Related papers: Partial regularity in nonlocal systems II

In this paper we present a family of second order in time nonlinear partial differential equations, which have only one higher symmetry. These equations are not integrable, but have a solution depending on one arbitrary function.

We study well-posedness, local and global, existence of solutions for a general class of physically meaningful nonlinear Schr\"odinger systems with magnetic field involving local and nonlocal nonlinearities.

This paper investigates the higher pointwise regularity of nonnegative classical solutions for fully fractional parabolic equations $(\partial_t -\Delta)^{s} u = f,$ where $s\in(0,1)$. We establish $C^{k+\alpha+2s}$ or $C^{k+\alpha+2s,\ln}…

We study the regularity of the solution of the double obstacle problem form for fully non linear parabolic and elliptic operators. We show that when the obstacles are sufficiently regular the solution is $C^{1,\alpha}$ in the interior for…

A method for solving a quasilinear nonelliptical equation of the second order is developed and we give classification and parametrization of simple elements of the equation.We find exact solutions of an equation for potential stationary…

We prove that manifold constrained $p(x)$-harmonic maps are $C^{1,\beta}$-regular outside a set of zero $n$-dimensional Lebesgue's measure, for some $\beta \in (0,1)$. We also provide an estimate from above of the Hausdorff dimension of the…

Higher Sobolev and H\"older regularity is studied for local weak solutions of the fractional $p$-Laplace equation of order $s$ in the case $p\ge 2$. Depending on the regime considered, i.e. $$0<s\le\tfrac{p-2}{p}\quad \text{or}…

We establish the interior $C^{1,\alpha}$-estimate for viscosity solutions of degenerate/singular fully nonlinear parabolic equations $$u_t = |Du|^{\gamma}F(D^2u) + f.$$ For this purpose, we prove the well-posedness of the regularized…

We establish H\"older estimates for the time derivative of solutions of fully non-linear parabolic equations that does not necessarily have $C^{2,\alpha}$ estimates.

We classify the stable solutions (positive or sign-changing, radial or not) to the following nonlocal Lane-Emden equation: $(-\Delta)^s u=|u|^{p-1}u$ in $\mathbb{R}^n$ for $2<s<3$.

In this work, we study a numeral system with a natural base $s \geq 2$ and a redundant alphabet $A_r=\{0,1, \dots, r\}$, where $s \leq r \leq 2s-2$. We investigate the topological, metric, and fractal properties of the set of numbers in the…

We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem $$ \Delta^2 u=|u|^{p-1}u\ \{in} \ \R^n,$$ where $ p>1$ and $n\ge1$. We give a complete classification of stable and finite Morse index…

Selection of 25 examples from extensive nontrivial families for different types of nonlinear PDEs and their formal general solutions are given. The main goal here is to show on examples the types of solvable PDEs and what their general…

Asymptotic reductions of a defocusing nonlocal nonlinear Schr\"{o}dinger model in $(3+1)$-dimensions, in both Cartesian and cylindrical geometry, are presented. First, at an intermediate stage, a Boussinesq equation is derived, and then its…

In the present paper, a class of fully non-linear elliptic equations are considered, which are degenerate as the gradient becomes small. H\"older estimates obtained by the first author (2011) are combined with new Lipschitz estimates…

This article is concerned with ``up to $C^{2, \alpha}$-regularity results'' about a mixed local-nonlocal nonlinear elliptic equation which is driven by the superposition of Laplacian and fractional Laplacian operators. First of all, an…

This paper concerns linear first-order hyperbolic systems in one space dimension of the type $$ \partial_tu_j + a_j(x,t)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x,t)u_k = f_j(x,t),\; x \in (0,1),\; j=1,\ldots,n, $$ with periodicity…

Consider a class of non-homogenous ultraparabolic differential equations with drift terms or lower order terms arising from some physical models, and we prove that weak solutions are H\"{o}lder continuous, which also generalizes the classic…

We study stable solutions to the fractional Allen-Cahn equation \linebreak $(-\Delta)^{s/2} u = u-u^3$, $|u|<1$ in $\mathbb{R}^n$. For every $s\in (0,1)$ and dimension $n\geq 2$, we establish sharp energy estimates, density estimates, and…

We extend the invariant manifold method for analyzing the asymptotics of dissipative partial differential equations on unbounded spatial domains to treat equations in which the linear part has order greater than two. One important example…