Related papers: A basic homogenization problem for the $p$-Laplaci…

We study solutions to variable-coefficient elliptic equations of the form $-\D(A(x) \nabla u) = \kappa u$, $\kappa>0$, in an exterior domain $\Om\subset \Rn$, where $A(x)$ is uniformly elliptic and asymptotically flat. Extending Rellich's…

We consider periodic homogenization with localized defects for semilinear elliptic equations and systems of the type $$ \nabla\cdot\Big(\Big(A(x/\varepsilon)+B(x/\varepsilon)\Big)\nabla u(x)+c(x,u(x)\Big)=d(x,u(x)) \mbox{ in } \Omega $$…

We study regularity criteria for the $d$-dimensional incompressible Navier-Stokes equations. We prove if $u\in L_{\infty}^tL_d^x((0,T)\times \mathbb{R}^d_+)$ is a Leray-Hopf weak solution vanishing on the boundary and the pressure $p$…

We present new $L^\infty$ a priori estimates for weak solutions of a wide class of subcritical $p$-laplacian equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method…

In this work, we study the existence of weak solution to the following quasi linear elliptic problem involving the fractional $p$-Laplacian operator, a Hardy potential and multiple critical Sobolev nonlinearities with singularities,…

Given a map $u : \Omega \subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$, the $\infty$-Laplacian is the system \[ \label{1} \Delta_\infty u \, :=\, \Big(\text{D}u \otimes \text{D}u + |\text{D}u|^2 [\text{D}u]^\bot \! \otimes I \Big) :…

A compactness theorem for volume-constrained almost-critical points of elliptic integrands is proven. The result is new even for the area functional, as almost-criticality is measured in an integral rather than in a uniform sense. Two main…

We consider the minimal discrete and continuous energy problems on the unit sphere $\mathbb{S}^d$ in the Euclidean space $\mathbb{R}^{d+1}$ in the presence of an external field due to finitely many localized charge distributions on…

To our knowledge, this paper is the first attempt to consider the existence issue for fractional $p$-Laplacian equation: $(-\Delta)_p^s u= \lambda f(u),\; u> 0 ~\text{in}~\Omega;\; u=0\;\text{in}~ \mathbb{R}^N\setminus\Omega$, where $p>1$,…

We consider the problem of proving $L^p$ bounds for eigenfunctions of the Laplacian in the high frequency limit in the presence of nonpositive curvature and more generally, manifolds without conjugate points. In particular, we prove…

We analyze various formulations of the $\infty$-Laplacian eigenvalue problem on graphs, comparing their properties and highlighting their respective advantages and limitations. First, we investigate the graph $\infty$-eigenpairs arising as…

The Dirichlet Laplacian in a curved three-dimensional tube built along a spatial (bounded or unbounded) curve is investigated in the limit when the uniform cross-section of the tube diminishes. Both deformations due to bending and twisting…

Let $\Omega$ be a bounded domain of $\mathbf{R}^{N},$ $N\geq2.$ Let, for $p>N,$ \[ \Lambda_{p}(\Omega):=\inf\left\{ \left\Vert \nabla u\right\Vert _{p}^{p}:u\in W_{0}^{1,p}(\Omega)\quad and\quad\left\Vert u\right\Vert _{\infty}=1\right\} .…

In this article, we prove an eigenvalue pinching theorem for the first eigenvalue of the Laplacian on compact hypersurfaces in a sphere. Let $(M^n,g)$ be a closed, connected and oriented Riemannian manifold isometrically immersed by $\phi$…

In this paper we analyze the asymptotic behaviour as $p\to 1^+$ of solutions $u_p$ to $$ \left\{ \begin{array}{rclr} -\Delta_pu&=&\lambda|\nabla u|^{p-2}\nabla u\cdot\frac{x}{|x|^2}+ f&\quad \mbox{ in } \Omega,\\ u_p&=&0 &\quad \mbox{ on…

Motivated by Thurston and Daskalopoulos--Uhlenbeck's approach to Teichm\"uller theory, we study the behavior of $q$-harmonic functions and their $p$-harmonic conjugates in the limit as $q \to 1$, where $1/p + 1/q = 1$. The $1$-Laplacian is…

Let $\Omega \Subset \mathbb R^n$, $f \in C^1(\mathbb R^{N\times n})$ and $g\in C^1(\mathbb R^N)$, where $N,n \in \mathbb N$. We study the minimisation problem of finding $u \in W^{1,\infty}_0(\Omega;\mathbb R^N)$ that satisfies \[ \big\|…

We study the relationship between the Regularity and Dirichlet boundary value problems for parabolic equations of the form $Lu=\text{div}(A \nabla u)-u_t=0$ in Lip$(1,1/2)$ time-varying cylinders, where the coefficient matrix $A = \left[…

We prove a stochastic homogenization result for a class of \emph{nonlinear} and \emph{nonlocal} variational problems in domains with many small randomly distributed (bilateral) obstacles. Our model case is a Dirichlet problem for the…

This work is devoted to study the asymptotic behavior of critical points $\{(u_\varepsilon,v_\varepsilon)\}_{\varepsilon>0}$ of the Ambrosio-Tortorelli functional. Under a uniform energy bound assumption, the usual $\Gamma$-convergence…