相关论文: Subellipticity of the $\bar\partial$-Neumann probl…

We study the behavior of weak solutions to the singular quasilinear elliptic problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$, in a bounded domain with the Dirichlet boundary condition, where $p>1$, $\gamma>0$,…

Let $X$ be a, possibly non-reduced, analytic space of pure dimension. We introduce a notion of $\overline{\partial}$-equation on $X$ and prove a Dolbeault-Grothendieck lemma. We obtain fine sheaves $\mathcal{A}_X^q$ of $(0,q)$-currents, so…

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.

This paper is a sequel to \cite{Choi} in Math. Ann. In that paper we studied the subharmonicity of K\"ahler-Einstein metrics on strongly pseudoconvex domains of dimension greater than or equal to $3$. In this paper, we study the variations…

We show that Property $(P)$ of $\partial\Omega$, compactness of the $\bar{\partial}$-Neumann operators $N_1$, and compactness of Hankel operator on a smooth bounded pseudoconvex Hartogs domain $\Omega={\{(z, w_1, w_2,\dots, w_n) \in…

We give a proof of the Greene-Krantz conjecture on convex domains in $\CC^2$. Curiously, the proof technique depends on subelliptic estimates for the $\bar{\partial}$ problem.

We prove several Sobolev-type inequalities related to the $\bar\partial$-operator on bounded domains in $\mathbb{C}^n$, which can be viewed as a $\bar\partial$-version of the classical Sobolev inequality and its various generalizations, and…

On a bounded smooth pseudoconvex domain in $\mathbb{C}^n$ with $n >2$, inspired by the compactness condition introduced by Yue Zhang, we present the new sufficient condition for the exact regularity of the $\overline\partial$-Neumann…

In this paper we study optimization problems for Neumann eigenvalues $\mu_k$ among convex domains with a constraint on the diameter or the perimeter. We work mainly in the plane, though some results are stated in higher dimension. We study…

Our main goal is to describe the integral kernel of the dbar-Neumann operator on certain non-smooth domains, the so-called Henkin-Leiterer domains. We do so by explicitly constructing an integral kernel which accounts for the main mapping…

We investigate partial symmetry of solutions to semi-linear and quasi-linear elliptic problems with convex nonlinearities, in domains that are either axially symmetric or radially symmetric.

In this preprint we consider fully nonlinear equations in thin domains with oblique boundary condition, finding some new phenomena, in particular the limit equation contains "new terms" of the second, first and zeroth order which don't have…

We establish the existence of solutions to the following semilinear Neumann problem for fractional Laplacian and critical exponent: \begin{align*}\left\{\begin{array}{l l} { (-\Delta)^{s}u+ \lambda u= \abs{u}^{p-1}u } & \text{in $ \Omega,$…

In this paper we are concerned with the initial boundary value problems of linear and semi-linear parabolic equations with mixed boundary conditions on non-cylindrical domains in spatial-temporal space. We obtain the existence of a weak…

We proceed with the investigation of the problem $(P_\lambda): $ $-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } \Omega, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial \Omega$, where $\Omega$ is a…

We derive estimates in a weighted Sobolev space $W^{k,p}_{\mu}(D)$ for a homotopy operator on a bounded strictly pseudoconvex domain $D$ of $C^2$ boundary in ${\C}^n$. As a result, we show that given any $2n < p < \infty$, $k > 1$, $q \geq…

In this article we focus on inverse problems for a semilinear elliptic equation. We show that a potential $q$ in $L^{n/2+\varepsilon}$, $\varepsilon>0$, can be determined from the full and partial Dirichlet-to-Neumann map. This extends the…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a bounded smooth domain and $\delta(x)=\text{dist}(x,\partial \Omega)$. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to $$…

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,…

In this paper, we mainly establish the existence of at least three non-trivial solutions for a class of nonhomogeneous quasilinear elliptic systems with Dirichlet boundary value or Neumann boundary value in a bounded domain…