Related papers: Weak observability estimates for 1-D wave equation…

The present paper is concerned with Schr\"odinger 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 (resp.…

In this paper we prove global bounds on the spatial gradient of viscosity solutions to second order linear and nonlinear parabolic equations in $(0,T) \times \R^N$. Our assumptions include the case that the coefficients be both unbounded…

This paper considers stochastic weakly convex optimization without the standard Lipschitz continuity assumption. Based on new adaptive regularization (stepsize) strategies, we show that a wide class of stochastic algorithms, including the…

We study global existence and decay estimates for quasilinear wave equations with dissipative terms in the Sobolev space $H^L \times H^{L-1}$, where $L \geq [d/2]+3$. The linear dissipative terms depend on space variable coefficient, and…

In the theory of second-order, nonlinear elliptic and parabolic equations, obtaining local or global gradient bounds is often a key step for proving the existence of solutions but it may be even more useful in many applications, for example…

In this paper, we continue the investigation on the connection between observability and inverse problems for a class of parabolic equations with unbounded first order coefficients. We prove new logarithmic stability estimates for a class…

We examine weak measurements of arbitrary observables where the object is prepared in a mixed state and on which measurements with imperfect detectors are made. The weak value of an observable can be expressed as a conditional expectation…

We establish local Calder\'on-Zygmund type estimates for weak solutions to nonlinear parabolic systems with $p$-growth and VMO coefficients. In particular, we prove that if the right-hand side belongs locally to $L^{\mu s}$, where the…

We study a class of partial differential equations in divergence form that admit highly irregular Lipschitz weak solutions. By reformulating these divergence-form equations as a first-order partial differential relation and adapting the…

We prove Strichartz estimates with a loss of derivatives for the Schr\"odinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates the on polygon…

We establish a relationship between the equations that constitute the so-called good-bad-ugly model, whose nonlinearities are known to mimic those present in the Einstein field equations in generalized harmonic gauge. This relationship…

We prove local-in-time a-priori estimates in $H^{-1}(\mathbb{R})$ for a family of generalized Korteweg--de Vries equations. This is the first estimate for any non-integrable perturbation of the KdV equation that matches the regularity of…

The sharp Wolff-type decoupling estimates of Bourgain--Demeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian manifolds,…

In $\C_z\times\R_t$ we consider the function $g=g(z)$, set $g_1=\di_z g$, $g_{1\bar 1}=\di_z\dib_zg$ and define the operator $L_g=\di_z+ig_1\di_t$. We discuss estimates with loss of derivatives, in the sense of Kohn, for the system $(\bar…

We give stability estimates in the Cauchy problem for general partial differential equation of the elliptic type similar to the Helmholtz equation. We do not impose any (pseudo)convexity assumptions on the domain or the operator. These…

We consider an inverse problem for the linear one-dimensional wave equation with variable coefficients consisting in determining an unknown source term from a boundary observation. A method to obtain approximations of this inverse problem…

Composite optimization problems, where the sum of a smooth and a merely lower semicontinuous function has to be minimized, are often tackled numerically by means of proximal gradient methods as soon as the lower semicontinuous part of the…

We investigate sufficient H\"older continuity conditions on Leray-Hopf (weak) solutions to the in unsteady Navier-Stokes equations in three dimensions guaranteeing energy conservation. Our focus is on the half-space case with homogeneous…

Datasets consisting of objects such as shapes, networks, images, or signals overlaid on such geometric objects permeate data science. Such datasets are often equipped with metrics that quantify the similarity or divergence between any pair…

For a class of weakly hyperbolic systems of the form D_t - A(t,x,D_x), where A(t,x,D_x) is a first-order pseudodifferential operator whose principal symbol degenerates like t^{l_*} at time t=0, for some integer l_* \geq 1, well-posedness of…