Related papers: Solution of the dbar-Neumann problem on a bi-disc
The spectrum of the d-bar-Neumann Laplacian on a polydisc in several complex variables is explicitly computed. The calculation exhibits that the spectrum consists of eigenvalues, some of which, in particular the smallest ones, are of…
We prove subelliptic estimates for the dbar-problem at the isolated singularity of the variety $z^2=xy$ in $\mathbb{C}^3$.
We study a semilinear elliptic problem with a singular nonlinear term of the type $g(u)=-u^{-1}$, using a variational approach. Note that the minus sign is important since the corresponding term in the Euler-Lagrange functional is concave.…
We study the $\bar{\partial}_b$-Neumann problem for domains $\Omega$ contained in a strictly pseudoconvex manifold M^{2n+1} whose boundaries are noncharacteristic and have defining functions depending solely on the real and imaginary parts…
he equation $-\Delta u = \lambda e^u$ posed in the unit ball $B \subseteq \R^N$, with homogeneous Dirichlet condition $u|_{\partial B} = 0$, has the singular solution $U=\log\frac1{|x|^2}$ when $\lambda = 2(N-2)$. If $N\ge 4$ we show that…
Related to a conjecture of Tom Wolff, we solve a singular Neumann problem for a linearized p-Laplace equation in the unit disk.
Let $\Sigma$ be a 2-dimensional subvariety in $C^3$ with an isolated simple (rational double point) singularity at the origin. The main objective of this paper is to solve the $\dbar$-equation on a neighbourhood of the origin in $\Sigma$,…
A mixed Dirichlet-Neumann problem is regularized with a family of singularly perturbed Neumann-Robin boundary problems, parametrized by $\varepsilon > 0$. Using an asymptotic development by Gamma-convergence, the asymptotic behavior of the…
We consider a variational problem with boundary singularity and Dirichlet condition. We give a blow-up analysis for sequences of solutions of an equation with exponential nonlinearity. Also, we derive a compactness criterion under some…
The aim of this paper is twofold. First, we establish the representation formula and the uniqueness of the solutions to a class of inhomogeneous biharmonic Dirichlet problems, and then prove the bi-Lipschitz continuity of the solutions.
We revisit the stability issue of determining the conductivity at the boundary from the corresponding Dirichlet-to-Neumann map. We discuss both the method based on singular solutions and the one built on the localized oscillating solutions.…
We consider the Neumann problem in $C^2$ bounded domains for fully nonlinear second order operators which are elliptic, homogenous with lower order terms. Inspired by \cite{bnv}, we define the concept of principal eigenvalue and we…
For a compact Riemannian surface $(M,g)$ with non-empty boundary $\Gamma$, the Dirichlet-to-Neumann operator (DtN-map) $\Lambda_g:C^\infty(\Gamma)\to C^\infty(\Gamma)$ is defined by $\Lambda_gf=\left.\frac{\partial…
The Dbar dressing method is extended to study the focusing/defocusing nonlinear Schr\"odinger (NLS) equation with nonzero boundary condition. A special type of complex function is considered. The function is meromorphic outside an annulus…
We give a potential theoretic characterization for compactness of the dbar-Neumann problem on smooth bounded pseudoconvex domains in C^n.
We study (0,2) two-dimensional theories in type IIB configurations with D5 branes wrapping blow-up ${\bf{P}}^1$ cycles of deformed resolutions for $A_n$ singularities or in T-dual IIA configurations with suspended D4 branes. We consider…
In this paper we consider some nodal solutions of the H\'enon problem in the unit disc with Dirichlet boundary conditions and we show that they are quasiradial, that is to say they are nonradial, they have two nodal regions and their nodal…
We investigate the existence and uniqueness of solutions for second-order semi-linear partial differential equations defined on a Riemannian manifold $M$. By combining differential geometry and analysis techniques, we establish the…
We first study the linear eigenvalue problem for a planar Dirac system in the open half-line and describe the nodal properties of its solution by means of the rotation number. We then give a global bifurcation result for a planar nonlinear…
We construct Delaunay-type solutions for the fractional Yamabe problem with an isolated singularity $(-\Delta)^\gamma w = c_{n, \gamma} w^{\frac{n+2\gamma}{n-2\gamma}}, w>0 \ \mbox{in} \ \mathbb{R}^n \backslash \{0\}$ We follow a…