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Unique Solvability of the Homogeneous Dirichlet Problem

In: Real Analysis Methods for Markov Processes

Author

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  • Kazuaki Taira

    (University of Tsukuba, The College of Mathematics)

Abstract

ThisDirichlet problem chapterHomogeneous Dirichlet problem isSecond order elliptic differential operator devotedElliptic differential operator to the study of the homogeneous Dirichlet problem for a second order, uniformly elliptic differential operator with vanishing mean oscillation (VMO) coefficients inExistence and uniqueness theorem theExistence theorem frameworkUniqueness theorem of Sobolev spaces of $$L^{p}$$ L p style. We proveVMO (vanishing mean oscillation) anVanishing mean oscillation (VMO) existence and uniqueness theorem for the Dirichlet problem (Theorem 15.1). Our proof is based on some interior and boundary a priori estimates for the solutions of problem (15.2) (Theorems 12.1 and 12.2). Both the interior and boundary a priori estimates are consequences of explicit representation formulas (13.1) and (14.1) for the solutions of problem (15.2) (Theorems 13.1 and 14.1) and also of the $$L^{p}$$ L p -boundedness of Calderón–Zygmund singular integral operators and boundary commutators appearing in those representation formulas (Theorems 14.2 and 14.5).

Suggested Citation

  • Kazuaki Taira, 2024. "Unique Solvability of the Homogeneous Dirichlet Problem," Springer Books, in: Real Analysis Methods for Markov Processes, chapter 0, pages 481-497, Springer.
    • Handle: RePEc:spr:sprchp:978-981-97-3659-1_15
    DOI: 10.1007/978-981-97-3659-1_15
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