Related papers: Nonlinear Evolution Equation Associated with Hyper…

Consider the following nonlinear elliptic equation of $p(x)$-Laplacian type with nonstandard growth \begin{equation*} \left\{ \begin{aligned} &{\rm div} a(Du, x)=\mu \quad &\text{in}& \quad \Omega, &u=0 \quad &\text{on}& \quad…

This work tackles an inverse boundary value problem for a $p$-Laplace type partial differential equation parametrized by a smoothening parameter $\tau \geq 0$. The aim is to numerically test reconstructing a conductivity type coefficient in…

We establish the existence of multiple solutions for a nonvariational elliptic systems involving $p(x)$-Laplacian operator. The approach combines the methods of sub-supersolution and Leray--Schauder topological degree.

In this paper we study the nonlinear Dirac (NLD) equation on noncompact metric graphs with localized Kerr nonlinearities, in the case of Kirchhoff-type conditions at the vertices. Precisely, we discuss existence and multiplicity of the…

The $p$-Laplacian evolution equation in metric measure spaces has been studied as the gradient flow in $L^2$ of the $p$-Cheeger energy (for $1 < p < \infty$). In this paper, using the first-order differential structure on a metric measure…

Laplace interpolation is a popular approach in image inpainting using partial differential equations. The classic approach considers the Laplace equation with mixed boundary conditions. Recently a more general formulation has been proposed…

We consider an elliptic differential operator $A_\varepsilon = - \frac{d}{dx} g(x/\varepsilon) \frac{d}{dx} + \varepsilon^{-2} V(x/\varepsilon)$, $\varepsilon > 0$, with periodic coefficients acting in $L_2(\mathbb{R})$. For the…

In this paper, with a fixed $p\in (1,+\infty)$ and a bounded domain $\Omega \subset \mathbb{R}^N$ whose boundary $\partial\Omega$ fulfills the $C^1$ regularity, we study a boundary value problem involving a nonlocal operator assigning to…

We introduce a mathematical model in $\mathbb{R}^{n}$ for evolution equations with modified generalized Hartree nonlinearity given by $S_{\alpha,p,q}(u)=I_{\alpha}(|u|^{p+q}).$ One can see that this nonlinearity is not integrable due to the…

We discuss one of the many topics that illustrate the interaction of Blaine Lawson's deep geometric and analytic insights. The first author is extremely grateful to have had the pleasure of collaborating with Blaine over many enjoyable…

In this article we implement a method for the computation of a nonlinear elliptic problem with nonstandard growth driven by the $p(x)-$Laplacian operator. Our implementation is based in the {\em decomposition--coordination} method that…

We prove weighted $L^p$-Liouville theorems for a class of second order hypoelliptic partial differential operators $\mathcal{L}$ on Lie groups $\mathbb{G}$ whose underlying manifold is $n$-dimensional space. We show that a natural weight is…

We study ultradistributional boundary values of zero solutions of a hypoelliptic constant coefficient partial differential operator $P(D) = P(D_x, D_t)$ on $\mathbb{R}^{d+1}$. Our work unifies and considerably extends various classical…

We develop a nonlinear spectral graph theory, in which the Laplace operator is replaced by the 1-Laplacian ?$\Delta_1$. The eigenvalue problem is to solve a nonlinear system involving a set valued function. In the study, we investigate the…

In $L_2(\mathbb{R}^d)$, we consider an elliptic differential operator $\mathcal{A}_\varepsilon = - \operatorname{div} g(\mathbf{x}/\varepsilon) \nabla + \varepsilon^{-2} V(\mathbf{x}/\varepsilon)$, $ \varepsilon > 0$, with periodic…

Lie algebras of systems of $2 g$ graded heat conduction operators $Q_{2k}$, where $k = 0,1, \ldots,2 g-1$, determining sigma functions $\sigma(z, \lambda)$ of genus $g = 1,2$, and $3$ hyperelliptic curves are constructed. As a corollary, it…

Let $\Omega$ be a bounded domain of $\mathbb{R}^{n+1}$ with $n \ge 1$. We assume that the boundary $\Gamma$ of $\Omega$ is Lipschitz. Consider the Dirichlet-to-Neumann operator $N_0$ associated with a system in divergence form of size $m$…

We study the character of dependence on the data and the nonlinear structure of the equation for the solutions of the homogeneous Dirichlet problem for the evolution $p(x,t)$-Laplacian with the nonlinear source \[…

We study Poincar\'e problem for a linear uniformly parabolic operator $\P$ in a cylinder $Q=\Omega\times (0,T).$ The boundary operator $\B$ is defined by an oblique derivative with respect to a tangential vector field $\l$ defined on the…

The purpose of this paper is to describe geometrically discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids. From a variational principle we derive the discrete Euler-Lagrange equations and we introduce a symplectic 2-section,…