The study demonstrates L^{2} solvability in boundary value problems with matrix coefficients, implying new methods in harmonic analysis.
We prove a Kato square root estimate with anisotropically degenerate matrix coefficients. We do so by doing the harmonic analysis, using an auxiliary Riemannian metric adapted to the operator. We also derive L² -solvability estimates for boundary value problems, for divergence form elliptic equations with matrix degenerate coefficients. Main tools are chain rules and Piola transformations, for fields in matrix weighted L² spaces, under W1,1 homeomorphism.
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Brocchi et al. (2025) studied this question.
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