We consider here a class of anisotropic elliptic equations, in a bounded domain $\Omega$ with Lipschitz continuous boundary $\partial \Omega$, of the type $$ -\sum_{i=1}^{N}\partial_{x_{i}}\big(a_{i}(x,u) |\partial_{x_{i}}u|^{p_{i}-2}\partial_{x_{i}} u\big) =f- {\rm div} g $$ with Dirichlet boundary conditions. Using the framework of renormalized solutions we prove the uniqueness of the solution under a very local Lipschitz condition on the coefficients $a_{i}(x,s)$ with respect to $s$ and with $f$ belonging to $L^1(\Omega)$.

In this paper, we establish sufficient conditions for a singular integral $T$ to be bounded from certain Hardy spaces $H^p_L$ to Lebesgue spaces $L^p$, $0 < p \le 1$, and for the commutator of $T$ and a BMO function to be weak-type bounded on Hardy space $H_L^1$. We then show that our sufficient conditions are applicable to the following cases: (i) $T$ is the Riesz transform or a square function associated with the Laplace--Beltrami operator on a doubling Riemannian manifold, (ii) $T$ is the Riesz transform associated with the magnetic Schr\"odinger operator on a Euclidean space, and (iii) $T = g(L) $ is a singular integral operator defined from the holomorphic functional calculus of an operator $L$ or the spectral multiplier of a non-negative self-adjoint operator $L$.

We study the convergence of a semidiscrete scheme for the forward-backward parabolic equation $u_t= (W'(u_x))_x$ with periodic boundary conditions in one space dimension, where $W$ is a standard double-well potential. We characterize the equation satisfied by the limit of the discretized solutions as the grid size goes to zero. Using an approximation argument, we show that it is possible to flow initial data ${\overline u}$ having regions where ${\overline u}_x$ falls within the concave region $\{W''<0\}$ of $W$, where the backward character of the equation manifests itself. It turns out that the limit equation depends on the way we approximate ${\overline u}$ in its unstable region. Our result can be viewed as a characterization, among all Young measure solutions of the equation, of the much smaller subset of those solutions which can be obtained as limit of the semidiscrete scheme.

The paper investigates overdetermined fully nonlinear problems on nonsmooth domains. Under natural regularity assumptions on solutions it is shown that overdetermined problems on reflectionally symmetric, bounded domains can have positive solutions only if the domain is a ball. These results are extensions of results of Serrin, who proved this statement for smooth solutions on smooth domains. The results for overdetermined problems are applied to a study of Dirichlet problems, specifically to the question when a nonnegative solution is positive or zero everywhere. As a consequence, an extension of symmetry results of Gidas--Ni--Nirenberg to nonnegative solutions is obtained.

We address global regularity of solutions of the Navier--Stokes equations in a periodic domain $Q=[0,1]^3$. We prove that, in the class of solutions oscillating in the vertical direction, the global solutions are smooth under natural conditions on the derivatives in the horizontal direction and the vertical and horizontal averages of the initial data. There are no restrictions on the size of the data.

In this work will use the Genus theory, introduced by Krasnolselskii, the mountain-pass theorem, introduced by Ambrosetti and Rabinowitz, and the concentration-compactness principle, due to Lions, to show results of existence and multiplicity of solutions for the bi-nonlocal equation $$ -M \Big (\int_{\Omega}|\nabla u|^{p} \, dx\ \Big )\Delta_{p} u = \lambda |u|^{q-2}u+ \mu g(x)|u|^{\gamma-2}u \Big [\frac{1}{\gamma}\int_{\Omega}g(x)|u|^{\gamma} \, dx \Big ]^{2r} \ \mbox{in} \ \ \Omega, $$ with Dirichlet boundary condition, where $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N}$, $1<p<N$, $1<\gamma< p^{*}$, and $M:[0,+\infty)\rightarrow \mathbb{R}$ and $g:\Omega\rightarrow \mathbb{R}$ are continuous functions. We consider $r$ a positive parameter, and we study several cases: $\lambda=0$, $\lambda=1$, $\mu=1$, $p(\alpha+1)<q\leq p^{*}$, and $p-1 < q <p(\alpha+1)$.