ŁOJASIEWICZ INEQUALITY IN P-MINIMAL STRUCTURES

Th purpose of this paper is to extend the Łojasiewicz inequality for functions definable in some subclass of P-minimal structures. More precisely, we prove that the Łojasiewicz inequality holds for functions definable in poptimal expansions of Qp. It is also shown that the Łojasiewicz exponent is a rational number in such p-optimal expansions.

On the asymptotic behavior of the solutions to parabolic variational inequalities

We consider various versions of the obstacle and thin-obstacle problems, we interpret them as variational inequalities, with non-smooth constraint, and prove that they satisfy a new constrained Łojasiewicz inequality. The difficulty lies in the fact that, since the constraint is non-analytic, the pioneering method of L. Simon ([22]) does not apply and we have to exploit a better understanding on the constraint itself. We then apply this inequality to two associated problems. First we combine it with an abstract result on parabolic variational inequalities, to prove the convergence at infinity of the strong global solutions to the parabolic obstacle and thin-obstacle problems to a unique stationary solution with a rate. Secondly, we give an abstract proof, based on a parabolic approach, of the epiperimetric inequality, which we then apply to the singular points of the obstacle and thin-obstacle problems.

Locality in the Fukaya category of a hyperkähler manifold

Let $(M,I,J,K,g)$ be a hyperkähler manifold. Then the complex manifold $(M,I)$ is holomorphic symplectic. We prove that for all real $x,y$, with $x^{2}+y^{2}=1$ except countably many, any finite-energy $(xJ+yK)$-holomorphic curve with boundary in a collection of $I$-holomorphic Lagrangians must be constant. By an argument based on the Łojasiewicz inequality, this result holds no matter how the Lagrangians intersect each other. It follows that one can choose perturbations such that the holomorphic polygons of the associated Fukaya category lie in an arbitrarily small neighborhood of the Lagrangians. That is, the Fukaya category is local. We show that holomorphic Lagrangians are tautologically unobstructed. Moreover, the Fukaya $A_{\infty }$ algebra of a holomorphic Lagrangian is formal. Our result also explains why the special Lagrangian condition holds without instanton corrections for holomorphic Lagrangians.

Using Landweber iteration to quantify source conditions – a numerical study

Source conditions of the type {x^{\dagger}\in\mathcal{R}((A^{\ast}A)^{\mu})} are a standard assumption in the theory of inverse problems to show convergence rates of regularized solutions as the noise in the data goes to zero. Unfortunately, it is rarely possible to verify these conditions in practice, rendering data-independent parameter choice rules unfeasible. In this paper we show that such a source condition implies a Kurdyka–Łojasiewicz inequality with certain parameters depending on μ. While the converse implication is unclear from a theoretical point of view, we propose an algorithm which represents a first attempt that allows to approximate the value of μ numerically. It is based on combining the Landweber iteration with the Kurdyka–Łojasiewicz inequality. We conduct several numerical experiments to demonstrate the potential and limitations of the current method. We also show that the source condition implies a lower bound on the convergence rate which is of optimal order and observable without the knowledge of μ.