Integral-equation method in the mixed oblique derivative problem for harmonic functions outside cuts on the plane

The mixed problem for the Laplace equation outside cuts on the plane is considered. As boundary conditions, the value of the desired function on one side of each of the cuts and the value of its oblique derivative on the other side are prescribed. This problem generalizes the mixed Dirichlet-Neumann problem. By using the potential method, the problem reduces to a uniquely solvable Fredholm integral equation of the second kind. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 115–135, 2006.     

Elementary methods for solving the plane harmonic dirichlet problem for some canonical domains

This paper is concerned with an explicit method for solving the Dirichlet problem in the unit circle for piecewise rational boundary values as well as in the half-plane. In the latter case also the solutions to mixed boundary conditions are given explicitly. The method is based on the Milne-Thomson theorem     

Quadrature domains for harmonic functions

This paper establishes a conjecture of Gustafsson, Sakai andShapiro by showing that any quadrature domain (for harmonicfunctions) with respect to a signed measure is also a quadraturedomain with respect to a positive measure.     

The Green function for the mixed problem for the linear Stokes system in domains in the plane

We construct the Green function for the mixed boundary value problem for the linear Stokes system in a two‐dimensional Lipschitz domain.     

Inverse boundary-value problems of cauchy type for harmonic functions

We apply two methods for solving the inverse boundary-value problem (the so-called problem (A)) in the Cauchy statement for an analytic function and an unknown curve ??. We obtain criteria for ?? to be the unit circle. We apply the proposed methods for solving a modified Hadamard example and generalize the obtained results for the case of doubly connected domains.     

Generalization of the Neumann problem to harmonic functions outside cuts on the plane

We consider a boundary value problem for harmonic functions outside cuts on the plane. The jump of the normal derivative and a linear combination of the normal derivative on one side with the jump of the unknown function are given on each cut. The problem is considered with three conditions at infinity, which lead to distinct results on the existence and number of solutions. We obtain an integral representation of the solution in the form of potentials whose density satisfies a uniquely solvable Fredholm integral equation of the second kind.     

A mixed boundary-value problem for a hyperbolic-parabolic equation

Let Ω be a bounded domain in the n-dimensional Euclidean space. In the cylindrical domain QT=Ω x [0, T] we consider a hyperbolic-parabolic equation of the form (1) $$Lu = k(x,t)u_{tt} + \sum\nolimits_{i = 1}^n {a_i u_{tx_i } - } \sum\nolimits_{i,j = 1}^n {\tfrac{\partial }{{\partial x_i }}} (a_{ij} (x,t)u_{x_j } ) + \sum\nolimits_{i = 1}^n {t_i u_{x_i } + au_t + cu = f(x,t),} $$ where \(k(x,t) \geqslant 0,a_{ij} = a_{ji} ,\nu |\xi |^2 \leqslant a_{ij} \xi _i \xi _j \leqslant u|\xi |^2 ,\forall \xi \in R^n ,\nu > 0\) . The classical and the “modified” mixed boundary-value problems for Eq. (1) are studied. Under certain conditions on the coefficients of the equation it is proved that these problems have unique solution in the Sobolev spaces W ₂ ¹ (QT) and W ₂ ² (QT).     

Conform and mixed finite element schemes for the Dirichlet problem for the Bilaplacian in plane domains with corners

In plane domains with corners for the Bilaplacian a uniquely solvable conform variational principle is studied on weighted Sobolev spaces which is equivalent to the standard Dirichlet problem in the weak form. Clamped plates under point forces near corners are handled by this approach. With weighted Hsieh-Clough-Tocher elements on regular triangulations as conform C¹-finite elements a new error analysis is performed without higher regularity assumptions on the exact solution than given by the data and the boundary. The rate of convergence of the error depends on the eigenvalue with smallest imaginary part of a clamped infinite wedge since this eigenvalue describes the singularity of the exact solution in a sector with same angle. Using different spaces of trial and test functions in the standard Galerkin procedure it is shown that the error in the weighted energy norm does not pollute. For convex corners asymptotic error estimates, are proved yielding convergence for a mixed method in hydrodynamics where the solution of a system of 2^nd order and its Laplacian are approximated simultaneously by C⁰-finite elements being piecewise polynomials.     

On boundary behavior of harmonic functions in Hölder domains

We analyze the boundary behavior of harmonic functions in a domain whose boundary is locally given by a graph of a Hölder continuous function. In particular we give a non-probabilistic proof of a Harnack-type principle, due to Bañuelos et al. and study some properties of the harmonic measure.