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The Dirichlet problem in arbitrary domains for a wide class of anisotropic elliptic equations of the second order with variable exponent nonlinearities and the right-hand side as a measure is considered. The existence of an entropy solution…
A unifying scheme of classical special functions of hypergeometric type obeying orthogonality or biorthogonality relations is described. It expands the Askey scheme of classical orthogonal polynomials and its $q$-analogue based on the…
We investigate Lawruk elliptic boundary-value problems for homogeneous differential equations in a two-sided refined Sobolev scale. These problems contain additional unknown functions in the boundary conditions of arbitrary orders. The…
We want to prove a Harnack type inequality for solutions of strongly degenerate parabolic, or elliptic-parabolic, equations. To do that, we first define a De Giorgi class of order $p = 2$ that contains the solutions of evolution equations…
We extend the work of Fuchs, Painlev\'e and Manin on a Calogero-like expression of the sixth Painlev\'e equation (the ``Painlev\'e-Calogero correspondence'') to the other five Painlev\'e equations. The Calogero side of the sixth Painlev\'e…
The state of art of electromagnetic integral equations has seen significant growth over the past few decades, overcoming some of the fundamental bottlenecks: computational complexity, low frequency and dense discretization breakdown,…
The Painleve-IV equation has two families of rational solutions generated respectively by the generalized Hermite polynomials and the generalized Okamoto polynomials. We apply the isomonodromy method to represent all of these rational…
A new method to construct algebro-geometric solutions of rank two Schlesinger systems is presented. For an elliptic curve represented as a ramified double covering of CP^1, a meromorphic differential is constructed with the following…
In this short note we give two examples of using the algebro-geometric theory of Painlev\'e equations to solve the Painlev\'e identification problem. The equations that we consider were recently obtained by M. van der Put and J. Top in…
We investigate and derive second solutions to linear homogeneous second-order difference equations using a variety of methods, in each case going beyond the purely formal solution and giving explicit expressions for the second solution. We…
Optimal higher-order Sobolev type embeddings are shown to follow via isoperimetric inequalities. This establishes a higher-order analogue of a well-known link between first-order Sobolev embeddings and isoperimetric inequalities. Sobolev…
It is well-known that differential Painlev\'e equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique -- there are many very different Hamiltonians that result in the same…
We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining…
The inverse problem of the calculus of variations consists in determining if the solutions of a given system of second order differential equations correspond with the solutions of the Euler-Lagrange equations for some regular Lagrangian.…
A starting point of this paper is a classification of quadratic polynomial transformations of the monodromy manifold for the 2x2 isomonodromic Fuchsian systems associated to the Painleve VI equation. Up to birational automorphisms of the…
Equivalence is established between a special class of Painleve VI equations parametrized by a conformal dimension $\mu$, time dependent Euler top equations, isomonodromic deformations and three-dimensional Frobenius manifolds. The…
We assign some kind of invariant manifolds to a given integrable PDE (its discrete or semi-discrete variant). First, we linearize the equation around its arbitrary solution $u$. Then we construct a differential (respectively, difference)…
We consider the Gauss-Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallelly to themselves. We reduce these equations modulo a prime integer $p$ and…
We consider matrix orthogonal polynomials related to Bessel type matrices of weights that can be defined in terms of a given matrix Pearson equation. From a Riemann-Hilbert problem we derive first and second order differential relations for…
For space-times with two spacelike isometries, we present infinite hierarchies of exact solutions of the Einstein and Einstein--Maxwell equations as represented by their Ernst potentials. This hierarchy contains three arbitrary rational…