Related papers: A note on elliptic coordinates on the Lie algebra …
This article focuses on Lp-estimates for the square root of elliptic systems of second order in divergence form on a bounded domain. We treat complex bounded measurable coefficients and allow for mixed Dirichlet/Neumann boundary conditions…
The notion of $p$-ellipticity has recently played a significant role in improving our understanding of issues of solvability of boundary value problems for scalar complex valued elliptic PDEs. In particular, the presence of $p$-ellipticity…
One of the most fascinating and technically demanding parts of the theory of two-dimensional integrable systems constitute the models with the spectral parameter on an elliptic curve, including Landau-Lifshitz and Krichever-Novikov…
We consider various generalizations of the Kepler problem to three-dimensional sphere $S^3$, a compact space of constant curvature. These generalizations include, among other things, addition of a spherical analog of the magnetic monopole…
A representation of the quadratic Dirac equation and the Maxwell equations in terms of the three-dimensional universal complex Clifford algebra is given. The investigation considers a subset of the full algebra, which is isomorphic to the…
Recent advances in quantitative unique continuation properties for solutions to uniformly elliptic, divergence form equations (with Lipschitz coefficients) has led to a good understanding of the vanishing order and size of singular and zero…
We prove that the Schweitzer complex is elliptic and its cohomologies define cohomological functors. As applications, we obtain finite dimensionality, a version of Serre duality, restrictions of the behaviour of cohomology in small…
Elliptic problems with additional unknown distributions in boundary conditions are investigated in Besov and Sobolev-Triebel-Lizorkin spaces of low regularity, specifically of an arbitrary negative order. We find that the problems induce…
Take an open domain $\Omega \subset \mathbb R^n$ whose boundary may be composed of pieces of different dimensions. For instance, $\Omega$ can be a ball on $\mathbb R^3$, minus one of its diameters $D$, or $\Omega \subset \mathbb R^3$ could…
We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three…
We introduce a class of multidimensional Schr\"odinger operators with elliptic potential which generalize the classical Lam\'e operator to higher dimensions. One natural example is the Calogero--Moser operator, others are related to the…
In this paper, we deal with an elliptic problem with the Dirichlet boundary condition. We operate in Sobolev spaces and the main analytic tool we use is the Lax-Milgram lemma. First, we present the variational approach of the problem which…
Petrovskii elliptic systems of linear differential equations given on a closed smooth manifold are investigated on the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to the…
Many PDEs (Burgers' equation, KdV, Camassa-Holm, Euler's fluid equations,...) can be formulated as infinite-dimensional Lie-Poisson systems. These are Hamiltonian systems on manifolds equipped with Poisson brackets. The Poisson structure is…
We consider Nambu-Poisson 3-algebras on three dimensional manifolds $ {\cal M}_{3} $, such as the Euclidean 3-space $R^{3}$, the 3-sphere $S^{3}$ as well as the 3-torus $T^{3}$. We demonstrate that in the Clebsch-Monge gauge, the Lie…
We consider the inverse problem of determining the coefficients of a general second-order elliptic operator in two dimensions by measuring the corresponding Cauchy data on an arbitrary open subset of the boundary. We show that one can…
First, we review the Dirac operator folklore about basic analytic and geometrical properties of operators of Dirac type on compact manifolds with smooth boundary and on closed partitioned manifolds and show how these properties depend on…
A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic…
We consider a slightly subcritical elliptic system with Dirichlet boundary conditions and a non-power nonlinearity in a bounded smooth domain. For this problem, standard compact embeddings cannot be used to guarantee the existence of…
We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space $\mathbb{R}^d_+$, where the coefficients are the product of $x_d^\alpha, \alpha \in (-\infty, 1),$ and a bounded uniformly elliptic…