Related papers: Sharp regularity estimates for solutions of the co…
We study the sharp constant in the Morrey inequality for fractional Sobolev-Slobodecki\u{\i} spaces on the whole $\mathbb{R}^N$. By generalizing a recent work by Hynd and Seuffert, we prove existence of extremals, together with some…
We find the nodes that minimise divided differences and use them to find the sharp constant in a sublevel set estimate. We also find the sharp constant in the first instance of the van der Corput Lemma using a complex mean value theorem for…
This paper presents regularity results and associated high-order numerical methods for one-dimensional Fractional-Laplacian boundary-value problems. On the basis of a factorization of solutions as a product of a certain edge-singular weight…
The primary purpose of this paper is to study the Wiener-type regularity criteria for non-linear equations driven by integro-differential operators, whose model is the fractional $p-$Laplace equation. In doing so, with the help of tools…
It is studied that pointwise estimates and continuities on Hardy spaces of pseudo-differential operators (PDOs for short) with the symbol in general H\"{o}rmander's classes. We get weighted weak-type $(1,1)$ estimate, weighted normal…
First of all, we establish compactness of continuous mappings of the Orlicz--Sobolev classes $W^{1,\varphi}_{\rm loc}$ with the Calderon type condition on $\varphi$ and, in particular, of the Sobolev classes $W^{1,p}_{\rm loc}$ for $p>n-1$…
The main goal of the paper is to establish time semidiscrete and space-time fully discrete maximal parabolic regularity for the lowest order time discontinuous Galerkin solution of linear parabolic equations with time-dependent…
In this paper we derive quantitative estimates for the Lagrangian flow associated to a partially regular vector field of the form $$ b(t,x_1,x_2) = (b_1(t,x_1),b_2(t,x_1,x_2)) \in {\mathbb R}^{n_1}\times{\mathbb R}^{n_2} \,, \qquad…
We obtain local pointwise second derivative estimates for $W^{2,p}$-strong solutions to a class of fully nonlinear elliptic equations on Euclidean domains, motivated by problems in conformal geometry.
We establish sharp quantitative stability estimates near finite sums of ground states. The results depend on the dimension and the order of nonlinearity.
We shall study special regularity properties of solutions to some nonlinear dispersive models. The goal is to show how regularity on the initial data is transferred to the solutions. This will depend on the spaces where regularity is…
Optimal second-order regularity in the space variables is established for solutions to Cauchy-Dirichlet problems for nonlinear parabolic equations and systems of $p$-Laplacian type, with square-integrable right-hand sides and initial data…
We prove weighted $L_{p,q}$-estimates for divergence type higher order elliptic and parabolic systems with irregular coefficients on Reifenberg flat domains. In particular, in the parabolic case the coefficients do not have any regularity…
We prove new velocity averaging results for second-order multidimensional equations of the general form, $\op(\nabla_x,v)f(x,v)=g(x,v)$ where $\op(\nabla_x,v):=\bba(v)\cdot\nabla_x-\nabla_x^\top\cdot\bbb(v)\nabla_x$. These results quantify…
Given an arbitrary planar $\infty$-harmonic function $u$, for each $\alpha>0$ we establish a quantitative local $W^{1,2}$-estimate of $|Du|^\alpha $, which is sharp as $\alpha\to0$. We also show that the distributional determinant of $u$ is…
We prove quantitative regularity estimates for the solutions to non-linear continuity equations and their discretized numerical approximations on Cartesian grids when advected by a rough force field. This allow us to recover the known…
We continue the development, by reduction to a first order system for the conormal gradient, of $L^2$ \textit{a priori} estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence form second…
We establish partial regularity for vector-valued solutions to parabolic systems where the coefficients are possibly discontinuous with respect to (x,t). More precisely, we assume a VMO-condition with respect to the (x,t) and continuity…
We prove a sharp $L^p$-Sobolev regularity results for a class of generalized Radon transforms for families of curves in a three dimensional manifold, with folding canonical relations. The proof relies on decoupling inequalities by Wolff and…
In this study, a new form of quadratic spline is obtained, where the coefficients are determined explicitly by variational methods. Convergence is studied and parity conservation is demonstrated. Finally, the method is applied to solve…