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

In this paper we give elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes equations in the class of H\"older continuous functions, relaxing some of the assumptions on the time variable (both…

We investigate linear parabolic, second-order boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness and ellipticity on the coefficient function and very mild conditions on the geometry of the…

In this work, we obtain an existence of nontrivial solutions to a minimization problem involving a fractional Hardy-Sobolev type inequality in the case of inner singularity. Precisely, for $\lambda>0$ we analyze the attainability of the…

Let $N\in\mathbb{N}$ and $u$ be a weak solution of equation $\displaystyle Lu\equiv - \sum_{i,j=1}^{N}\frac{\partial}{\partial x_{j}}(\frac{\partial u}{\partial x_{i}}b^{ij})= f$ in $\Omega\subset \mathbb{R}^{N}$. We obtain functions $G$…

The first-order approach to boundary value problems for second-order elliptic equations in divergence form with transversally independent complex coefficients in the upper half-space rewrites the equation algebraically as a first-order…

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{cases} \displaystyle - \Delta_{1} u = h(u)f & \text{in}\, \Omega,\newline…

Locally bounded, local weak solutions to a special class of quasilinear, anisotropic, $p$-Laplacian type elliptic equations, are shown to be locally H\"older continuous. Homogeneous local upper bounds are established for local weak…

The aim of this paper is to consider the linear ultraparabolic equation with bounded and VMO coefficients $a_{ij} (z)$. Assume that the operator $L_0$ obtained by freezing the coefficients $a_{ij}(z)$ at any point ${z_0} \in {\mathbb{R}^{N…

Motivated by recent results on the (possibly conditional) regularity for time-dependent hypoelliptic equations, we prove a parabolic version of the Poincar\'e inequality, and as a consequence, we deduce a version of the classical Moser…

We construct multibump nodal solutions of the elliptic equation $$ -\Delta u=a^+[\lambda u+ f(\, \cdot\,, u)]-\mu a^- g(\, \cdot\,, u) $$ in $H^1_0(\Omega)$, when $\mu$ is large, under appropriate assumptions, for $f$ superlinear and…

This article studies the continuity of bounded nonnegative weak solutions to inhomogeneous doubly nonlinear parabolic equations. A model equation is \begin{equation*}\partial_t u-\operatorname{div}(u^{m-1}|Du|^{p-2}Du)=f\qquad…

This paper is focused on the local interior $W^{1,\infty}$-regularity for weak solutions of degenerate elliptic equations of the form $\text{div}[\mathbf{a}(x,u, \nabla u)] +b(x, u, \nabla u) =0$, which include those of $p$-Laplacian type.…

In this paper, we characterize all the distributions $F \in \mathcal{D}'(U)$ such that there exists a continuous weak solution $v \in C(U,\mathbb{C}^{n})$ (with $U \subset \Omega$) to the divergence-type equation…

We show that any weak solution to elliptic equations in divergence form is continuously differentiable provided that the modulus of continuity of coefficients in the $L^1$-mean sense satisfies the Dini condition. This in particular answers…

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…

In the present paper we investigate the following semilinear singular elliptic problem: \begin{equation*} (\rm P)\qquad \left \{\begin{array}{l} -\Delta u = \dfrac{p(x)}{u^{\alpha}}\quad \text{in} \Omega \\ u = 0\ \text{on} \Omega,\ u>0…

We shall establish the interior H\"older continuity for locally bounded weak solutions to a class of parabolic singular equations whose prototypes are \begin{equation} u_t= \nabla \cdot \bigg( |\nabla u|^{p-2} \nabla u \bigg), \quad \text{…

We establish that the elliptic equation $\Delta u+f(x,u)+g(| x|)x\cdot \nabla u=0$, where $x\in\mathbb{R}^{n}$, $n\geq3$, and $| x|>R>0$, has a positive solution which decays to 0 as $| x|\to +\infty$ under mild restrictions on the…

We study the boundary value problem $-{\rm div}((|\nabla u|^{p\_1(x) -2}+|\nabla u|^{p\_2(x)-2})\nabla u)=f(x,u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a smooth bounded domain in $\RR^N$. We focus on the cases when…

We study spectral stability estimates of elliptic operators in divergence form $-\textrm{div} [A(w) \nabla g(w)]$ with the Neumann boundary condition in non-Lipschitz domains $\Omega \subset \mathbb C$. The suggested method is based on…