Related papers: The Hopf-Laplace equation: harmonicity and regular…

The Convex Envelope of a given function was recently characterized as the solution of a fully nonlinear Partial Differential Equation (PDE). In this article we study a modified problem: the Dirichlet problem for the underlying PDE. The main…

We extend recent work of Gurel-Gurevich--Jerison--Nachmias (2020) and Bou-Rabee--Gwynne (2024) by showing that as the mesh of our lattice tends to $0$, we have a polynomial rate of convergence for the Dirichlet problem on orthodiagonal maps…

Hein and Pr\"{u}ss [J. Differential Equations, 261(2016)4709-4727] presented a version of Hartman-Grobman type $C^{0}$ linearization result for semilinear hyperbolic evolution equations. They showed that the linearising map (homomorphism)…

We establish interior and up to the boundary H\"older regularity estimates for weak solutions of the Dirichlet problem for the fractional $g-$Laplacian with bounded right hand side and $g$ convex. These are the first regularity results…

We prove several results for the Dirichlet, Neumann and Regularity problems for the Laplace equation in graph Lipschitz domains in the plane, considering $A_{\infty}$-measures on the boundary. More specifically, we study the…

In this paper, we study the homogenization of elliptic equations that combine a local part, given by the Laplacian with Neumann boundary conditions, and its nonlocal version, defined through an integral operator with a smooth kernel. These…

We introduce techniques for turning estimates on the infinitesimal behavior of solutions to nonlinear equations (statements concerning tangent cones and blow ups) into more effective control. In the present paper, we focus on proving…

We consider a family of self-adjoint Ornstein--Uhlenbeck operators $L_{\alpha} $ in an infinite dimensional Hilbert space H having the same gaussian invariant measure $\mu$ for all $\alpha \in [0,1]$. We study the Dirichlet problem for the…

The Laplace equation in three dimensional Euclidean space is $R$-separable in bi-cyclide coordinates leading to harmonic functions expressed in terms of Lam\'e-Wangerin functions called internal and external bi-cyclide harmonics. An…

We prove a Hopf-type lemma for antisymmetric super-solutions to the Dirichlet problem for the fractional Laplacian with zero-th order terms. As an application, we use such a Hopf-type lemma in combination with the method of moving planes to…

We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range amongst those with unbalanced…

We consider minimising $p$-harmonic maps from three-dimensional domains to the real projective plane, for $1<p<2$. These maps arise as least-energy configurations in variational models for nematic liquid crystals. We show that the singular…

For minimizers in a geometrically nonlinear Cosserat model for micropolar elasticity of continua, we prove interior H\"older regularity, up to isolated singular points that may be possible if the exponent $p$ from the model is $2$ or in…

In this article, we investigate the H\"{o}lder regularity of the fractional $p$-Laplace equation of the form $(-\Delta_p)^s u=f$ where $p>1, s\in (0, 1)$ and $f\in L^\infty_{\rm loc}(\Omega)$. Specifically, we prove that $u\in C^{0,…

In this paper, we study the stability problem of exponentially subelliptic harmonic maps from sub-Riemannian manifolds to Riemannian manifolds. We derive the rst and second variation formulas for exponentially subelliptic harmonic maps, and…

In this paper we are interested in integro-differential elliptic and parabolic equations involving nonlocal operators with order less than one, and a gradient term whose coercivity growth makes it the leading term in the equation. We obtain…

In this paper, we study the Dirichlet problem for Laplace's equation in an open disk. The uniqueness of solutions is ensured by the well-known weak maximum principle. We introduce a novel approach to demonstrate the existence of a solution…

We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative,…

The aim of the present paper is to investigate the behavior of the spectrum of the Neumann Laplacian in domains with little holes excised from the interior. More precisely, we consider the eigenvalues of the Laplacian with homogeneous…

We study the asymptotic Dirichlet problem for $\mathcal{A}$-harmonic functions on a Cartan-Hadamard manifold whose radial sectional curvatures outside a compact set satisfy an upper bound $$ K(P)\le - \frac{1+\varepsilon}{r(x)^2 \log r(x)}…