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where are real smooth vector fields satisfying Hörmander’s condition in some bounded domain (), and the coefficients , are real valued, bounded measurable functions defined in , satisfying the uniform positivity conditions
for a.e. , every , and some constant .
We prove that if the coefficients , belong to the Hölder space with respect to the distance induced by the vector fields, local Schauder estimates of the following kind hold:
for any .
If the coefficients , belong to the space with respect to the distance induced by the vector fields, local estimates of the following kind hold, for every :
This paper is concerned with Schrödinger equations with variable coefficients and unbounded electromagnetic potentials, where the kinetic energy part is a long-range perturbation of the flat Laplacian and the electric (respectively magnetic) potential can grow subquadratically (respectively sublinearly) at spatial infinity. We prove sharp (local-in-time) Strichartz estimates, outside a large compact ball centered at the origin, for any admissible pair including the endpoint. Under the nontrapping condition on the Hamilton flow generated by the kinetic energy, global-in-space estimates are also studied. Finally, under the nontrapping condition, we prove Strichartz estimates with an arbitrarily small derivative loss without asymptotic flatness on the coefficients.
We show an energy convexity along any harmonic map heat flow with small initial energy and fixed boundary data on the unit -disk. In particular, this gives an affirmative answer to a question raised by W. Minicozzi asking whether such harmonic map heat flow converges uniformly in time strongly in the -topology, as time goes to infinity, to the unique limiting harmonic map.
Subfactors where the initial branching point of the principal graph is -valent are subject to strong constraints called triple point obstructions. Since more complicated initial branches increase the index of the subfactor, triple point obstructions play a key role in the classification of small index subfactors. There are two strong triple point obstructions, called the triple-single obstruction and the quadratic tangles obstruction. Although these obstructions are very closely related, neither is strictly stronger. In this paper we give a more general triple point obstruction which subsumes both. The techniques are a mix of planar algebraic and connection-theoretic techniques with the key role played by the rotation operator.
In this paper, we consider the wave equation in 3-dimensional space with an energy-subcritical nonlinearity, either in the focusing or defocusing case. We show that any radial solution of the equation which is bounded in the critical Sobolev space is globally defined in time and scatters. The proof depends on the compactness/rigidity argument, decay estimates for radial, “compact” solutions, gain of regularity arguments and the “channel of energy” method.
In this paper, we prove that there exists some small such that the derivative nonlinear Schrödinger equation (DNLS) is globally well-posed in the energy space, provided that the initial data satisfies . This result shows us that there are no blow-up solutions whose masses slightly exceed , even if their energies are negative. This phenomenon is much different from the behavior of the nonlinear Schrödinger equation with critical nonlinearity. The technique used is a variational argument together with the momentum conservation law. Further, for the DNLS on the half-line , we show the blow-up for the solution with negative energy.
We consider Calderón’s inverse problem with partial data in dimensions . If the inaccessible part of the boundary satisfies a (conformal) flatness condition in one direction, we show that this problem reduces to the invertibility of a broken geodesic ray transform. In Euclidean space, sets satisfying the flatness condition include parts of cylindrical sets, conical sets, and surfaces of revolution. We prove local uniqueness in the Calderón problem with partial data in admissible geometries, and global uniqueness under an additional concavity assumption. This work unifies two earlier approaches to this problem—one by Kenig, Sjöstrand, and Uhlmann, the other by Isakov—and extends both. The proofs are based on improved Carleman estimates with boundary terms, complex geometrical optics solutions involving reflected Gaussian beam quasimodes, and invertibility of (broken) geodesic ray transforms. This last topic raises questions of independent interest in integral geometry.