We study linear and quasilinear Venttsel boundary value problems involving elliptic operators with discontinuous coefficients. On the base of the a priori estimates obtained, maximal regularity and strong solvability in Sobolev spaces are proved.
We use novel integral representations developed by the second author to prove certain rigorous results concerning elliptic boundary value problems in convex polygons. Central to this approach is the so-called global relation, which is a non-local equation in the Fourier space that relates the known boundary data to the unknown boundary values. Assuming that the global relation is satisfied in the weakest possible sense, i.e. in a distributional sense, we prove there exist solutions to Dirichlet, Neumann and Robin boundary value problems with distributional boundary data. We also show that the analysis of the global relation characterises in a straightforward manner the possible existence of both integrable and non-integrable corner-singularities.
In this paper we develop the global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator. For this, we also establish elements of a non-self-adjoint distribution theory and the corresponding biorthogonal Fourier analysis. We give applications of the developed analysis to obtain a-priori estimates for solutions of operators that are elliptic within the constructed calculus.
In many cases, groundwater flow in an unconfined aquifer can be simplified to a one-dimensional Sturm-Liouville model of the form: begin{equation*} x(t)+lambda x(t)=h(t)+varepsilon f(x(t)),hspace{.1in}tin(0,pi) end{equation*} subject to non-local boundary conditions begin{equation*} x(0)=h_1+varepsiloneta_1(x)text{ and } x(pi)=h_2+varepsiloneta_2(x). end{equation*} In this paper, we study the existence of solutions to the above Sturm-Liouville problem under the assumption that $varepsilon$ is a small parameter. Our method will be analytical, utilizing the implicit function theorem and its generalizations.
In this paper, we present a unified analysis of the superconvergence property for a large class of mixed discontinuous Galerkin methods. This analysis applies to both the Poisson equation and linear elasticity problems with symmetric stress formulations. Based on this result, some locally postprocess schemes are employed to improve the accuracy of displacement by order min(k+1, 2) if polynomials of degree k are employed for displacement. Some numerical experiments are carried out to validate the theoretical results.
We construct an expression for the Green function of a differential operator satisfying nonlocal, homogeneous boundary conditions starting from the fundamental solution of the differential operator. This also provides the solution to the boundary value problem of an inhomogeneous partial differential equation with inhomogeneous, nonlocal, and linear boundary conditions. The construction generally applies for all types of linear partial differential equations and linear boundary conditions.
Darya E. Apushkinskaya
,Alexander I. Nazarov
,Dian K. Palagachev
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(2019)
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"Venttsel boundary value problems with discontinuous data"
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Dian Palagachev K
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