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Some generic properties of nonlinear second order diffusional type problem

Vladimír Ďurikovič, Mária Ďurikovičová (1999)

Archivum Mathematicum

We are interested of the Newton type mixed problem for the general second order semilinear evolution equation. Applying Nikolskij’s decomposition theorem and general Fredholm operator theory results, the present paper yields sufficient conditions for generic properties, surjectivity and bifurcation sets of the given problem.

Some Gradient Estimates on Covering Manifolds

Nick Dungey (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

Let M be a complete Riemannian manifold which is a Galois covering, that is, M is periodic under the action of a discrete group G of isometries. Assuming that G has polynomial volume growth, we provide a new proof of Gaussian upper bounds for the gradient of the heat kernel of the Laplace operator on M. Our method also yields a control on the gradient in case G does not have polynomial growth.

Spectral shift and multiplicity of the first eigenvalue of the magnetic Schrödinger operator in two dimensions

László Erdős (2002)

Annales de l’institut Fourier

We show that the lowest eigenvalue of the magnetic Schrödinger operator on a line bundle over a compact Riemann surface M is bounded by the L 1 -norm of the magnetic field B . This implies a similar bound on the multiplicity of the ground state. An example shows that this degeneracy can indeed be comparable with ∫ M | B | even in case of the trivial bundle.

Stability results for Harnack inequalities

Alexander Grigor'yan, Laurent Saloff-Coste (2005)

Annales de l’institut Fourier

We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack inequalities under certain non-uniform changes of the weight. We also prove necessary and sufficient conditions for the Harnack inequalities to hold on complete non-compact manifolds having non-negative Ricci curvature outside a compact set and a finite first Betti number or just having asymptotically...

Submersions and equivariant Quillen metrics

Xiaonan Ma (2000)

Annales de l'institut Fourier

In this paper, we calculate the behaviour of the equivariant Quillen metric by submersions. We thus extend a formula of Berthomieu-Bismut to the equivariant case.

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