Related papers: The unique continuation property for a nonlinear e…

The purpose of this paper is to investigate the time behavior of the solution of a weighted $p$-Laplacian evolution equation, given by \begin{align} \label{eveq} \begin{cases} u_{t} = \text{div} \left(\gamma |\nabla u|^{p-2}\nabla u \right)…

The notion of tree entropy was introduced by the author as a normalized limit of the number of spanning trees in finite graphs, but is defined on random infinite rooted graphs. We give some new expressions for tree entropy; one uses…

In the present paper we prove uniqueness results for solutions to a class of Neumann boundary value problems whose prototype is --div((1 + |$\nabla$u| 2) (p--2)/2 $\nabla$u) -- div(c(x)|u| p--2 u) = f in $\Omega$, (1 + |$\nabla$u| 2)…

In this paper, we study the following singular nonlinear elliptic problem \begin{equation}\label{eq:1} \left\{ \begin{array}{ll} \displaystyle (-\Delta)^{\frac \alpha 2} u=\lambda |u|^{r-2}u+\mu\frac{|u|^{q-2}u}{|x|^{s}}\quad &{\rm in…

This paper examines the behavior of a positive solution $u\in C^{1,\alpha}(\Bar{\Omega})$ of the $(p,q)$ Laplace equation with a singular term and zero Dirichlet boundary condition. Specifically, we consider the equation: \begin{equation*}…

We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation $u_t-\Delta _{p(x)}u = f(x,u)$ in $ (0,T)\times\Omega$; $u = 0$ on $(0,T)\times\partial\Omega$; $u(0,x)=u_0(x)$ in $\Omega$;…

In this paper we prove unique continuation principles for some systems of elliptic partial differential equations satisfying a suitable superlinearity condition. As an application, we obtain nonexistence of nontrivial (not necessarily…

We look for nonconstant, positive, radially nondecreasing solutions of the quasilinear equation $-\Delta_p u+u^{p-1}=f(u)$ with $p>2$, in the unit ball $B$ of $\mathbb R^N$, subject to homogeneous Neumann boundary conditions. The…

We consider a nonlinear eigenvalue problem driven by the sum of $p$ and $q$-Laplacian. We show that the problem has a continuous spectrum. Our result reveals a discontinuity property for the spectrum of a parametric ($p,q$)-differential…

We study the Dirichlet problem of the following discrete infinity Laplace equation on a subgraph with finite width $$\Delta_{\infty} u(x) = \inf_{y \sim x}u(y)+\sup_{y \sim x}u(y)-2u(x) = f(x).$$ We say that a subgraph has finite width if…

The Unique Games Conjecture (UGC) constitutes a highly dynamic subarea within computational complexity theory, intricately linked to the outstanding P versus NP problem. Despite multiple insightful results in the past few years, a proof for…

As an application of the theory of linear parabolic differential equations on noncompact Riemannian manifolds, developed in earlier papers, we prove a maximal regularity theorem for nonuniformly parabolic boundary value problems in…

We prove a unique continuation property for the fractional Laplacian $(-\Delta)^s$ when $s \in (-n/2,\infty)\setminus \mathbb{Z}$. In addition, we study Poincar\'e-type inequalities for the operator $(-\Delta)^s$ when $s\geq 0$. We apply…

In this paper, we investigate the existence and uniqueness of a non-trivial solution for a class of nonlocal equations involving the fractional $p$-Laplacian operator defined on compact Riemannian manifold, namely,…

We consider the (viscosity) solution $u(x,t)$ of the nonlinear evolution equation $u_t-\Delta^G_p u=0$ in a (not necessarily bounded) domain $\Omega$, such that $u=0$ in $\Omega$ at time $t=0$ and $u=1$ on the boundary of $\Omega$ at all…

We consider N-player non-zero sum games played on finite trees (i.e., sequential games), in which the players have the right to repeatedly update their respective strategies (for instance, to improve the outcome wrt to the current strategy…

We introduce a new method for the analysis of singularities in the unstable problem $$\Delta u = -\chi_{\{u>0\}},$$ which arises in solid combustion as well as in the composite membrane problem. Our study is confined to points of…

We provide explicit upper bounds for the eigenvalues of the Laplacian on a finite metric tree subject to standard vertex conditions. The results include estimates depending on the average length of the edges or the diameter. In particular,…

In this paper we study the Poisson problem, \[ \begin{cases} -{\rm div}(d^\beta\nabla u)=f&{\rm in}\ \Omega\\ u=0&{\rm on}\ \partial\Omega, \end{cases} \] where $\Omega\subset\mathbb R^N$, $N\ge2$ is a smooth bounded domain, $f$ is a…

Let $u$ be a non-negative super-solution to a $1$-dimensional singular parabolic equation of $p$-Laplacian type ($1<p<2$). If $u$ is bounded below on a time-segment $\{y\}\times(0,T]$ by a positive number $M$, then it has a power-like decay…