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.
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