A universal boundary value problem for partial differential equations

TL;DR

This paper introduces a new universal boundary value problem for PDEs, deriving integral equations that boundary data must satisfy without predefined boundary conditions, with applications to physics and other fields.

Contribution

It formulates a universal boundary value problem for PDEs, providing integral equations for boundary data, a novel approach that broadens the understanding of boundary conditions in physics.

Findings

Derived universal boundary value equations for PDE boundary data

Presented solutions for Laplace and heat equations within this framework

Discussed applications to cosmology and quantum mechanics

Abstract

A new boundary value problem for partial differential equations is discussed. We consider an arbitrary solution of an elliptic or parabolic equation in a given domain and no boundary conditions are assumed. We study which restrictions the boundary values of the solution and its normal derivatives must satisfy. Linear integral equations for the boundary values of the solution and its normal derivatives are obtained, which we call the universal boundary value equations. A universal boundary value problem is defined as a partial differential equation together with the boundary data which specify the values of the solution on the boundary and its normal derivatives and satisfy to the universal boundary value equations. For the equations of mathematical physics such as Laplace's and the heat equation the solution of the universal boundary value problem is presented. Applications to cosmology and quantum mechanics are mentioned.