Assuming existence of a transversal mixing Misiurewicz-Thurston parameter , we find a set of CE parameters accumulating at , a constant , and a function , so that C vertical bar t - t(0)vertical bar(1/2) >= vertical bar integral A(0)d mu(t) - integral A(0)d mu(t0) vertical bar >= C-1 vertical bar t - t(0)vertical bar(1/2), for all t is an element of Delta(MT)'.

For a large class of non-uniformly hyperbolic attractors of dissipative diffeomorphisms, we prove that there are no holes in the basin of attraction: stable manifolds of points in the attractor fill-in a full Lebesgue measure subset. Then, Lebesgue almost every point in the basin is generic for the SRB (Sinai-Ruelle-Bowen) measure of the attractor. This solves a problem posed by Sinai and by Ruelle, for this class of systems.

5. Borcea, Julius

et al.

Bränden, Petter

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

In 1952 Lee and Yang proposed the program of analyzing phase transitions in terms of zeros of partition functions. Linear operators preserving non-vanishing properties are essential in this program and various contexts in complex analysis, probability theory, combinatorics, and matrix theory. We characterize all linear operators on finite or infinite-dimensional spaces of multivariate polynomials preserving the property of being non-vanishing whenever the variables are in prescribed open circular domains. In particular, this solves the higher dimensional counterpart of a long-standing classification problem originating from classical works of Hermite, Laguerre, Hurwitz and Polya-Schur on univariate polynomials with such properties.

Hedge integrals and Gromov-Witten theory2000In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 139, no 1, p. 173-199Article in journal (Refereed)

7. Favre, C

et al.

Jonsson, Mattias

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

We show that valuations on the ring R of holomorphic germs in dimension 2 may be naturally evaluated on plurisubharmonic functions, giving rise to generalized Lelong numbers in the sense of Demailly. Any plurisubharmonic function thus defines a real-valued function on the set V of valuations on R and - by way of a natural Laplace operator defined in terms of the tree structure on V - a positive measure on V. This measure contains a great deal of information on the singularity at the origin. Under mild regularity assumptions, it yields an exact formula for the mixed Monge-Ampere mass of two plurisubharmonic functions. As a consequence, any generalized Lelong number can be interpreted as an average of valuations. Using our machinery we also show that the singularity of any positive closed ( 1, 1) current T can be attenuated in the following sense: there exists a finite composition of blowups such that the pull-back of T decomposes into two parts, the first associated to a divisor with normal crossing support, the second having small Lelong numbers.

New bounds on the Lieb-Thirring constants2000In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 140, no 3, p. 693-704Article in journal (Refereed)

Abstract [en]

Improved estimates on the constants L(gamma,)d, foT 1/2 < gamma < 3/2, d epsilon N, in the inequalities for the eigenvalue moments of Schrodinger operators are established.

9.

Petersen, Dan

et al.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

We prove that for N equal to at least one of the integers 8, 12, 16, 20 the tautological ring is not Gorenstein. In fact, our N equals the smallest integer such that there is a non-tautological cohomology class of even degree on . By work of Graber and Pandharipande, such a class exists on , and we present some evidence indicating that N is in fact 20.

We consider the question of future global non-linear stability in the case of Einstein's equations coupled to a non-linear scalar field. The class of potentials V to which our results apply is defined by the conditions V(0)> 0, V'(0)=0 and V ''(0)> 0. Thus Einstein's equations with a positive cosmological constant represents a special case, obtained by demanding that the scalar field be zero. In that context, there are stability results due to Helmut Friedrich, the methods of which are, however, not so easy to adapt to the presence of matter. The goal of the present paper is to develop methods that are more easily adaptable. Due to the extreme nature of the causal structure in models of this type, it is possible to prove a stability result which only makes local assumptions concerning the initial data and yields global conclusions in time. To be more specific, we make assumptions in a set of the form B4r(0) (p) for some r(0)> 0 on the initial hypersurface, and obtain the conclusion that all causal geodesics in the maximal globally hyperbolic development that start in Br-0 (p) are future complete. Furthermore, we derive expansions for the unknowns in a set that contains the future of Br-0 (p). The advantage of such a result is that it can be applied regardless of the global topology of the initial hypersurface. As an application, we prove future global non-linear stability of a large class of spatially locally homogeneous spacetimes with compact spatial topology.