Related papers: Quantitative uniqueness estimates for the general …
We investigate the robustness of singularity avoidance mechanisms in nonrelativistic quantum mechanics on the discretised real line when lattice points are allowed to approach a singularity of the classical potential. We consider the…
In this article, we study the order of vanishing and a quantitative form of Landis' conjecture in the plane for solutions to second-order elliptic equations with variable coefficients and singular lower order terms. Precisely, we let $A$ be…
We revisit the classical theory of linear second-order uniformly elliptic equations in divergence form whose solutions have H\"older continuous gradients, and prove versions of the generalized maximum principle, the $C^{1,\alpha}$-estimate,…
We consider the two-dimensional incompressible Navier-Stokes equations with measure initial vorticity. By means of improved Nash inequalities, we establish quantitative estimates for the enstrophy depending on the absolute vorticity decay…
We provide new results on the existence of nonzero positive weak solutions for a class of second order elliptic systems. Our approach relies on a combined use of iterative techniques and classical fixed point index. Some examples are…
We establish the uniqueness of large solutions to the non-cutoff Boltzmann equation with moderate soft potentials. Specifically, the weak solution $F=\mu+\mu^{\frac{1}{2}}f$ is unique as long as it has finite energy, in the sense that the…
In this article we consider the discretely self-similar singular solutions of the Euler equations, and the possible velocity profiles concerned not only have decaying spatial asymptotics, but also have unconventional non-decaying…
Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the…
A Carleman estimate and the unique continuation property of solutions for a multi-terms time fractional diffusion equation up to order $\alpha\,\,(0<\alpha<2)$ and general time dependent second order strongly elliptic time elliptic operator…
We consider the Cauchy problem for wave equations with unbounded damping coefficients in the whole space. For a general class of unbounded damping coefficients, we derive uniform total energy decay estimates together with a unique existence…
As explained in detail in the prologue to this manuscript, boundedness of weak solutions for general classes of elliptic equations in divergence form is a classic tool for achieving higher regularity. We propose here some global boundedness…
In this paper we use the method of layer potentials to study $L^2$ boundary value problems in a bounded Lipschitz domain $\Omega$ for a family of second order elliptic systems with rapidly oscillating periodic coefficients, arising in the…
We derive local boundedness estimates for weak solutions of a large class of second order quasilinear equations. The structural assumptions imposed on an equation in the class allow vanishing of the quadratic form associated with its…
In recent years considerable advances have been made in quantitative homogenization of partial differential equations in the periodic and non-periodic settings. This monograph surveys the theory of quantitative homogenization for…
We study via Carleman estimates the sharpest possible exponential decay for {\it waveguide} solutions to the Laplace equation $$(\partial^2_t+\triangle)u=Vu+W\cdot(\partial_t,\nabla)u,$$ and find a necessary quantitative condition on the…
In this paper, we focus on the quantitative unique continuation property of solutions to \begin{equation*} \Delta^2u=Vu, \end{equation*} where $V\in W^{1,\infty}$. We show that the maximal vanishing order of the solutions is not large than…
We give an example of quasiderivatives constructed by random time change, Girsanov's Theorem and Levy's Theorem. As an application, we investigate the smoothness and estimate the derivatives up to second order for the probabilistic solution…
We study the 2D-wave equation with a scaling-critical electromagnetic potential. This problem is doubly critical, because of the scaling invariance of the model and the singularities of the potentials, which are not locally integrable. In…
Weighted quadratic estimates are proved for certain bisectorial firstorder differential operators with bounded measurable coefficients which are (not necessarily pointwise) accretive, on complete manifolds with positive injectivity radius.…
In this manuscript we present an approach to analyze the discontinuous Galerkin solution for general quasilinear elliptic problems. This approach is sufficiently general to extend most of the well-known discretization schemes, including…