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Abstract <p> This paper first demonstrates the existence and uniqueness of solutions to homogeneous Dirichlet boundary value problems for second-order linear elliptic equations with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^2\)</EquationSource> </InlineEquation>-drifts of negative divergence and positive <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^1\)</EquationSource> </InlineEquation>-zero-order terms, based on a functional analytic approach, including weak convergence methods and duality arguments. By improving the previous contraction properties, which may not be effective when the zero-order term is very small, this paper introduces a general <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L^2\)</EquationSource> </InlineEquation>-“contraction” property for any positive zero-order term, leading to remarkable results regarding <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(L^2\)</EquationSource> </InlineEquation>-stability. These stability results are applicable to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(L^2\)</EquationSource> </InlineEquation>-error analysis for physics-informed neural networks, and can also be applied to stationary Schrödinger operators with <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(L^2\)</EquationSource> </InlineEquation>-zero-order terms. We emphasize that all the constants arising in the estimates of this paper can be explicitly computed. </p>

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On the Stability of Linear Elliptic Equations with \(L^2\)-Drifts of Negative Divergence and Singular Zero-Order Terms

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • Haesung Lee

摘要

Abstract

This paper first demonstrates the existence and uniqueness of solutions to homogeneous Dirichlet boundary value problems for second-order linear elliptic equations with \(L^2\) -drifts of negative divergence and positive \(L^1\) -zero-order terms, based on a functional analytic approach, including weak convergence methods and duality arguments. By improving the previous contraction properties, which may not be effective when the zero-order term is very small, this paper introduces a general \(L^2\) -“contraction” property for any positive zero-order term, leading to remarkable results regarding \(L^2\) -stability. These stability results are applicable to \(L^2\) -error analysis for physics-informed neural networks, and can also be applied to stationary Schrödinger operators with \(L^2\) -zero-order terms. We emphasize that all the constants arising in the estimates of this paper can be explicitly computed.