Regularizing effect of absorption terms in singular problems

A comprehensive framework for hard-magnetic beams: Reduced-order theory, 3D simulations, and experiments

In this article, we consider mixed local and nonlocal Sobolev $(q,p)$-inequalities with extremal in the case $0<q<1<p<\infty$. We prove that the extremal of such inequalities is unique up to a multiplicative constant that is associated with a singular elliptic problem involving the mixed local and nonlocal $p$-Laplace operator. Moreover, it is proved that the mixed Sobolev inequalities are necessary and sufficient condition for the existence of weak solutions of such singular problems. As a consequence, a relation between the singular $p$-Laplace and mixed local and nonlocal $p$-Laplace equation is established. Finally, we investigate the existence, uniqueness, regularity and symmetry properties of weak solutions for such problems.

Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle - \Delta _{1} u = h(u)f &{} \text {in}\, \Omega , \\ u\ge 0&{} \text {in}\ \Omega ,\\ u=0 &{} \text {on}\ \partial \Omega \,. \end{array}\right. } \end{aligned}$$
 Here $$\Delta _{1} $$
 is the 1-Laplace operator, $$\Omega $$
 is a bounded open subset of $$\mathbb {R}^N$$
 with Lipschitz boundary, h(s) is a continuous function which may become singular at $$s=0^{+}$$
 , and f is a nonnegative datum in $$L^{N,\infty }(\Omega )$$
 with suitable small norm. Uniqueness of solutions is also shown provided h is decreasing and $$f>0$$
 . As a preparatory tool for our method a general theory for the same problem involving the p-Laplacian (with $$p>1$$
 ) as principal part is also established. The main assumptions are further discussed in order to show their optimality.

The Dirichlet problem for singular elliptic equations with general nonlinearities

In the regime of lubrication approximation, we look at spreading phenomena under the action of mildly singular potentials of the form 𝑃 ( ℎ ) ≈ ℎ 1 − 𝑚 as ℎ → 0 + with 1 < 𝑚 < 3, modeling repulsion between liquid-gas interface and substrate. We assume zero slippage at the contact line. Based on formal analysis arguments, here we report that for any 𝑚 ∈ ( 1 , 3 ) and any value of the speed (both positive and negative) there exists a two-parameter, hence generic, family of fronts (i.e., traveling-wave solutions with a contact line): they have ﬁnite energy, ﬁnite rate of dissipation, and microscopic contact angle equal to 𝜋 / 2, in agreement with energy minimizers. A one-parameter family of advancing “linear-log” fronts also exists, having a logarithmically corrected linear behaviour in the liquid bulk. These facts indicate that mildly singular potentials stand as an alternative solution to the contact-line paradox. We also propose a selection criterion for the fronts, based on thermodynamically consistent contact-line conditions modeling friction at the contact line. So as contact-angle conditions do in the case of slippage models, this criterion selects a unique linear-log front for each positive speed. Numerical evidences suggest that, ﬁxed the speed and the frictional coeﬃcient, its shape depends on the spreading coeﬃcient, with steeper fronts in partial wetting and a more prominent precursor region in dry complete wetting.

Thin-ﬁlm equations with mildly singular potentials: an alternative solution to the contact-line paradox

We study existence and uniqueness of solutions to a nonlinear elliptic boundary value problem with a general, and possibly singular, lower order term, whose model is $$\begin{cases} 
-\Delta_p u = H(u)\mu & \text{in}\ \Omega,\\ u>0 &\text{in}\ \Omega,\\ u=0 &\text{on}\ \partial\Omega. \end{cases}$$ Here $\Omega$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge2$), $\Delta_p u:= \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ ($1<p<N$) is the $p$-laplacian operator, $\mu$ is a nonnegative bounded Radon measure on $\Omega$ and $H(s)$ is a continuous, positive and finite function outside the origin which grows at most as $s^{-\gamma}$, with $\gamma\ge0$, near zero.

Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data

In the regime of lubrication approximation, we look at spreading phenomena under the action of singular potentials of the form $P(h)\approx h^{1-m}$ as $h\to 0^+$ with $m>1$, modeling repulsion between the liquid-gas interface and the substrate. We assume zero slippage at the contact line. Based on formal analysis arguments, we report that for any $m>1$ and any value of the speed (both positive and negative) there exists a three-parameter, hence generic, family of fronts (i.e., traveling-wave solutions with a contact line). A two-parameter family of advancing"linear-log"fronts also exists, having a logarithmically corrected linear behaviour in the liquid bulk. All these fronts have finite rate of dissipation, indicating that singular potentials stand as an alternative solution to the contact-line paradox. In agreement with steady states, fronts have microscopic contact angle equal to $\pi/2$ for all $m>1$ and finite energy for all $m<3$. We also propose a selection criterion for the fronts, based on thermodynamically consistent contact-line conditions modeling friction at the contact line. So as contact-angle conditions do in the case of slippage models, this criterion selects a unique (up to translation) linear-log front for each positive speed. Numerical evidence suggests that, fixed the speed and the frictional coefficient, its shape depends on the spreading coefficient, with steeper fronts in partial wetting and a more prominent precursor region in dry complete wetting.

Thin-film equations with singular potentials: an alternative solution to the contact-line paradox

We study the asymptotic behavior, as $\gamma$ tends to infinity, of solutions for the homogeneous Dirichlet problem associated to singular semilinear elliptic equations whose model is $$ -\Delta u=\frac{f(x)}{u^\gamma}\,\text{ in }\Omega\,, $$ where $\Omega$ is an open, bounded subset of $\mathbb{R}^N$ and $f$ is a bounded function. We prove existence and nonexistence of a limit equation under two different assumptions on $f$: either strictly positive on every compactly contained subset of $\Omega$ or only nonnegative. Through this study we deduce optimal existence and nonexistence results of positive solutions for the homogeneous Dirichlet problem associated to $$ -\Delta v + \frac{|\nabla v|^2}{v} = f\,\text{ in }\Omega\,. $$

Existence and Nonexistence results for singular elliptic equations

We deal with existence, uniqueness and regularity of nonnegative solutions to a Dirichlet problem for equations as \begin{equation*} \displaystyle -\operatorname{div}\left(\frac{|\nabla u|^{p-2}\nabla u}{(1+u)^{\theta(p-1)}}\right) = h(u)f \quad \text{in }\Omega, \end{equation*} where $\Omega$ is an open bounded subset of $\mathbb{R}^N$ ($N\ge 2$), $p>1$, $\theta\ge 0$, $f\geq 0$ belongs to a suitable Lebesgue space and $h$ is a continuous, nonnegative function which may blow up at zero and it is bounded at infinity.

Riccardo Durastanti

Papers

Thin-ﬁlm equations with mildly singular potentials: an alternative solution to the contact-line paradox

Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data

Thin-film equations with singular potentials: an alternative solution to the contact-line paradox

Existence and Nonexistence results for singular elliptic equations

The Dirichlet problem for possibly singular elliptic equations with degenerate coercivity.