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We develop a Poisson geometric framework for studying the representation theory of all contragredient quantum super groups at roots of unity. This is done in a uniform fashion by treating the larger class of quantum doubles of bozonizations…
We settle the issue of well-posedness for the Dirichlet problem for a higher order elliptic system ${\mathcal L}(x,D_x)$ with complex-valued, bounded, measurable coefficients in a Lipschitz domain $\Omega$, with boundary data in Besov…
The structure of Poisson polynomial algebras of the type obtained as semiclassical limits of quantized coordinate rings is investigated. Sufficient conditions for a rational Poisson action of a torus on such an algebra to leave only…
We establish well posedness of the Poisson problem in weak local John domains, for linear second order elliptic equations with real coefficients, and with data in weighted Lebesgue spaces with a very broad range of acceptable parameters.
We consider the Dirichlet and Neumann problems for second-order linear elliptic equations: \[ -\triangle u +\mathrm{div}(u\mathbf{b}) =f \quad\text{ and }\quad -\triangle v -\mathbf{b} \cdot \nabla v =g \] in a bounded Lipschitz domain…
The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into…
We use the method of layer potentials to study the $R_2$ Regularity problem and the $D_2$ Dirichlet problem for second order elliptic equations of the form $\mathcal{L}u=0$, with lower order coefficients, in bounded Lipschitz domains. For…
This paper is concerned with boundary regularity estimates in the homogenization of elliptic equations with rapidly oscillating and high-contrast coefficients. We establish uniform nontangential-maximal-function estimates for the Dirichlet,…
Let $M$ be an $n$-dimensional Hadamard manifold of pinched negative curvature $-b^2 \leq K_M \leq -a^2$. The solution of the Dirichlet problem at infinity for $M$ leads to the construction of a family of mutually absolutely continuous…
We present a simple discretization by radial basis functions for the Poisson equation with Dirichlet boundary condition. A Lagrangian multiplier using piecewise polynomials is used to accommodate the boundary condition. This simplifies…
We show that a bilinear estimate for biharmonic functions in a Lipschitz domain $\Omega$is equivalent to the solvability of the Dirichlet problem for the biharmonic equationin $\Omega$. As a result, we prove that for any given bounded…
Given a domain above a Lipschitz graph, we establish solvability results for strongly elliptic second-order systems in divergence-form, allowed to have lower-order (drift) terms, with $L^p$-boundary data for $p$ near $2$ (more precisely, in…
We present the converse to a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz. More precisely, for $n\geq 2$, for an ADR domain $\Omega\subset \re^{n+1}$ which satisfies the Harnack Chain condition plus…
We study the polyharmonic Neumann and mixed boundary value problems on the Kor\'{a}nyi ball in the Heisenberg group $\H_n$. Necessary and sufficient solvability conditions are obtained for the nonhomogeneous polyharmonic Neumann problem and…
The solutions of boundary value problems for the Laplacian and the bilaplacian exhibit very different qualitative behaviors. Particularly, the failure of general maximum principles for the bilaplacian implies that solutions of higher-order…
This paper consists of two parts. In the first part, which is of more abstract nature, the notion of quasi boundary triples and associated Weyl functions is developed further in such a way that it can be applied to elliptic boundary value…
We present a new complexification scheme based on the classical double layer potential for the solution of the Helmholtz equation with Dirichlet boundary conditions in compactly perturbed half-spaces in two and three dimensions. The kernel…
We derive two-sided bounds for the Newton and Poisson kernels of the $W$-invariant Dunkl Laplacian in geometric complex case when the multiplicity $k(\alpha)=1$, i.e. for flat complex symmetric spaces. For the invariant Dunkl-Poisson kernel…
The problem of finding superintegrable Hamiltonians and their integrals of motion can be reduced to solving a series of compatibility equations that result from the overdetermination of the commutator or Poisson bracket relations. The…
The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with $t$-independent complex bounded measurable coefficients ($t$ being the transversal direction to the boundary). To be…