Best-Possible Estimates of Solutions of the Dirichlet Problem for Linear Elliptic Nondivergence Equations of Second Order in a Neighborhood of a Conical Point of the Boundary

Best possible estimates of solutions to the Robin boundary value problem for linear elliptic nondivergence second-order equations in a neighborhood of the conical point

“…We obtain best possible estimates of the strong solutions of the problem (L) near a conical boundary point. Analogous results were established in [2] for the Dirichlet problem. Many mathematicians have considered the third boundary value problem [6].…”

Solvability of Elliptic Problems in Domains with Conic Point

“…Taking into account Remark 2, as a corollary of Theorem 3 in [4] one can obtain the following statement: …”

“…To prove the lemma, it is necessary to verify the conditions of Theorem 3 in [4], taking into account that the constant C in this theorem is independent of u and depends only on the value of sup G |u|, which can be estimated according to Remark 2.…”

“…Furthermore, the parameters ϑ i , i = 1, 2, 3, 4, are constants dependent on λ, k, β, and n; in particular, they satisfy the conditions of Theorem 1. Then problem (3), (4) has at least one solution in the space X.…”

Dini continuity of the first derivatives of generalized solutions to the Dirichlet problem for linear elliptic second order equations in nonsmooth domains

“…The conclusion of Theorem 4 is best possible. This is shown for A(r) = r α , λ ≥ 1 + α, α ∈ (0, 1), in [6] (see also examples in [2]). …”