A Foundational Criterion in Boundary Value Problems for Pseudo-Differential Operators

• Dr. Lenric A. Voswain

  Department of Mathematics, University of Vienna, Austria

  Author

• Dr. Tharuni M. Kavelin

  School of Mathematical Sciences, University of Hyderabad, India

  Author

• Dr. Jorvan E. Milbrant

  Department of Mathematical Sciences, Chalmers University of Technology, Sweden

  Author

This article explores the fascinating world of non-local operators and the challenges they pose when confined to domains with boundaries. We focus on a class of operators that, while powerful, lack the convenient symmetries of well-studied examples like the fractional Laplacian. These operators often fail to meet the stringent "transmission condition" required by classical theories. Instead, they satisfy a more forgiving criterion: the principal transmission condition. The central discovery we present is that even with this weaker condition, a robust and elegant analytical framework can be built. We show that these operators act naturally on a special family of function spaces—the µ-transmission spaces—which are tailor-made to handle the singular way solutions behave at a boundary. For the important case of strongly elliptic operators, we find that these spaces are not just convenient, but are the exact solution spaces for the Dirichlet problem, leading to precise predictions about solution regularity. A cornerstone of our work is a new, generalized integration by parts formula, a versatile tool that holds even for non-elliptic operators. This formula unlocks further insights, including a Green's formula for "large" solutions that blow up at the boundary. Our approach marks a departure from standard methods, relying on the classic Wiener-Hopf factorization technique to navigate the complexities introduced by the weaker symbolic properties. The result is a unified theory that extends our understanding of a broad and important class of non-local boundary value problems.

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