Related papers: Partial data inverse problems for semilinear ellip…

In this article, we investigate a density problem coming from the linearization of Calder\'on's problem with partial data. More precisely, we prove that the set of products of harmonic functions on a bounded smooth domain $\Omega$ vanishing…

We show that the knowledge of the Dirichlet-to-Neumann maps given on an arbitrary open non-empty portion of the boundary of a smooth domain in $\mathbb{R}^n$, $n\ge 2$, for classes of semilinear and quasilinear conductivity equations,…

Weakly harmonic maps from a domain $\Omega$ (the upper half-space $\Rd$ or a bounded $C^{1,\alpha}$ domain, $\alpha\in (0,1]$) into a smooth closed manifold are studied. Prescribing small Dirichlet data in either of the classes…

We establish the existence of positive solutions to a general class of overdetermined semilinear elliptic boundary problems on suitable bounded open sets $\Omega\subset\mathbb{R}^n$. Specifically, for $n\leq 4$ and under mild technical…

We show that tensor products of $k$ gradients of harmonic functions, with $k$ at least three, are dense in $C(\overline{\Omega})$, for any bounded domain $\Omega$ in dimension 3 or higher. The bulk of the argument consists in showing that…

In this article we discuss density of products of biharmonic functions vanishing on an arbitrarily small part of the boundary. We prove that one can use three or more such biharmonic functions to construct a dense subset of smooth symmetric…

We study existence of nontrivial solutions to problem \begin{equation*} \left\lbrace \begin{array}{rcll} -\Delta u &=& \lambda u+f(u)&\text{ in }\Omega,\\ u&=&0&\text{ on }\partial \Omega, \end{array}\right. \end{equation*} where $\Omega…

A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in $\mathbb{R}^n$ is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for…

Let $n\geq2$ and $ \Omega\subset \mathbb{R}^{n+1}$ be a Lipschitz wedge- like domain . We construct positive weak solutions of the problem $$\Delta u + u^p = 0 \quad\hbox{in}\, \Omega,$$ which vanish in a suitable trace sense on…

In this paper we develop a reconstruction algorithm for the solution of an inverse boundary value problem dealing with a semilinear elliptic partial differential equation of interest in cardiac electrophysiology. The goal is the detection…

We study semilinear elliptic equations with Hardy potential $\mathrm{(E)} \; -L_\mu u+u^q=0$ in a bounded smooth domain $\Omega\subset \mathbb R^N$. Here $q>1$, $L_\mu=\Delta+\frac{\mu}{\delta_\Omega^2}$ and…

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$ and let $m$ be a possibly discontinuous and unbounded function that changes sign in $\Omega$. Let $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ be a continuous…

In this paper, we consider semilinear elliptic problems in a bounded domain $\Omega$ contained in a given unbounded Lipschitz domain $\mathcal C \subset \mathbb R^N$. Our aim is to study how the energy of a solution behaves with respect to…

We consider a semilinear elliptic equation on a smooth bounded domain $\Om$ in $\R^2$, assuming that both the domain and the equation are invariant under reflections about one of the coordinate axes, say the y-axis. It is known that…

This paper deals with the lack of compactness in nonlinear elliptic problems $(P)$. In particular, a domain $\Omega$ is provided where not converging Palais-Smale sequences exist at every energy level. Nevertheless, it is proved that…

This paper deals with a class of singularly perturbed nonlinear elliptic problems $(P_\e)$ with subcritical nonlinearity. The coefficient of the linear part is assumed to concentrate in a point of the domain, as $\e\to 0$, and the domain is…

We show that the knowledge of the Dirichlet-to-Neumann map on an arbitrary open portion of the boundary of a domain in $\mathbb{R}^n$, $n\ge 2$, for a class of semilinear elliptic equations, determines the nonlinearity uniquely.

Let $\textbf{A}$ be a symmetric convex quadratic form on $\mathbb{R}^{Nn}$ and $\Omega\Subset \mathbb{R}^n$ a bounded convex domain. We consider the problem of existence of solutions $u: \Omega \subset \mathbb{R}^n \longrightarrow…

We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $\Omega$ in…

It was recently shown by the authors that a semilinear elliptic equation can be represented as an infinite-dimensional dynamical system in terms of boundary data on a shrinking one-parameter family of domains. The resulting system is…