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Backward stochastic partial differential equations of parabolic type with variable coefficients are considered in smooth domains. Existence and uniqueness results are given in weighted Sobolev spaces allowing the derivatives of the…
We prove the existence and a new representation formula for the gradient of the semigroup associated to an Ornstein-Uhlenbeck in a bounded convex domain in d dimensions.
This article is devoted the semiclassical spectral analysis of the Neumann magnetic Laplacian on a smooth bounded domain in three dimensions. Under a generic assumption on the variable magnetic field (involving a localization of the…
In this paper, we prove that the identity map for the smoothly bounded pseudoconvex domain of finite type in $\mathbb{C}^2$ extends to a bi-H\"{o}lder map between the Euclidean boundary and Gromov boundary. As an application, we show the…
We prove a sharp lower bound for the fundamental gap on convex domains in Gaussian spaces, the difference between the first two eigenvalues of the Ornstein-Uhlenbeck operator with Dirichlet boundary conditions. Our main result establishes…
We consider mixed Steklov-Dirichlet eigenvalue problem on smooth bounded domains in Riemannian manifolds. Under certain symmetry assumptions on multiconnected domains in $\mathbb{R}^{n}$ with a spherical hole, we obtain isoperimetric…
Witten has argued that in $2+1$ dimensions local supersymmetry can ensure the vanishing of the cosmological constant without requiring the equality of bose and fermi masses. We find that this mechanism is implemented in a novel fashion in…
Given the Laplacian on a planar, convex domain with piecewise linear boundary subject to mixed Dirichlet-Neumann boundary conditions, we provide a sufficient condition for its lowest eigenvalue to dominate the lowest eigenvalue of the…
We show some lower estimates for the Kobayashi-Royden metric on a class of smooth bounded pseudoconvex domains.
The classification of Fatou components for rational functions was concluded with Sullivan's proof of the No Wandering Domains Theorem in 1985. In 2016 it was shown, in joint work of the first and last author with Buff, Dujardin and Raissy,…
Comparing Neumann and Dirichlet eigenvalues of the Laplacian on a bounded domain $\Omega\subseteq\Rbb^n$ is a topic that goes back at least to the work of P\'olya \cite{polya}. We study the effect of the isoperimetric ratio of $\Omega$ on…
We show that if the closure of a smooth pseudoconvex domain admits a "sufficiently nice" Stein neighborhood basis, then the d-bar Neumann problem on the domain is globally regular.
Given a Schr\"odinger operator with a real-valued potential on a bounded, convex domain or a bounded interval we prove inequalities between the eigenvalues corresponding to Neumann and Dirichlet boundary conditions, respectively. The…
We study some special almost complex structures on strictly pseudoconvex domains. They appear naturally as limits under a nonisotroping scaling procedure and play a role of model objects in the geometry of almost complex manifolds with…
We introduce a new integral representation formula in the d-bar Neumann Theory on weakly pseudoconvex domains which satisfies certain estimates analogous to the basic L^2 estimate. It is expected that more complete estimates can be obtained…
We prove that all locally exposable points in a Stein compact in a complex space can be exposed along a given curve to a given real hypersurface. Moreover, the exposing map for a boundary point can be sufficiently close to the identity map…
We prove that in a strongly pseudoconvex domain with smooth boundary, then the length of a geodesic for the Kobayashi-Royden infinitesimal metric between two points is bounded by a constant multiple of the Euclidean distance between the…
We obtain $L^p$ regularity for the Bergman projection on some Reinhardt domains. We start with a bounded initial domain $\Omega$ with some symmetry properties and generate successor domains in higher {dimensions}. We prove: If the Bergman…
We treat the boundary problem for complex varieties (with isolated singularities) of dimension greater than one, which are contained in a suitable class of strictly pseudoconvex, unbounded domains of C^n.
We show that on bounded Lipschitz pseudoconvex domains that admit good weight functions the $\overline{\partial}$-Neumann operators $N_q, \overline{\partial}^* N_{q}$, and $\overline{\partial} N_{q}$ are bounded on $L^p$ spaces for some…