Initial blow-up rates and universal bounds for nonlinear heat equations

We establish a universal upper bound on the initial blow-up rate for all positive classical solutions of the Dirichlet problem for the nonlinear heat equation

     

Universal bound for global positive solutions of a superlinear parabolic problem

We prove universal a priori estimates of global positive solutions of the parabolic problem in , on . Here is a bounded domain in , , and p < 5 if n=3. Received April 6, 2000 / Accepted September 21, 2000 / Published online February 5, 2001     

An interior second derivative bound for solutions of Hessian equations

In previous work we showed that weak solutions in of the k-Hessian equation have locally bounded second derivatives if g is positive and sufficiently smooth and p > kn/2. Here we improve this result to p > k(n-1)/2, which is known to be sharp in the Monge-Ampère case k=n > 2. Received June 21, 1999 / Accepted June 12, 2000 / Published online November 9, 2000     

Some interior regularity results for solutions of Hessian equations

We prove monotonicity formulae related to degenerate k-Hessian equations which yield Morrey type estimates for certain integrands involving the second derivatives of the solution. In the special case k=2 we deduce that weak solutions in , , have locally H?lder continuous gradients. In the nondegenerate case we also show that weak solutions in , , have locally bounded second derivatives. Received February 25, 1999 / Accepted June 11, 1999 / Published online April 6, 2000     

On universal subsets of Banach spaces

We prove that to most of the known hypercyclic operators A on separable Banach spaces there exist compact (compact convex, compact connected) subsets K of E such that each compact (compact convex, compact connected) subset of E can be approximated with respect to Hausdorff's distance by for suitable . Received July 8, 1997, in final form October 17, 1997     

On quasilinear parabolic equations which admit global solutions for initial data with unrestricted growth

Three classes of quasilinear parabolic equations which have the common feature that their principal coefficients decay as the solution or its gradient blows up are studied. Long time existence of solutions for their Cauchy problems for initial data with arbitrary growth is established. Received September 9, 1999 / Accepted May 9, 2000 / Published online September 14, 2000     

Local isometries of function spaces

Abstract. We show that for a large class of function spaces any isometry that coincides locally with a surjective isometry must be automatically surjective. This class includes finite-codimensional subspaces of and spaces of E-valued continuous functions for finite-dimensional or uniformly convex and algebraically reflexive E. Received: 5 November 2001 / Published online: 14 February 2003 Thanks: Research of both authors partially supported by a grant \# 1102386 from the NSF and a grant \# DST/INT/US(NSF-RP041)/2000 from the DST     

Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Stratonovitch's case

In this article we prove new results concerning the long-time behavior of random fields that are solutions in some sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise. Our main results state that these random fields eventually homogeneize with respect to the spatial variable and finally converge to a non-random global attractor which consists of two spatially and temporally homogeneous asymptotic states. More precisely, we prove that the random fields either stabilize exponentially rapidly with probability one around one of the asymptotic states, or that they set out to oscillate between them. In the first case we can also determine exactly the corresponding Lyapunov exponents. In the second case we prove that the random fields are in fact recurrent in that they can reach every point between the two asymptotic states in a finite time with probability one. In both cases we also interpret our results in terms of stability properties of the global attractor and we provide estimates for the average time that the random fields spend in small neighborhoods of the asymptotic states. Our methods of proof rest upon the use of a suitable regularization of the Brownian motion along with a related Wong-Zaka? approximation procedure. Received: 8 April 1997/Revised version: 30 January 1998     

Sobolev spaces and harmonic maps between singular spaces

Recently Korevaar and Schoen developed a Sobolev theory for maps from smooth (at least ) manifolds into general metric spaces by proving that the weak limit of appropriate average difference quotients is well behaved. Here we extend this theory to functions defined over Lipschitz manifold. As an application we then prove an existence theorem for harmonic maps from Lipschitz manifolds to NPC metric spaces. Received December 6, 1996 / Accepted March 4, 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     

Scaling identity for crossing Brownian motion in a Poissonian potential

We consider d-dimensional Brownian motion in a truncated Poissonian potential (d≥ 2). If Brownian motion starts at the origin and ends in the closed ball with center y and radius 1, then the transverse fluctuation of the path is expected to be of order |y|^ξ, whereas the distance fluctuation is of order |y|^χ. Physics literature tells us that ξ and χ should satisfy a scaling identity 2ξ− 1 = χ. We give here rigorous results for this conjecture. Received: 31 December 1997 / Revised version: 14 April 1998     

Berezin transform on¶compact Hermitian symmetric spaces

Let X=G ^* be a compact Hermitian symmetric space. We study the Berezin transform on L ²(X) and calculate its spectrum under the decomposition of L ²(X) into the irreducible representations of G ^*. As applications we find the expansion of powers of the canonical polynomial (Bergman reproducing kernel for the canonical line bundle) in terms of the spherical polynomials on X, and we find the irreducible decomposition of tensor products of Bergman spaces on X. Received: 10 September 1996 / Revised version: 10 September 1997