In this paper, we study existence of solutions to the $1$--Laplacian elliptic equation with inhomogeneous Robin boundary conditions. It is also analyzed from the point of view of the Euler--Lagrange equation of a lower semicontinuous functional. We see the equivalence between the solutions of the elliptic problem and the minimizers of the functional.

The object of study is the Klein-Gordon equation in $1+1$ dimensions, with integer power non-linearities. In particular, of interest is the spectral stability/instability (with respect to perturbations of the same period) of traveling-standing periodic solitons, which are of cnoidal ($p=2$), dnoidal ($p=3$) or more general type ($p=5$). The corresponding linearized problem for this two-parameter family of solutions fits the general abstract framework of spectral stability for second order Hamiltonian systems, recently developed by the last two authors and Bronski-Johnson-Kapitula. It is worth noting that the spatial periodicity however, forces a relation between the speed and the phase, which results in some unique challenges in the computations of the quantities involved in the stability index. Our results generalize recent work on the simpler case of standing waves of Natali-Pastor, [9] and Natali-Cardoso, [10].

In this paper, we study the derivation of a certain type of NLS from many-body interactions of bosonic particles in $d=1,2$. We consider a model with a finite linear combination of $n$-body interactions and obtain that the $k$-particle marginal density of the BBGKY hierarchy converges when particle number goes to infinity. Moreover, the limit solves a corresponding infinite Gross-Pitaevskii hierarchy. We prove the uniqueness of factorized solution to the Gross-Pitaevskii hierarchy based on a priori space time estimates. The convergence is established by adapting the arguments originated or developed in [6], [15] and [2]. For the uniqueness part, we expand the procedure in [16] by introducing a different board game argument to handle the factorial in the number of terms from Duhamel expansion. The space time bound assumption in [16] is removed in our proof.

The principal purpose of this note is to provide a reconstruction procedure for distributional matrix-valued potential coefficients of Schrödinger-type operators on a half-line from the underlying Weyl--Titchmarsh function.

The aim of this note is to investigate the size of the singular set of a general class of free interface problems. We show porosity of the singular set, obtaining as a corollary that both its Hausdorff and Minkowski dimensions are strictly smaller than $n-1$.

We give a regularity criterion for a $Q$-tensor system modeling a nematic Liquid Crystal, under homogeneous Neumann boundary conditions for the tensor $Q$. Starting of a criterion only imposed on the velocity field ${{\textbf{u}}}$ two results are proved; the uniqueness of weak solutions and the global in time weak regularity for the time derivative $(\partial_t {{\textbf{u}}},\partial_t Q)$. This paper extends the work done in [8] for a nematic Liquid Crystal model formulated in $({{\textbf {u}}},{{\textbf {d}}})$, where ${{\textbf {d}}}$ denotes the orientation vector of the liquid crystal molecules.

In this paper, we investigate the lack of compactness of the Sobolev embedding of $H^1(\mathbb R^2)$ into the Orlicz space $L^{{\phi}_p}(\mathbb R^2)$ associated to the function $\phi_p$ defined by $ \phi_p(s):={\rm{e}^{s^2}}-\sum_{k=0}^{p-1} \frac{s^{2k}}{k!}$. We also undertake the study of a nonlinear wave equation with exponential growth where the Orlicz norm $\|.\|_{L^{\phi_p}}$ plays a crucial role. This study includes issues of global existence, scattering and qualitative study.

We study well-posedness issues in Gevrey classes for the Cauchy problem for wave equations of the form $\partial_t^2 u -a(t) \partial_x^2 u=0$. In the strictly hyperbolic case $a (t)\geq c(> 0)$, Colombini, De Giorgi and Spagnolo have shown that it is sufficient to assume H$\ddot { \rm o}$lder regularity of the coefficients in order to prove Gevrey well-posedness. Recently, assumptions bearing on the oscillations of the coefficient have been imposed in the literature in order to guarantee well-posedness. In the weakly hyperbolic case $a(t) \geq 0$, Colombini, Jannelli and Spagnolo proved well-posedness in Gevrey classes of order $1\leq s<s_0=1+(k+\alpha)/2$. In this paper, we put forward condition $ |a'(t)| \leq Ca(t)^p/t^q $ that bears both on the oscillations and the degree of degeneracy of the coefficient. we show that under such a condition, Gevrey well-posedness holds for $1\leq s<qs_0/\{ (k+\alpha) (1-p) +q-1\}$ if $q \geq 3-2p$. In particular, this improves on the result (corresponding to the case $p=0$) of Colombini, Del Santo and Kinoshita.