For $d\geq 2$ and $\frac{2d+2}{d+2} < p < \infty $, we prove a strict Faber-Krahn type inequality under polarizations for the first eigenvalue $\lambda _1(\Omega )$ of the $p$-Laplace operator on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ with mixed boundary conditions. We apply this inequality to the obstacle problems on domains of the form $\Omega \setminus \mathscr{O}$, where $\mathscr{O}\subset \subset \Omega $ is an obstacle. Under some geometric assumptions on $\Omega $ and $\mathscr{O}$, we prove the strict monotonicity of $\lambda _1 (\Omega \setminus \mathscr{O})$ with respect to certain translations and rotations of $\mathscr{O}$ in $\Omega$.

The aim of this paper is to study the long-time behavior of impulsive evolution processes. We obtain qualitative properties for impulsive evolution processes, and we prove an existence result of impulsive pullback attractors. Additionally, we provide sufficient conditions to obtain the upper semicontinuity at zero for a family of impulsive pullback attractors. As an application, we study the asymptotic dynamics of the following non-autonomous coupled wave system subject to impulsive effects at variable times, given by the following evolution system $$ \begin{cases} u_{tt} - \Delta u + u + \eta(-\Delta)^{\frac{1}{2}}u_t + a_{\epsilon}(t)(-\Delta)^{\frac{1}{2}}v_t = f(u), & (x, t) \in \Omega \times (\tau, \infty), \\ v_{tt} - \Delta v + \eta(-\Delta)^{\frac{1}{2}}v_t - a_{\epsilon}(t)(-\Delta)^{\frac{1}{2}}u_t = 0, & (x, t) \in \Omega \times (\tau, \infty), \\ u = v = 0, & (x, t) \in \partial\Omega\times (\tau, \infty), \\ \left\{I_t\colon M(t)\subset Y_0\to Y_0\right\}_{t \in \mathbb R}, \end{cases} $$ with initial conditions $u(\tau, x) = u_0(x)$, $u_t(\tau, x) = u_1(x)$, $v(\tau, x) = v_0(x)$, $v_t(\tau, x) = v_1(x)$, $x \in \Omega$, $\tau \in \mathbb{R}$, where $\Omega$ is a bounded smooth domain in $\mathbb{R}^n$ $(n \geq 3)$ with boundary $\partial\Omega$ assumed to be regular enough, $\eta > 0$ is constant, $a_{\epsilon}$ and $f$ are suitable functions, the family $\hat{M} = \{M(t)\}_{t \in \mathbb R}$ is an impulsive family, $\hat{I} = \{I_t\colon M(t)\subset Y_0\to Y_0\}_{t \in \mathbb R}$ is an impulse function and $Y_0$ is a Hilbert space.

We consider a Ginzburg-Landau type equation in $\mathbb R^2$ of the form $-\Delta u = u J'(1-|u|^{2})$ with a potential function $J$ satisfying weak conditions allowing for example a zero of infinite order in the origin. We extend in this context the results concerning quantization of finite potential solutions of H. Brezis, F. Merle, T. Rivière from [10] who treat the case when $J$ behaves polinomially near $0,$ as well as a result of Th. Cazenave, found in the same reference, and concerning the form of finite energy solutions.

This paper treats second order fully nonlinear degenerate elliptic equations having a family of subunit vector fields satisfying a full-rank bracket condition. It studies Liouville properties for viscosity sub- and supersolutions in the whole space, namely, under a suitable bound at infinity from above and, respectively, from below, they must be constants. In a previous paper, we proved an abstract result and discussed operators on the Heisenberg group. Here, we consider various families of vector fields: the generators of a Carnot group, with more precise results for those of step 2, in particular H-type groups and free Carnot groups, the Grushin and the Heisenberg-Greiner vector fields. All these cases are relevant in sub-Riemannian geometry and have in common the existence of a homogeneous norm that we use for building Lyapunov-like functions for each operator. We give explicit sufficient conditions on the size and sign of the first and 0-th order terms in the equations and discuss their optimality. We also outline some applications of such results to the problem of ergodicity of multidimensional degenerate diffusion processes in the whole space.

This work is concerned with new results on long-time dynamics of a class of hyperbolic evolution equations related to extensible beams with three distinguished nonlocal nonlinear damping terms. In the first possibly degenerate case, the results feature the existence of a family of compact global attractors and a thickness estimate for their Kolmogorov’s $\varepsilon$-entropy. Then, in the non-degenerate context, the structure of the helpful nonlocal damping leads to the existence of finite-dimensional compact global and exponential attractors. Lastly, in a degenerate and critical framework, it is proved the existence of a bounded closed global attractor but not compact. To the proofs, we provide several new technical results by means of refined estimates that open up perspectives for a new branch of nonlinearly damped problems.