Related papers: On the best Hoelder exponent for two dimensional e…

We consider weak solutions of the equation $$-\Delta_p^H u+a(x,u)H^q(\nabla u)=f(x,u) \quad \text{in } \Omega,$$ where $H$ is in some cases called Finsler norm, $\Omega$ is a domain of $\mathbb R^N$, $p>1$, $q\ge \max\{p-1,1\}$, and…

We give a proof of H\"older continuity for bounded local weak solutions to the equation $u_t= \sum_{i=1}^N (|u_{x_i}|^{p_i-2} u_{x_i})_{x_i}$, in $\Omega \times [0,T]$, with $\Omega \subset \subset \mathbb{R}^N$, under the condition $…

We consider Lipschitz solutions to the possibly highly degenerate elliptic equation $ {\rm div} G(\nabla u)=0$ in $B_1\subset\mathbb{R}^2 $, for any continuous strictly monotone vector field $G \colon \mathbb{R}^2 \to \mathbb{R}^2$. We show…

We prove H\"older continuity of weak solutions of the uniformly elliptic and parabolic equations %$\Delta u-\frac{A}{|x|^{2+\beta}}u=0,\,\,(\beta\geq 0)$, and variable second order term coefficients case $%% \begin{equation}\label{01}…

We consider solutions $u\in W^{1,p}\big(\Omega;\mathbb{R}^{N}\big)$ of the $p$-Laplacian PDE \begin{equation} \nabla\cdot\big(a(x)|Du|^{p-2}Du\big)=0,\notag \end{equation} for $x\in\Omega\subseteq\mathbb{R}^{n}$, where $\Omega$ is open and…

We consider divergence form elliptic operators $L=-\dv A(x)\nabla$, defined in $\mathbb{R}^{n+1}=\{(x,t)\in\mathbb{R}^{n}\times\mathbb{R}\}, n \geq 2$, where the $L^{\infty}$ coefficient matrix $A$ is $(n+1)\times(n+1)$, uniformly elliptic,…

We prove a multiplicity result for non-constant weak solutions $u \in H^1(\Omega)$ for the quasilinear elliptic equation \[ \begin{cases} \displaystyle-\text{div}(A(x,u)\nabla u) + \frac{1}{2} D_sA(x,u)\nabla u \cdot \nabla u = g(x,u) -…

We consider second-order uniformly elliptic operators subject to Dirichlet boundary conditions. Such operators are considered on a bounded domain $\Omega$ and on the domain $\phi(\Omega)$ resulting from $\Omega$ by means of a bi-Lipschitz…

Building on the recent work of C. De Lellis and L. Sz\'{e}kelyhidi, we construct global weak solutions to the three-dimensional incompressible Euler equations which are zero outside of a finite time interval and have velocity in the…

We establish the local H\"older regularity of the spatial gradient of bounded weak solutions $u\colon E_T\to\R^k$ to the non-linear system of parabolic type \begin{equation*} \partial_tu-\Div\Big(…

We study the higher gradient integrability of distributional solutions $u$ to the equation $div(\sigma \nabla u) = 0$ in dimension two, in the case when the essential range of $\sigma$ consists of only two elliptic matrices, i.e.,…

We establish new, optimal gradient continuity estimates for solutions to a class of 2nd order partial differential equations, $\mathscr{L}(X, \nabla u, D^2 u) = f$, whose diffusion properties (ellipticity) degenerate along the \textit{a…

We investigate the weak solvability and properties of weak solutions to the Dirichlet problem for a scalar elliptic equation $-\Delta u + b^{(\alpha)}\cdot \nabla u= f$ in a bounded domain $\Omega\subset {\mathbb R^2}$ containing the…

This paper studies the Sobolev regularity estimates for weak solutions of a class of degenerate, and singular quasi-linear elliptic problems of the form $\text{div}[\mathbf{A}(x,u, \nabla u)]= \text{div}[\mathbf{F}]$ with non-homogeneous…

We establish local boundedness and local H\"older continuity of weak solutions to the following prototype problem: $$ -\operatorname{div}\left(|x|^{-2 \beta}|\nabla u|^{\mathbf{q}-2} \nabla u\right)+(-\Delta)_{p(\cdot, \cdot),…

This paper studies the Sobolev regularity estimates of weak solutions of a class of singular quasi-linear elliptic problems of the form $u_t - \mbox{div}[\mathbb{A}(x,t,u,\nabla u)]= \mbox{div}[{\mathbf F}]$ with homogeneous Dirichlet…

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded, Lipschitz domain. We consider bounded, weak solutions ($u\in W^{1, 2}\cap L^{\infty}(\Omega;\mathbb{R}^N)$) of the vector-valued, Euler-Lagrange system: \text{div } \big( A(x, u)Du\big)=g(x,…

This paper is a generalization of the author's previous work [14]. We extend the argument [14] for any uniformly elliptic operator in divergence form $\mathcal{L}u=-div(A(x)\nabla u)$, more precisely, we study a fractional type degenerate…

In this paper we develop some new techniques to study the multiscale elliptic equations in the form of $-\text{div} \big(A_\varepsilon \nabla u_{\varepsilon} \big) = 0$, where $A_\varepsilon(x) = A(x, x/\varepsilon_1,\cdots,…

We develop a discrete counterpart of the De Giorgi-Nash-Moser theory, which provides uniform H\"older-norm bounds on continuous piecewise affine finite element approximations of second-order linear elliptic problems of the form $-\nabla…