Related papers: On the continuity of the solution operator to the …

We describe continuity properties of the multivalued inverse of the numerical range map $f_A:x \mapsto \left\langle Ax, x \right\rangle$ associated with a linear operator $A$ defined on a complex Hilbert space $\mathcal{H}$. We prove in…

We construct counterexamples to inverse problems for the wave operator on domains in $\mathbb{R}^{n+1}$, $n \ge 2$, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which…

This paper concerns piecewise-smooth maps on $\mathbb{R}^d$ that are continuous but not differentiable on switching manifolds (where the functional form of the map changes). The stability of fixed points on switching manifolds is…

This paper ascertains the global behavior of the forward and backward branches of solutions provided by the Leray-Schauder continuation theorem for orientable $\mathcal{C}^1$ Fredholm maps, as developed by the authors in [54]. Under…

We investigate the continuity of boundary operators, such as the Neumann-to-Dirichlet map, with respect to the coefficient matrices of the underlying elliptic equations. We show that for nonsmooth coefficients the correct notion of…

This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e. the original problem is first reformulated as a nonconvex optimization problem, its well-posedness…

This is the first part of a two-paper series that establishes the uniqueness and regularity of a threshold energy wave map that does not scatter in both time directions. Consider the two-sphere valued equivariant energy critical wave maps…

This note concerns the topology of the connected components of the zero sets of monochromatic random waves on compact Riemannian manifolds without boundary. In [SW] it is shown that these are distributed according to a universal measure on…

We discuss the essential self-adjointness of wave operators, as well as the limiting absorption principle, in generalizations of asymptotically Minkowski settings. This is obtained via using a Fredholm framework for inverting the spectral…

This article examines a family of smooth mappings between Banach spaces and establishes conditions for the existence of their zeros. Applications to fixed-point problems and the Implicit Function Theorem are also discussed.

This paper discusses a general and useful stability principle which, roughly speaking, says that given a uniformly continuous function defined on an arbitrary metric space, if the function is bounded on the constraint set and we slightly…

The classical Minkowski problem in Minkowski space asks, for a positive function $\phi$ on $\mathbb{H}^d$, for a convex set $K$ in Minkowski space with $C^2$ space-like boundary $S$, such that $\phi(\eta)^{-1}$ is the Gauss--Kronecker…

In this paper we prove the nonlinear stability of Minkowski space-time with a translation Killing field. In the presence of such a symmetry, the 3 + 1 vacuum Einstein equations reduce to the 2 + 1 Einstein equations with a scalar field. We…

A general overview of the phenomenon of automatic continuity of homomorphisms between Polish groups is given. In particular, we study variants and improvements of the closed graph theorem, applying these to the problem of continuity of…

For Besov spaces $B^s_{p,r}(\rr)$ with $s>\max\{ 2 + \frac1p , \frac52\} $, $p \in (1,\infty]$ and $r \in [1 , \infty)$, it is proved that the data-to-solution map for the FORQ equation is not uniformly continuous from $B^s_{p,r}(\rr)$ to…

We obtain large deviation results for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space…

This work addresses the regularity of solutions for a nonlocal wave equation over the space of periodic distributions. The spatial operator for the nonlocal wave equation is given by a nonlocal Laplace operator with a compactly supported…

For a large class of nonuniformly expanding maps of $\Bbb R^m$, with indifferent fixed points and unbounded distorsion and non necessarily Markovian, we construct an absolutely continuous invariant measure. We extend to our case techniques…

This article addresses the regularity issue for minimizing fractional harmonic maps of order $s\in(0,1/2)$ from an interval into a smooth manifold. H\"older continuity away from a locally finite set is established for a general target. If…

We show the existence of non-trivial self-expanding harmonic map flows starting from non-energy-minimizing 0-homogeneous maps to a regular ball or a closed hemisphere. In particular, given a non-minimizing but stationary 0-homogeneous…