Related papers: Positive Harmonic Functions on Denjoy Domains in t…

If $\mathcal{B}\subset \mathbb{R}^d$ ($d\geqslant 2$) is a compact convex domain with a smooth boundary of finite type, we prove that for almost every rotation $\theta\in SO(d)$ the remainder of the lattice point problem, $P_{\theta…

We study properties of positive functions satisfying (E) --$\Delta$u + u p -- M |$\nabla$u| q = 0 is a domain $\Omega$ or in R N + when p > 1 and 1 < q < min{p, 2}. We concentrate our research on the solutions of (E) vanishing on the…

Let $S(C_0(X))^+$ and $S(C_0(Y))^+$ denote the positive parts of the unit spheres of $C_0(X)$ and $C_0(Y)$, where $X$ and $Y$ are locally compact Hausdorff spaces. We prove that every surjective isometry from $S(C_0(X))^+$ onto…

Conformal mapping may be the best-known topic in complex analysis. Any simply connected nonempty domain $\Omega$ in the complex plane ${{\mathbb{C}}}$ (assuming $\Omega\ne {{\mathbb{C}}}$) can be mapped bijectively to the unit disk by an…

This work establishes the well-posedness and a priori error analysis for the mixed FEEC-type finite element approximation of the three-dimensional vector Laplace boundary value problem subject to the Dirichlet boundary condition. The…

We study the double obstacle problem for p-harmonic functions on arbitrary bounded nonopen sets E in quite general metric spaces. The Dirichlet and single obstacle problems are included as special cases. We obtain Adams' criterion for the…

We consider the Dirichlet problem for a class of quasilinear elliptic systems in domain with irregular boundary. The principal part satisfies componentwise coercivity condition and the nonlinear terms are Carath\'eodory maps having Morrey…

It is established that if a harmonic function $u$ on the unit disk $\mathbb D$ in $\mathbb C$ has angular limits on a measurable set $E$ of the unit circle $\partial\mathbb D$, then its conjugate harmonic function $v$ in $\mathbb D$ also…

For odd dimensional Poincar\'e-Einstein manifolds $(X^{n+1},g)$, we study the set of harmonic $k$-forms (for $k<\ndemi$) which are $C^m$ (with $m\in\nn$) on the conformal compactification $\bar{X}$ of $X$. This is infinite dimensional for…

In this paper we study upper and lower bounds of the index and the nullity for sequences of harmonic maps with uniformly bounded Dirichlet energy from a two-dimensional Riemann surface into a compact target manifold. The main difficulty…

We take an open regular domain $\Omega$ in $\mathbb{R}^n$ with $n\ge 3$. We introduce a pair of positive parameters $\epsilon_1$ and $\epsilon_2$ and we set $\epsilon\equiv(\epsilon_1,\epsilon_2)$. Then we define the perforated domain…

We provide closed formulas for (unique) solutions of nonhomogeneous Dirichlet problems on balls involving any positive power $s>0$ of the Laplacian. We are able to prescribe values outside the domain and boundary data of different orders…

We study biharmonic maps between Riemannian manifolds with finite energy and finite bi-energy. We show that if the domain is complete and the target of non-positive curvature, then such a map is harmonic. We then give applications to…

The object of the paper is to find complete systems of inequalities relating the perimeter $P$, the area $|\cdot|$ and the Cheeger constant $h$ of planar sets. To do so, we study the so called Blaschke--Santal\'o diagram of the triplet…

Any planar shape $P\subset \mathbb{C}$ can be embedded isometrically as part of the boundary surface $S$ of a convex subset of $\mathbb{R}^3$ such that $\partial P$ supports the positive curvature of $S$. The complement $Q = S \setminus P$…

It is shown that any non-degenerate $\mathbb C$-convex domain in $\mathbb C^n$ is uniformly squeezing. It is also found the precise behavior of the squeezing function near a Dini-smooth boundary point of a plane domain.

Here is an example of a plane set of vanishing area and consisting of line-segments whose directions cover an angle : let E be a Cantor set of dissection ratio 1/4 (therefore dimension 1/2) carried by the horizontal axis and E' the image of…

Let $\Omega\subset \mathbb{C}^n$ for $n\geq 2$ be a bounded pseudoconvex domain with a $C^2$-smooth boundary. We study the compactness of composition operators on the Bergman spaces of smoothly bounded convex domains. We give a partial…

We consider the conformal mapping of the Bunimovich stadium, a region enclosed by a Jordan curve with four smooth corners, primarily in the context of a particle undergoing Brownian motion within its closed geometry with Dirichlet boundary…

We study harmonic maps from Riemannian manifolds into arbitrary non-positively curved and CAT(-1) metric spaces. First we discuss the domain variation formula with special emphasis on the error terms. Expanding higher order terms of this…