Ideal weights: Asymptotically optimal versions of doubling, absolute continuity, and bounded mean oscillation

Sharp inequalities between weight bounds (from the doubling, A_p, and reverse Hölder conditions) and the BMO norm are obtained when the former are near their optimal values. In particular, the BMO norm of the logarithm of a weight is controlled by the square root of the logarithm of its A_∞ bound. These estimates lead to a systematic development of asymptotically sharp higher integrability results for reverse Hölder weights and extend Coifman and Fefferman's formulation of the A_∞ condition as an equivalence relation on doubling measures to the setting in which all bounds become optimal over small scales.     

Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory

We study the general asymptotic behavior of critical points, including those of non-minimal energy type, of the functional for the van der Waals-Cahn-Hilliard theory of phase transitions. We prove that the interface is close to a hypersurface with mean curvature zero when no Lagrange multiplier is present, and with locally constant mean curvature in general. The energy density of the limiting measure has integer multiplicity almost everywhere modulo division by a surface energy constant. Received March 16, 1999 / Accepted June 11, 1999     

On the Inverse and Determinant of Certain Truncated Wiener-Hopf Operators

The solution of a problem arising in integrable systems requires sharp asymptotics for the inverses and determinants of truncated Wiener-Hopf operators, both in the regular case (where the non-truncated Wiener-Hopf operator is invertible) and in singular cases. This paper treats two cases where the symbol of the Wiener-Hopf operator has Fisher-Hartwig singularities, one double zero or two simple zeros. We find formulas for the inverse that hold uniformly throughout the underlying interval with very small error, and formulas for the determinant with very small error.     

Weakly Gibbsian representations for joint measures of quenched lattice spin models

Can the joint measures of quenched disordered lattice spin models (with finite range) on the product of spin-space and disorder-space be represented as (suitably generalized) Gibbs measures of an “annealed system”? - We prove that there is always a potential (depending on both spin and disorder variables) that converges absolutely on a set of full measure w.r.t. the joint measure (“weak Gibbsianness”). This “positive” result is surprising when contrasted with the results of a previous paper [K6], where we investigated the measure of the set of discontinuity points of the conditional expectations (investigation of “a.s. Gibbsianness”). In particular we gave natural “negative” examples where this set is even of measure one (including the random field Ising model). Further we discuss conditions giving the convergence of vacuum potentials and conditions for the decay of the joint potential in terms of the decay of the disorder average over certain quenched correlations. We apply them to various examples. From this one typically expects the existence of a potential that decays superpolynomially outside a set of measure zero. Our proof uses a martingale argument that allows to cut (an infinite-volume analogue of) the quenched free energy into local pieces, along with generalizations of Kozlov's constructions. Received: 11 November 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000 RID="*" ID="*" Work supported by the DFG Schwerpunkt `Wechselwirkende stochastische Systeme hoher Komplexit?t'     

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     

On the Curvature of the Free Boundary for the Obstacle Problem in Two Dimensions

We give a new proof of the fact that the free boundary for the obstacle problem in two dimensions satisfies a natural and sharp inner ball condition.     

Bernoulli jets and the zero mean curvature equation

We consider an elliptic PDE problem related with fluid mechanics. We show that level sets of rescaled solutions satisfy the zero mean curvature equation in a suitable weak viscosity sense. In particular, such level sets cannot be touched from below (above) by a convex (concave) paraboloid in a suitably small neighborhood.     

The Gauss-Green theorem and removable sets for PDEs in divergence form

Applying a very general Gauss-Green theorem established for the generalized Riemann integral, we obtain simple proofs of new results about removable sets of singularities for the Laplace and minimal surface equations. We treat simultaneously singularities with respect to differentiability and continuity.     

Spatial decay estimates for the Maxwell-Cattaneo equations with mixed boundary conditions

In this paper the authors derive spatial decay bounds for the temperature and heat flux as defined by the Generalized Maxwell-Cattaneo equations for heat conduction in a semi-infinite cylinder when the temperature and the tangential components of the heat flux vector vanish on the lateral surface of the cylinder. The results here supplement those previously found by the authors [5] when the heat flux vector was assumed to be zero on the lateral surface but no condition was imposed on the temperature.Received: February 7, 2002; revised: June 3, 2002     

Boundary partial regularity for the high dimensional Navier–Stokes equations

We consider suitable weak solutions of the incompressible Navier–Stokes equations in two cases: the 4D time-dependent case and the 6D stationary case. We prove that up to the boundary, the two-dimensional Hausdorff measure of the set of singular points is equal to zero in both cases.     

Quasianalytic and nonquasianalytic solutions for a class of weakly hyperbolic Cauchy problems

In the weakly hyperbolic Cauchy problem, we investigate the relation between the modulus of continuity in the time variable of the coefficients and the well-posedness in Beurling-Roumieu classes of ultradifferentiable functions and functionals. We find a sharp condition on the modulus of continuity assuring the well-posedness in nonquasianalytic classes.     

Inertial manifolds and normal hyperbolicity

Our aim in this article is to derive an existence theorem of inertial manifolds for fairly general equations with a self-adjoint or nonself-adjoint linear operator in a Banach space setting. A sharp form of the spectral gap condition is given. Many other properties are proven including an interesting characterization of the inertial manifold and the normal hyperbolicity of the inertial manifold.     

Global s-solvability and global s-hypoellipticity for certain perturbations of zero order of systems of constant real vector fields

The main purpose of this paper is to show, in the two-dimensional torus, a necessary and sufficient condition in order to certain perturbations of zero order of a system of constant real vector fields to be globally s-solvable. We are also interested in studying its global s-hypoellipticity. We present connections between these global concepts and a priori estimates. We also present two applications of our results for systems of operators with variable coefficients.     

Dimension gap under Sobolev mappings

We prove an essentially sharp estimate in terms of generalized Hausdorff measures for the images of boundaries of Hölder domains under continuous Sobolev mappings, satisfying suitable Orlicz–Sobolev conditions. This estimate marks a dimension gap, which was first observed in [2] for conformal mappings.     

Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates

Recently Wolff [25] obtained a nearly sharp bilinear restriction theorem for bounded subsets of the cone in general dimension. We obtain the endpoint of Wolff's estimate and generalize to the case when one of the subsets is large. As a consequence, we are able to deduce some nearly-sharp null form estimates. Received: 24 September 1999 / in final form: 25 April 2000 / Published online: 4 May 2001     

Nontrivial solutions of elliptic semilinear equations¶at resonance

We find nontrivial solutions for semilinear boundary value problems having resonance both at zero and at infinity. Received: 14 January 1999 / Revised version: 17 May 1999     

Expansion of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition

This paper studies a linear hyperbolic system with static boundary condition that was first studied in Neves et al. [J. Funct. Anal. 67(1986) 320-344]. It is shown that the spectrum of the system consists of zeros of a sine-type function and the generalized eigenfunctions of the system constitute a Riesz basis with parentheses for the root subspace. The state space thereby decomposes into topological direct sum of root subspace and another invariant subspace in which the associated semigroup is superstable: that is to say, the semigroup is identical to zero after a finite time period.     

Spectral optimization for the Stekloff–Laplacian: The stability issue

We consider the problem of minimizing the first non-trivial Stekloff eigenvalue of the Laplacian, among sets with given measure. We prove that the Brock–Weinstock inequality, asserting that optimal shapes for this spectral optimization problem are balls, can be improved by means of a (sharp) quantitative stability estimate. This result is based on the analysis of a certain class of weighted isoperimetric inequalities already proved in Betta et al. (1999) [2]: we provide some new (sharp) quantitative versions of these, achieved by means of a suitable calibration technique.