We consider the problem of locating and reconstructing the geometry of a penetrable obstacle from time-domain measurements of causal waves. More precisely, we assume that we are given the scattered field due to point sources placed on a surface enclosing the obstacle and that the scattered field is measured on the same surface. From these multistatic scattering data we wish to determine the position

We study a class of Monge–Ampère-type functionals arising from the Lp dual Minkowski problem in convex geometry. As an application, we obtain some existence and nonuniqueness results for the problem.

Let A( ⋅) be an (n+1)×(n+1) uniformly elliptic matrix with Hölder continuous real coefficients and let ℰA(x,y) be the fundamental solution of the PDE divA( ⋅)∇u = 0 in ℝn+1 . Let μ be a compactly supported n-AD-regular measure in ℝn+1 and consider the associated operator Tμf(x) = ∫ ∇xℰA(x,y)f(y)dμ(y). We show that if Tμ is bounded in L2(μ), then μ is uniformly n-rectifiable. This extends the solution

In view of classical results of Masur and Veech almost every element in the moduli space of compact translation surfaces is recurrent; i.e., its Teichmüller positive semiorbit returns to a compact subset infinitely many times. We focus on the problem of recurrence for elements of smooth curves in the moduli space. We give an effective criterion for the recurrence of almost every element of a smooth

This paper establishes a theory of nonlinear spectral decompositions by considering the eigenvalue problem related to an absolutely one-homogeneous functional in an infinite-dimensional Hilbert space. This approach is motivated by works for the total variation, where interesting results on the eigenvalue problem and the relation to the total variation flow have been proven previously, and by recent

We estimate the L2 norm of the restriction to a totally geodesic submanifold of the eigenfunctions of the Laplace–Beltrami operator on the standard flat torus 𝕋d, d ≥ 2. We reduce getting correct bounds to counting lattice points in the intersection of some ν-transverse bands on the sphere. Moreover, we prove the correct bounds for rational totally geodesic submanifolds of arbitrary codimension. In

We consider approximations introduced by Sacks and Uhlenbeck of the harmonic energy for maps from S2 into S2. We continue our previous analysis (J. Differential Geom. 116:2 (2020), 321–348) on limits of α-harmonic maps with uniformly bounded energy. Using a recent energy identity of Li and Zhu (Ann. Inst. H. Poincaré Anal. Non Linéaire 36:1 (2019), 103–118), we obtain an optimal gap theorem for the

We consider a family of nonlocal problems that model the effects of transport and vortex stretching in the incompressible Euler equations. Using modulation techniques, we establish stable self-similar blow-up near a family of known self-similar blow-up solutions.

We consider the following nonlinear Schrödinger equation of derivative type: i∂tu + ∂x2u + i|u|2∂ xu + b|u|4u = 0,(t,x) ∈ ℝ × ℝ,b ∈ ℝ. If b = 0, this equation is known as a standard derivative nonlinear Schrödinger equation (DNLS), which is mass-critical and completely integrable. The equation above can be considered as a generalized equation of DNLS while preserving mass-criticality and Hamiltonian

We develop a novel decomposition of the monomials in order to study the set of monomial convergence for spaces of holomorphic functions over ℓr for 1 < r ≤ 2. For Hb(ℓr), the space of entire functions of bounded type in ℓr, we prove that monHb(ℓr) is exactly the Marcinkiewicz sequence space mΨr, where the symbol Ψr is given by Ψr(n) := log(n + 1)1−1∕r for n ∈ ℕ0. For the space of m-homogeneous polynomials

For distributions, we build a theory of higher-order pointwise differentiability comprising, for order zero, Łojasiewicz’s notion of point value. Results include Borel regularity of differentials, higher-order rectifiability of the associated jets, a Rademacher–Stepanov-type differentiability theorem, and a Lusin-type approximation. A substantial part of this development is new also for zeroth order

We are concerned with the short-time observability constant of the heat equation from a subdomain ω of a bounded domain ℳ. The constant is of the form e𝔎∕T , where 𝔎 depends only on the geometry of ℳ and ω. Luc Miller (J. Differential Equations 204:1 (2004), 202–226) conjectured that 𝔎 is (universally) proportional to the square of the maximal distance from ω to a point of ℳ. We show in particular

We prove the global stability of the Minkowski space viewed as the trivial solution of the Einstein–Vlasov system. To estimate the Vlasov field, we use the vector field and modified vector field techniques we previously developed in 2017. In particular, the initial support in the velocity variable does not need to be compact. To control the effect of the large velocities, we identify and exploit several

We consider the following perturbed critical Dirichlet problem involving the Hardy–Schrödinger operator: −Δu − γ(u∕|x|2) − 𝜖u = |u| 4 N−2 u in Ω, u = 0  on ∂Ω, when 𝜖 > 0 is small, γ < (N−2)2 4 , and where Ω ⊂ ℝN , N ≥ 3, is a smooth bounded domain with 0 ∈ Ω. We show that there exists a sequence (γj)j=1∞ in (−∞,0] with limj→∞γj = −∞ such that, if γ≠γj for any j and γ ≤ (N−2)2 4 − 1, then the above

We study the regularity of minima of scalar variational integrals of p-growth, 1 < p < ∞, in the Heisenberg group and prove the Hölder continuity of horizontal gradient of minima.

We derive local C2 estimates for complete noncompact translating solitons of the Gauss curvature flow in ℝ3 which are graphs over a convex domain Ω. This is closely is related to deriving local C1,1 estimates for the degenerate Monge–Ampère equation. As a result, given a weakly convex bounded domain Ω, we establish the existence of a Cloc1,1 translating soliton. In particular, when the boundary ∂Ω

We consider the dispersive logarithmic Schrödinger equation in a semiclassical scaling. We extend the results of Carles and Gallagher (Duke Math. J. 167:9 (2018), 1761–1801) about the large-time behavior of the solution (dispersion faster than usual with an additional logarithmic factor and convergence of the rescaled modulus of the solution to a universal Gaussian profile) to the case with semiclassical

We establish optimal local and global Besov–Lipschitz and Triebel–Lizorkin estimates for the solutions to linear hyperbolic partial differential equations. These estimates are based on local and global estimates for Fourier integral operators that span all possible scales (and in particular both Banach and quasi-Banach scales) of Besov–Lipschitz spaces Bp,qs(ℝn) and certain Banach and quasi-Banach

Żuk proved that if a finitely generated group admits a Cayley graph such that the Laplacian on the links of this Cayley graph has a spectral gap > 1 2, then the group has property (T), or equivalently, every affine isometric action of the group on a Hilbert space has a fixed point. We prove that the same holds for affine isometric actions of the group on a uniformly curved Banach space (for example

We consider a compact manifold with a piece isometric to a (finite-length) cylinder. By making the length of the cylinder tend to infinity, we obtain an asymptotic gluing formula for the zeta determinant of the Hodge Laplacian and an asymptotic expansion of the torsion of the corresponding long exact sequence of cohomology equipped with L2-metrics. As an application, we give a purely analytic proof

We study the local propagation of conormal singularities for solutions of semilinear wave equations □u = P(y,u), where P(y,u) is a polynomial of degree N ≥ 3 in u with C∞(ℝy3) coefficients. We know from the work of Melrose and Ritter and Bony that if u is conormal to three waves which intersect transversally at point q, then after the triple interaction u(y) is a conormal distribution with respect

We establish interior Schauder estimates for kinetic equations with integrodifferential diffusion. We study equations of the form ft + v ⋅∇xf = ℒvf + c, where ℒv is an integrodifferential diffusion operator of order 2s acting in the v-variable. Under suitable ellipticity and Hölder continuity conditions on the kernel of ℒv, we obtain an a priori estimate for f in a properly scaled Hölder space.

Internal waves describe the (linear) response of an incompressible stably stratified fluid to small perturbations. The inclination of their group velocity with respect to the vertical is completely determined by their frequency. Therefore the reflection on a sloping boundary cannot follow Descartes’ laws, and it is expected to be singular if the slope has the same inclination as the group velocity

We construct Green’s functions for elliptic operators of the form ℒu = −div(A∇u + bu) + c∇u + du in domains Ω ⊆ ℝn , under the assumption d ≥ divb or d ≥ divc. We show that, in the setting of Lorentz spaces, the assumption b − c ∈ Ln,1(Ω) is both necessary and optimal to obtain pointwise bounds for Green’s functions. We also show weak-type bounds for the Green’s function and its gradients. Our estimates

We complete the microlocal study of the geodesic x-ray transform on Riemannian manifolds with Anosov geodesic flow initiated by Guillarmou (J. Differential Geom. 105:2 (2017), 177–208) and pursued by Guillarmou and Lefeuvre in (Ann. of Math. (2) 190:1 (2019), 321–344). We prove new stability estimates and clarify some properties of the operator Πm — the generalized x-ray transform. These estimates

We study the phenomenon of invasion for heterogeneous reaction-diffusion equations in periodic domains with monostable and combustion reaction terms. We give an answer to a question raised by Berestycki, Hamel and Nadirashvili concerning the connection between the speed of invasion and the critical speed of fronts. To do so, we extend the classical Freidlin–Gärtner formula to such equations and we

We present an alternative approach to the theory of free Gibbs states with convex potentials. Instead of solving SDEs, we combine PDE techniques with a notion of asymptotic approximability by trace polynomials for a sequence of functions on MN(ℂ)sam to prove the following. Suppose μN is a probability measure on MN(ℂ)sam given by uniformly convex and semiconcave potentials V N, and suppose that the

We demonstrate a parametrix construction, together with associated pseudodifferential operator calculus, for an operator of sum-of-squares type with semiclassical parameter. The form of operator we consider includes the generator of kinetic Brownian motion on the cosphere bundle of a Riemannian manifold. Regularity estimates in semiclassical Sobolev spaces are proven by establishing mapping properties

We study some properties of propagation of regularity of solutions of the dispersive generalized Benjamin–Ono (BO) equation. This model defines a family of dispersive equations that can be seen as a dispersive interpolation between the Benjamin–Ono equation and the Korteweg–de Vries (KdV) equation. Recently, it has been shown that solutions of the KdV and BO equations satisfy the following property:

Regularity estimates in time and space for solutions to the porous medium equation are shown in the scale of Sobolev spaces. In addition, higher spatial regularity for powers of the solutions is obtained. Scaling arguments indicate that these estimates are optimal. In the linear limit, the proven regularity estimates are consistent with the optimal regularity of the linear case.

Let p ∈ (1,∞)∖{2}. We show that every homomorphism from a C∗-algebra 𝒜 into B(lp(J)) satisfies a compactness property where J is any set. As a consequence, we show that a C∗-algebra 𝒜 is isomorphic to a subalgebra of B(lp(J)), for some set J, if and only if 𝒜 is residually finite-dimensional.

We prove that 𝒵-stable, simple, separable, nuclear, nonunital C∗-algebras have nuclear dimension at most 1. This completes the equivalence between finite nuclear dimension and 𝒵-stability for simple, separable, nuclear, nonelementary C∗-algebras.

We develop refined Strichartz estimates at L2 regularity for a class of time-dependent Schrödinger operators. Such refinements quantify near-optimizers of the Strichartz estimate and play a pivotal part in the global theory of mass-critical NLS. On one hand, the harmonic analysis is quite subtle in the L2-critical setting due to an enormous group of symmetries, while on the other hand, the space-time

We consider the Cauchy problem for the defocusing power-type nonlinear wave equation in (1+ 3)-dimensions for energy subcritical powers p in the superconformal range 3 < p < 5. We prove that any solution is global-in-time and scatters to free waves in both time directions as long as its critical Sobolev norm stays bounded on the maximal interval of existence.

The goal of the paper is to prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary 1-Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense (more precisely in essentially nonbranching MCP(K,N)-spaces). Such a representation formula makes apparent the classical upper bounds together

We consider the volume-preserving geometric evolution of the boundary of a set under fractional mean curvature. We show that smooth convex solutions maintain their fractional curvatures bounded for all times, and the long-time asymptotics approach round spheres. The proofs are based on a priori estimates on the inner and outer radii of the solutions.

We study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge–Ampère type arising from optimal mass transport. Our main result is an exponential rate of convergence for solutions of this evolution equation to the stationary solution of the optimal transport problem. We derive a differential Harnack inequality for a special class of functions that solve

We revisit the question of regularity for minimizers of scalar autonomous integral functionals with so-called (p,q)-growth. In particular, we establish Lipschitz regularity under the condition q p < 1 + 2 n−1 for n ≥ 3, improving a classical result due to Marcellini (J. Differential Equations 90:1 (1991), 1–30).

In 2006, Dafermos and Holzegel formulated the so-called AdS instability conjecture, stating that there exist arbitrarily small perturbations to AdS initial data which, under evolution by the Einstein vacuum equations for Λ < 0 with reflecting boundary conditions on conformal infinity ℐ, lead to the formation of black holes. The numerical study of this conjecture in the simpler setting of the spherically

We study uniqueness of Dirichlet problems of second-order divergence-form elliptic systems with transversally independent coefficients on the upper half-space in the absence of regularity of solutions. To this end, we develop a substitute for the fundamental solution used to invert elliptic operators on the whole space by means of a representation via abstract single-layer potentials. We also show

We study eigenvalues of non-self-adjoint Schrödinger operators on nontrapping asymptotically conic manifolds of dimension n ≥ 3. Specifically, we are concerned with the following two types of estimates. The first one deals with Keller-type bounds on individual eigenvalues of the Schrödinger operator with a complex potential in terms of the Lp-norm of the potential, while the second one is a Lieb–Thirring-type

We solve the Dirichlet problem for the quaternionic Monge–Ampère equation with a continuous boundary data and the right-hand side in Lp for p > 2. This is the optimal bound on p. We prove also that the local integrability exponent of quaternionic plurisubharmonic functions is 2, which turns out to be less than an integrability exponent of the fundamental solution.

Columnar vortices are stationary solutions of the three-dimensional Euler equations with axial symmetry, where the velocity field only depends on the distance to the axis and has no component in the axial direction. Stability of such flows was first investigated by Lord Kelvin in 1880, but despite a long history the only analytical results available so far provide necessary conditions for instability

Some exotic compact objects, including supersymmetric microstate geometries and certain boson stars, possess evanescent ergosurfaces: time-like submanifolds on which a Killing vector field, which is time-like everywhere else, becomes null. We show that any manifold possessing an evanescent ergosurface but no event horizon exhibits a linear instability of a peculiar kind: either there are solutions

The theory of second-order complex-coefficient operators of the form ℒ = divA(x)∇ has recently been developed under the assumption of p-ellipticity. In particular, if the matrix A is p-elliptic, the solutions u to ℒu = 0 will satisfy a higher integrability, even though they may not be continuous in the interior. Moreover, these solutions have the property that |u|p∕2−1u ∈ Wloc1,2. These properties

We construct an example showing that the upper bound [w]A1 log(e+[w]A1) for the L1(w) → L1,∞(w) norm of the Hilbert transform cannot be improved in general.

We prove an 𝜖-regularity theorem for vector-valued p-harmonic maps, which are critical with respect to a partially free boundary condition, namely that they map the boundary into a round sphere. This does not seem to follow from the reflection method that Scheven used for harmonic maps with free boundary (i.e., the case p = 2): the reflected equation can be interpreted as a p-harmonic map equation

We consider a wide class of functionals with the property of changing their growth and ellipticity properties according to the modulating coefficients in the framework of Musielak–Orlicz spaces. In particular, we provide an optimal condition on the modulating coefficient to establish the Hölder regularity and Harnack inequality for quasiminimizers of the generalized double phase functional with (G

We prove uniform Sobolev estimates for the resolvent of Schrödinger operators with large scaling-critical potentials without any repulsive condition. As applications, global-in-time Strichartz estimates including some nonadmissible retarded estimates, a Hörmander-type spectral multiplier theorem, and Keller-type eigenvalue bounds with complex-valued potentials are also obtained.

We are interested in several questions raised mainly by Mancini and Martinazzi (2017) (see also work of McLeod and Peletier (1989) and Pruss (1996)). We consider the perturbed Moser–Trudinger inequality Iαg(Ω) at the critical level α = 4π, where g, satisfying g(t) → 0 as t → +∞, can be seen as a perturbation with respect to the original case g ≡ 0. Under some additional assumptions, ensuring basically

We address the local well-posedness of the hydrostatic Navier–Stokes equations. These equations, sometimes called reduced Navier–Stokes/Prandtl, appear as a formal limit of the Navier–Stokes system in thin domains, under certain constraints on the aspect ratio and the Reynolds number. It is known that without any structural assumption on the initial data, real-analyticity is both necessary and sufficient

We prove variation-norm estimates for certain oscillatory integrals related to Carleson’s theorem. Bounds for the corresponding maximal operators were first proven by Stein and Wainger. Our estimates are sharp in the range of exponents, up to endpoints. Such variation-norm estimates have applications to discrete analogues and ergodic theory. The proof relies on square function estimates for Schrödinger-like

Federer’s characterization of sets of finite perimeter states (in Euclidean spaces) that a set is of finite perimeter if and only if the measure-theoretic boundary of the set has finite Hausdorff measure of codimension 1. In complete metric spaces that are equipped with a doubling measure and support a Poincaré inequality, the “only if” direction was shown by Ambrosio (2002). By applying fine potential

We extend the results of our paper “Attractors for two-dimensional waves with homogeneous Hamiltonians of degree 0,” written with Laure Saint-Raymond, to the case of forced linear wave equations in any dimension. We prove that, in dimension 2, if the foliation on the boundary at infinity of the energy shell is Morse–Smale, we can apply Mourre’s theory and hence get the asymptotics of the forced solution

We show that the derivative nonlinear Schrödinger (DNLS) equation is globally well-posed in the weighted Sobolev space H2,2(ℝ). Our result exploits the complete integrability of the DNLS equation and removes certain spectral conditions on the initial data required by our previous work, thanks to Zhou’s analysis (Comm. Pure Appl. Math. 42:7 (1989), 895–938) on spectral singularities in the context of

We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension m ≥ 3 starting from any smooth, conformally hyperbolic initial metric. We do not require initial completeness or curvature bounds. With the same methods, we show rigidity of hyperbolic space under the Yamabe flow.

Motivated by the inequality ∥f + g∥22 ≤∥f∥22 + 2∥fg∥1 + ∥g∥22, Carbery (2009) raised the question of what is the “right” analogue of this estimate in Lp for p≠2. Carlen, Frank, Ivanisvili and Lieb (2018) recently obtained an Lp version of this inequality by providing upper bounds for ∥f + g∥pp in terms of the quantities ∥f∥pp, ∥g∥pp and ∥fg∥p∕2p∕2 when p ∈ (0,1] ∪ [2,∞), and lower bounds when p ∈ (−∞