Related papers: Sequential weak continuity of null Lagrangians at …
We make explicit the $p$-dependence of $C$ in the gradient estimate $\left\Vert \nabla u\right\Vert _{\infty}^{p-1}\leq C\left\Vert f\right\Vert _{N,1}$ by Cianchi and Maz'ya (2011). In such inequality, the constant $C$ is uniform with…
Using an approach by contradiction we prove the existence and uniqueness of a weak solution to a quasi-linear elliptic boundary value problem with singular convection term and Hardy Potential. Whose simplest model is \begin{equation*}…
We establish the pointwise continuity of bounded weak solutions to of a class of scalar parabolic equations and strongly coupled parabolic systems. Our approach to the regularity theory of parabolic scalar equations is quite elementary and…
We estimate contrasts $\int_0 ^1 \rho(F^{-1}(u)-G^{-1}(u))du$ between two continuous distributions $F$ and $G$ on $\mathbb R$ such that the set $\{F=G\}$ is a finite union of intervals, possibly empty or $\mathbb{R}$. The non-negative…
We prove that any weakly differentiable function with square integrable gradient can be extended to a capacitary boundary of any simply connected plane domain $\Omega\ne\mathbb R^2$ except a set of a conformal capacity zero. For locally…
In this paper, we prove that for each closed differential form $u \in L^1(\mathbb{R}^N,(\mathbb{R}^N)^{\ast} \wedge ... \wedge (\mathbb{R}^N)^{\ast})$, which is almost in $L^{\infty}$ in the sense that \[ \int_{\{y \in \mathbb{R}^N \colon…
It is well known that a weak solution $\varphi$ to the initial boundary value problem for the uniformly parabolic equation $\partial_t\varphi-\mbox{div}(A\nabla \varphi) +\omega\varphi= f $ in $\Omega_T\equiv\Omega\times(0,T)$ satisfies the…
We study the boundary value problem $-{\rm div}(\log(1+ |\nabla u|^q)|\nabla u|^{p-2}\nabla u)=f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\RR^N$ with smooth boundary. We distinguish the cases where…
The purpose of this paper is to characterize weak supercyclicity for Hilbert-space contractions, which is shown to be equivalent to characterizing weak supercyclicity for unitary operators$.$ This is naturally motivated by an open question…
We present an elementary approach for the proof of Schauder estimates for the equation $(-\Delta)^s u(x)=f(x), \,0<s<1$, with $f$ having a modulus of continuity $\omega_f$, based on the Poisson representation formula and dyadic ball…
In this paper, we study infinite-dimensional Lagrangian systems where the potential functions are periodic, rearrangement invariant and weakly upper semicontinuous. And we prove that there exists a calibrated curve for every $M\in…
This work is devoted to study the existence of infinitely many weak solutions to nonlocal equations involving a general integrodifferential operator of fractional type. These equations have a variational structure and we find a sequence of…
We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $p\ge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $\Omega$. By means of barriers, a nonlocal…
In this paper we study the following singular perturbation problem for the $p_\varepsilon(x)$-Laplacian: \[ \Delta_{p_\varepsilon(x)}u^\varepsilon:=\mbox{div}(|\nabla u^\varepsilon(x)|^{p_\varepsilon(x)-2}\nabla…
A uniformly continuously integrable sequence of real-valued measurable functions, defined on some probability space, is relatively compact in the $\sigma(L^1,L^\infty)$ topology. In this paper, we link such a result to weak convergence…
Demanding that charged Nariai black holes in (quasi-)de Sitter space decay without becoming super-extremal implies a lower bound on the masses of charged particles, known as the Festina Lente (FL) bound. In this paper we fix the…
Based on an optimal rate wavelet series representation, we derive a local modulus of continuity result with a refined almost sure upper bound for fractional Brownian motion. \sloppy The obtained upper bound of the small fractional Brownian…
We study the regularity of a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_t=\nabla\cdot(u\nabla (-\Delta)^{-1/2}u).$ For definiteness, the problem is posed…
In this article we provide existence, uniqueness and regularity results of a degenerate singular elliptic boundary value problem whose prototype is given by \begin{gather*} \begin{cases} -\operatorname{div}(w(x)|\nabla u|^{p-2}\nabla…
We find general non-linear lagrangians of a U(1) field invariant under electric-magnetic duality. They are characterized by an arbitrary function and go to the Maxwell theory in the weak field limit. We give some explicit examples which are…