TY - CHAP
T1 - Regularity for the dirichlet problem
AU - Shen, Zhongwei
PY - 2018
Y1 - 2018
N2 - This chapter is devoted to the study of uniform boundary regularity estimates for the Dirichlet problem [Formula presented.] where Lε =div(A(x/ε)∇). Assuming that the coefficient matrix A = A(y) is elliptic, periodic, and belongs to VMO(ℝd), we establish uniform boundary Hölder and W1, p estimates in C1 domains Ω. We also prove uniform boundary Lipschitz estimates in C1, α domains under the assumption that A is elliptic, periodic, and Hölder continuous. As in the previous chapter for interior estimates, boundary Hölder and Lipschitz estimates are proved by a compactness method. The boundaryW1, p estimates are obtained by combining the boundary Hölder estimates with the interior W1, p estimates, via the real-variable method introduced in Section 4.2.
AB - This chapter is devoted to the study of uniform boundary regularity estimates for the Dirichlet problem [Formula presented.] where Lε =div(A(x/ε)∇). Assuming that the coefficient matrix A = A(y) is elliptic, periodic, and belongs to VMO(ℝd), we establish uniform boundary Hölder and W1, p estimates in C1 domains Ω. We also prove uniform boundary Lipschitz estimates in C1, α domains under the assumption that A is elliptic, periodic, and Hölder continuous. As in the previous chapter for interior estimates, boundary Hölder and Lipschitz estimates are proved by a compactness method. The boundaryW1, p estimates are obtained by combining the boundary Hölder estimates with the interior W1, p estimates, via the real-variable method introduced in Section 4.2.
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U2 - 10.1007/978-3-319-91214-1_5
DO - 10.1007/978-3-319-91214-1_5
M3 - Chapter
AN - SCOPUS:85053039208
T3 - Operator Theory: Advances and Applications
SP - 99
EP - 134
BT - Operator Theory
ER -