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On the Existence of L2 Boundary Values of Solutions to an Elliptic Equation

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Abstract

The behavior of solutions of a second-order elliptic equation near a distinguished piece of the boundary is studied. On the remaining part of the boundary, the solutions are assumed to satisfy the homogeneous Dirichlet conditions. A necessary and sufficient condition is established for the existence of an L2 boundary value on the distinguished part of the boundary. Under the conditions of this criterion, estimates for the nontangential maximal function of the solution hold, the solution belongs to the space of (n − 1)-dimensionally continuous functions, and the boundary value is taken in a much stronger sense.

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  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    A. K. Gushchin

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Russian Text © The Author(s), 2019, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2019, Vol. 306, pp. 56–74.

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Gushchin, A.K. On the Existence of L2 Boundary Values of Solutions to an Elliptic Equation. Proc. Steklov Inst. Math. 306, 47–65 (2019). https://doi.org/10.1134/S0081543819050067

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