Related papers: $p-$Harmonic functions in the upper half-space

Harmonic maps are nonlinear extensions of harmonic functions. They are critical points of natural energy functionals between Riemannian manifolds. Such type of problems appear in Physics, Geometry of Finance and the study of regularity and…

Let $u$ be a bounded positive solution to the problem $-\Delta_p u = f(u)$ in $\mathbb{R}^N_+$ with zero Dirichlet boundary condition, where $p>1$ and $f$ is a locally Lipschitz continuous function. Among other things, we show that if…

Let $p,q$ be functions on $\mathbb{R}^{N}$ satisfying $1\ll q\ll p\ll N$, we consider $p(x)$-Laplacian problems of the form \[ \left\{ \begin{array} [c]{l}% -\Delta_{p(x)}u+V(x)\vert u\vert ^{p(x)-2}u=\lambda\vert u\vert…

These notes are concerned with harmonic and holomorphic functions on Euclidean spaces, using quaternions and Clifford algebras in higher dimensions. The main themes are weak solutions, the mean-value property, and subharmonicity.

We give a proof of Lipschitz continuity of p-harmonious functions, that are tug-of-war game analogies of ordinary p-harmonic functions. This result is used to obtain a new proof of Harnack's inequality for p-harmonic functions in the case…

Generalizing the well-known mean-value property of harmonic functions, we prove that a p-harmonic function of two variables satisfies, in a viscosity sense, two asymptotic formulas involving its local statistics. Moreover, we show that…

We prove an uniform boundary Harnack inequality for nonnegative functions harmonic with respect to $\alpha$-stable process on the Sierpi{\'n}ski triangle, where $\alpha \in (0, 1)$. Our result requires no regularity assumptions on the…

We consider weighted banach spaces of holomorphic functions on the upper half plane that are determined by $ \|f\|=\sup_{y>0,-\infty<x<\infty}p(y)|f(x+iy)|<\infty $ for a very large class of weight functions p. We completely solve the…

We investigate the existence of nonnegative solutions for a nonlinear problem involving the fractional p-Laplacian operator. The problem is set on a unbounded domain, and compactness issues have to be handled.

We show that any periodic with respect to normal subgroups (of the group representation of the Cayley tree) of finite index $p$-harmonic function is a constant. For some normal subgroups of infinite index we describe a class of…

Let $ \mathbb{R}^{n} $ denote Euclidean $ n $ space and given $k$ a positive integer let $ \Lambda_k \subset \mathbb{R}^{n} $, $ 1 \leq k < n - 1, n \geq 3, $ be a $k$-dimensional plane with $ 0 \in \Lambda_k.$ If $n-k < p <\infty$, we…

We study the holomorphic functions with Hadamard gaps in $\mathcal{N}_p$-spaces on the unit ball of $\mathbb{C}^n$ when $0<p \le n$ and $p>n$. A corollary on analytic functions with Hadamard gaps on $\mathcal{N}_p$-spaces on the unit disk…

We investigate the positive solutions of the semilinear elliptic equation \begin{align*} \sum^{N}_{i=1}\left(-\partial_{ii}\right)^{s}u=u^{p} \end{align*} with one-dimensional symmetric $2s$-stable operators. Firstly, in the whole space…

The existence or non-existence of positive orthogonal functions for subspaces of almost periodic functions has important applications in studying the oscillatory behavior of vibrations. Cazenave, Haraux and Komornik have obtained a number…

We extend to the case of a system involving p-Laplacians, the monotonicity and symmetry results of Damascelli and Pacella obtained in the case of a scalar p-Laplace equation with $1<p<2$. For this purpose, we use the moving hyperplanes…

There are ten chapters in this dissertation, which focuses on nine contents: growth estimates for a class of subharmonic functions in the half plane; growth estimates for a class of subharmonic functions in the half space; a generalization…

We prove partial and full boundary regularity for manifold constrained $p(x)$-harmonic maps.

We study the Dirichlet problem for p-harmonic functions on metric spaces with respect to arbitrary compactifications. A particular focus is on the Perron method, and as a new approach to the invariance problem we introduce Sobolev-Perron…

We obtain new uniqueness theorems for harmonic functions defined on the unit disc or in the half plane. These results are applied to obtain new resolvent descriptions of spectral subspaces of polynomially bounded groups of operators on…

We show that there are harmonic functions on a ball ${\mathbb{B}_n}$ of $\mathbb{R}^n$, $n\ge 2$, that are continuous up to the boundary (and even H\"older continuous) but not in the Sobolev space $H^s(\mathbb{B}_n)$ for any $s$…