Let $\Omega$ be a bounded simply connected domain in the complex plane, $ℂ$. Let $N$ be a neighborhood of $\partial \Omega$, let $p$ be fixed, $1\<p\<\infty ,$ and let $\widehat{u}$ be a positive weak solution to the $p$ Laplace equation in $\Omega \cap N.$ Assume that $\widehat{u}$ has zero boundary values on $\partial \Omega$ in the Sobolev sense and extend $\widehat{u}$ to $N\setminus \Omega$ by putting $\widehat{u}\equiv 0$ on $N\setminus \Omega .$ Then there exists a positive finite Borel measure $\widehat{\mu }$ on $ℂ$ with support contained in $\partial \Omega$ and such that$$\begin{array}{c}\hfill \int |\nabla \widehat{u}{|}^{p-2}\langle \nabla \widehat{u},\nabla \phi \rangle dA=-\int \phi d\widehat{\mu }\end{array}$$whenever $\phi \in C_0^{\infty }\left(N\right).$ If $p=2$ and if $\widehat{u}$ is the Green function for $\Omega$ with pole at $x\in \Omega \setminus \overline{N}$ then the measure $\widehat{\mu }$ coincides with harmonic measure...

A second order elliptic problem with axisymmetric data is solved in a finite element space, constructed on a triangulation with curved triangles, in such a way, that the (nonhomogeneous) boundary condition is fulfilled in the sense of a penalty. On the basis of two approximate solutions, extrapolates for both the solution and the boundary flux are defined. Some a priori error estimates are derived, provided the exact solution is regular enough. The paper extends some of the results of J.T. King...

We consider the problem$$\left\{\begin{array}{cc}-\Delta u=u^{\frac{N+2}{N-2}+\lambda }\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u\>0\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u=0\hfill & \text{on}\partial \left(\Omega \setminus \epsilon \omega \right),\hfill \end{array}\right.$$where $\Omega$ and $\omega$ are smooth bounded domains in ${ℝ}^N$, $N\ge 3$, $\epsilon \>0$ and $\lambda \in ℝ.$ We prove that if the size of the hole $\epsilon$ goes to zero and if, simultaneously, the parameter $\lambda$ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.

We study the existence of positive solutions for the $p$-Laplace Emden-Fowler equation. Let $H$ and $G$ be closed subgroups of the orthogonal group $O\left(N\right)$ such that $HG\subset O\left(N\right)$. We denote the orbit of $G$ through $x\in {ℝ}^N$ by $G\left(x\right)$, i.e., $G\left(x\right):=\{gx:g\in G\}$. We prove that if $H\left(x\right)G\left(x\right)$ for all $x\in \overline{\Omega }$ and the first eigenvalue of the $p$-Laplacian is large enough, then no $H$ invariant least energy solution is $G$ invariant. Here an $H$ invariant least energy solution means a solution which achieves the minimum of the Rayleigh quotient among all $H$ invariant functions. Therefore...

We consider linear elliptic equations $-\Delta u+q\left(x\right)u=\lambda u+f$ in bounded Lipschitz domains $D\subset {ℝ}^N$ with mixed boundary conditions $\partial u/\partial n=\sigma \left(x\right)\lambda u+g$ on $\partial D$. The main feature of this boundary value problem is the appearance of $\lambda$ both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient $\sigma \left(x\right)$. We study positivity principles and anti-maximum principles. One of our main results states that if $\sigma$ is somewhere negative, $q\ge 0$ and ${\int }_{D}q\left(x\right)dx>0$ then there exist two eigenvalues ${\lambda }_{-1}$, ${\lambda }_1$ such the positivity principle...