相关论文: Solving the Kerzman's problem on the sup-norm esti…

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…

Kohn introduced in 1979 the algorithm of multipliers to study the subelliptc estimate of the $\bar\partial$-Neumann problem for a smooth weakly pseudoconvex domain in a complex Euclidean space which satisfies D'Angelo's finite type…

In this paper we improve an earlier result by Bukhgeim and Uhlmann, by showing that in dimension larger than or equal to three, the knowledge of the Cauchy data for the Schr\"odinger equation measured on possibly very small subsets of the…

The problem of giving a (CR-)geometric description of the best possible order of a subelliptic estimate at a boundary point in the $\bar\partial$-Neumann problem is largely open. In this paper, we introduce a novel technique based on a…

First, we obtain a new formula for Bremermann type upper envelopes, that arise frequently in convex analysis and pluripotential theory, in terms of the Legendre transform of the convex- or plurisubharmonic-envelope of the boundary data.…

Solution operators for the equation $\bar \partial u=f$ are constructed on general product domains in $\mathbb{C}^n$. When the factors are one-dimensional, the operator is a simple integral operator: it involves specific derivatives of $f$…

In this article,we first give a modified Schwarz-Pompeiu formula in a general sector ring by proper conformal mappings, and obtain the solution of the Schwarz problem for the Cauchy-Riemann equation in explicit forms. Furthermore, a class…

We study the $\overline{\partial}$-Neumann problem using the Sobolev space inner product. We show that the problem can be solved on any smoothly bounded, pseudoconvex domain. We further formulate estimates and the basic results of a Sobolev…

We study the Cauchy problem in the space $H^1(\Sigma)$ for a nonlinear damped Schr\"odinger equation of the form \begin{equation}\tag{NLS-$\zeta$}\label{nls} i u_t + \Delta u + i \lambda u \, \zeta(|u|+1) = 0, \quad u(0,x) = u_0,…

The note concerns the $\bar\partial$ problem on product domains in $\mathbb C^2$. We show that there exists a bounded solution operator from $C^{k, \alpha}$ into itself, $k\in \mathbb Z^+\cup \{0\}, 0<\alpha< 1$. The regularity result is…

The Cauchy problem for a scalar conservation laws admits a unique entropy solution when the data $u_0$ is a bounded measurable function (Kruzhkov). The semi-group $(S_t)_{t\ge0}$ is contracting in the $L^1$-distance. For the…

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…

The first part of the course is devoted to the study of solutions to the Laplace equation in $\Omega\setminus K$, where $\Omega$ is a two-dimensional smooth domain and $K$ is a compact one-dimensional subset of $\Omega$. The solutions are…

The $k$-Cauchy-Fueter operators and complexes are quaternionic counterparts of the Cauchy-Riemann operator and the Dolbeault complex in the theory of several complex variables. To develop the function theory of several quaternionic…

In this paper we consider the overdetermined boundary problem for a general second order semilinear elliptic equation on bounded domains of $\mathbf{R}^n$, where one prescribes both the Dirichlet and Neumann data of the solution. We are…

We prove that the Cauchy data of Dirichlet or Neumann $\Delta$- eigenfunctions of Riemannian manifolds with concave (diffractive) boundary can only achieve maximal sup norm bounds if there exists a self-focal point on the boundary, i.e. a…

We investigate the overdetermined problem given by \begin{equation*} \Delta u=0 \text{ in } \Omega,\quad \frac{\partial u}{\partial\nu} =\sigma_1 u \text{ on } \partial \Omega, \quad |\nabla u|=\text{constant on } \partial \Omega,…

We make systematic developments on Lawson-Osserman constructions relating to the Dirichlet problem (over unit disks) for minimal surfaces of high codimension in their 1977 Acta paper. In particular, we show the existence of boundary…

It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex $n$-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem…

We give two results about Harnack type inequalities. First, on compact smooth Riemannian surface without boundary, we have an estimate of the type $\sup +\inf$. The second result concerns the solutions of prescribed scalar curvature…