Analysis and Applications

In this work, we prove existence and multiplicity of solutions to a Kirchhoff elliptic problem in the whole space ${ℝ}^N$. More specifically, we consider the following nonlocal elliptic problem:

This paper is dedicated to the homogenization and dimension reduction for a nonlinear elasticity problem of a textile structure. The structure is represented as a squared piece of cloth and modeled as a woven canvas made of long and thin fibers, crossing each other in a periodic pattern. The squared domain is partially clamped on the left and bottom. The fibers are assumed to be glued, a condition that allows to extend the woven structure to a non-perforate plate. The nonlinear elasticity problem is stated via minimization over the energy functional. The homogenization is made via the periodic unfolding method, with an additional dimension reduction. Since the existence of a minimum of the limit energy functional is not ensured, sufficient conditions for the existence are given by assuming that the material is homogeneous and isotropic.

In this paper, the Cauchy problem of the modified short pulse (mSP) equation with initial conditions in weighted Sobolev space is studied by using $\partial$-steepest descent method. Based on the spectral analysis of Lax pair and the transformations of field variables and independent variables, a basic Riemann–Hilbert problem (RHP) is established, and the solution of Cauchy problem for the mSP equation is transformed into the corresponding RHP. The asymptotic expansion of the solution of mSP equation is derived in a fixed space-time cone

The main focus of this contribution is on the harmonic Bergman spaces ${{ℬ}}_{\alpha }^p$ on the $q$-homogeneous tree ${{𝒳}}_q$ endowed with a family of measures ${\sigma }_{\alpha }$ that are constant on the horocycles tangent to a fixed boundary point and turn out to be doubling with respect to the corresponding horocyclic Gromov distance. A central role is played by the reproducing kernel Hilbert space ${{ℬ}}_{\alpha }^p$ for which we find a natural orthonormal basis and formulae for the kernel. We also consider the atomic Hardy space and the bounded mean oscillation space. Appealing to an adaptation of Calderón–Zygmund theory and to standard boundedness results for integral operators on $L_{\alpha }^p$ spaces with Hörmander-type kernels, we determine the boundedness properties of the Bergman projection. This work was inspired by [J. M. Cohen, F. Colonna, M. A. Picardello and D. Singman, Bergman spaces and Carleson measures on homogeneous isotropic trees, Potential Anal. 44(4) (2016) 745–766, doi:10.1007/s11118-015-9529-7; F. De Mari, M. Monti and M. Vallarino, Harmonic Bergman projectors on homogeneous trees, Potential Anal. 61 (2024) 153–182].

In this paper, we consider a planar viscous shock of moderate strength for a scalar viscous conservation law in multi-D. We consider a strictly convex flux, as a small perturbation of the Burgers flux, along the normal direction to the shock front. However, for the transversal directions, we do not have any restrictions on flux function. We first show the contraction property for any large perturbations in $L^2$ of the planar viscous shock. If the initial $L^2$-perturbation is also in $L^1$, the large perturbation converges to zero in $L^2$ as time goes to infinity with $t^{-1/4}$ decay rate. The contraction and decay estimates hold up to dynamical shift. For the results, we do not impose any smallness conditions on the initial value. This result extends the 1D case [M.-J. Kang and A. F. Vasseur, $L^2$-contraction for shock waves of scalar viscous conservation laws, Ann. Inst. H. Poincaré C Anal. Non Linéaire 34 (2017) 139–156] by the first author and Vasseur to the multi-dimensional case.