Exact Controllability of Linear Stochastic Differential Equations and Related Problems

A notion of $L^p$-exact controllability is introduced for linear controlled (forward) stochastic differential equations, for which several sufficient conditions are established. Further, it is proved that the $L^p$-exact controllability, the validity of an observability inequality for the adjoint equation, the solvability of an optimization problem, and the solvability of an $L^p$-type norm optimal control problem are all equivalent

LpL^p improving bounds for averages along curves

We establish local $(L^p,L^q)$ mapping properties for averages on curves. The exponents are sharp except for endpoints.Comment: 37 pages, simplified argument (no further need for algebraic complexity theory!), to appear, JAM

A Duality Exact Sequence for Legendrian Contact Homology

We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself. In this sequence, the singular homology H_* maps to linearized contact cohomology CH^* which maps to linearized contact homology CH_* which maps to singular homology. In particular, the sequence implies a duality between the kernel of the map (CH_*\to H_*) and the cokernel of the map (H_* \to CH^*). Furthermore, this duality is compatible with Poincare duality in L in the following sense: the Poincare dual of a singular class which is the image of a in CH_* maps to a class \alpha in CH^* such that \alpha(a)=1. The exact sequence generalizes the duality for Legendrian knots in Euclidean 3-space [24] and leads to a refinement of the Arnold Conjecture for double points of an exact Lagrangian admitting a Legendrian lift with linearizable contact homology, first proved in [6].Comment: 57 pages, 10 figures. Improved exposition and expanded analytic detai

Dimension reduction for functionals on solenoidal vector fields

We study integral functionals constrained to divergence-free vector fields in $L^p$ on a thin domain, under standard $p$-growth and coercivity assumptions, $1<p<\infty$. We prove that as the thickness of the domain goes to zero, the Gamma-limit with respect to weak convergence in $L^p$ is always given by the associated functional with convexified energy density wherever it is finite. Remarkably, this happens despite the fact that relaxation of nonconvex functionals subject to the limiting constraint can give rise to a nonlocal functional as illustrated in an example.Comment: 25 page