Some remarks on the duality method for Integro-Differential equations with measure data

We deal with existence, uniqueness, and regularity for solutions of the boundary value problem $$\begin{cases} {\mathcal L}^s u = \mu &\quad \text{in $\Omega$}, u(x)=0 \quad &\text{on} \ \ \mathbb{R}^N\backslash\Omega, \end{cases}$$ where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $\mu$ is a bounded radon measure on $\Omega$, and ${\mathcal L}^s$ is a nonlocal operator of fractional order $s$ whose kernel $K$ is comparable with the one of the factional laplacian

Non-stationary Magnetic Microstructures in Stellar Thin Accretion Discs

We examine the morphology of magnetic structures in thin plasma accretion discs, generalizing a stationary ideal MHD model to the time-dependent visco-resistive case. Our analysis deals with small scale perturbations to a central dipole-like magnetic field, which give rise -- as in the ideal case -- to the periodic modulation of magnetic flux surfaces along the radial direction, corresponding to the formation of a toroidal current channels sequence. These microstructures suffer an exponential damping in time because of the non-zero resistivity coefficient, allowing us to define a configuration lifetime which mainly depends on the midplane temperature and on the length scale of the structure itself. By means of this lifetime we show that the microstructures can exist within the inner region of stellar discs in a precise range of temperatures, and that their duration is consistent with local transient processes (minutes to hours).Comment: 14 text pages, 5 figures, to be published on PR

On singular elliptic equations with measure sources

We prove existence of solutions for a class of singular elliptic problems with a general measure as source term whose model is $$\begin{cases} -\Delta u = \frac{f(x)}{u^{\gamma}} +\mu & \text{in}\ \Omega, u=0 &\text{on}\ \partial\Omega, u>0 &\text{on}\ \Omega, \end{cases}$$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N$. Here $\gamma > 0$, $f$ is a nonnegative function on $\Omega$, and $\mu$ is a nonnegative bounded Radon measure on $\Omega$

A Lazer-McKenna type problem with measures

In this paper we are concerned with a general singular Dirichlet boundary value problem whose model is the following $$\begin{cases} -\Delta u = \frac{\mu}{u^{\gamma}} & \text{in}\ \Omega, u=0 &\text{on}\ \partial\Omega, u>0 &\text{on}\ \Omega\,. \end{cases}$$ Here $\mu$ is a nonnegative bounded Radon measure on a bounded open set $\Omega\subset\mathbb{R}^N$, and $\gamma>0$