Optimal estimates for the gradient of harmonic functions in the multidimensional half-space

Abstract

A representation of the sharp constant in a pointwise estimate of the gradient of a harmonic function in a multidimensional half-space is obtained under the assumption that function's boundary values belong to L p . This representation is concretized for the cases p = 1, 2, and ∞. 2000 MSC. Primary: 35B30; Secondary: 35J05

Generalized Poisson integral and sharp estimates for harmonic and biharmonic functions in the half-space

“…In the paper [2] (see also [5]) a representation for the sharp coefficient C p (x) in the inequality |∇v(x)| ≤ C p (x) v p was found, where v is harmonic function in the half-space R n + = x = (x ′ , x n ) : x ′ ∈ R n−1 , x n > 0 , represented by the Poisson integral with boundary values in L p (R n−1 ), || · || p is the norm in L p (R n−1 ), 1 ≤ p ≤ ∞, x ∈ R n + . It was shown that C p (x) = C p x (n−1+p)/p n and explicit formulas for C 1 , C 2 and C ∞ were found.…”

A proof of Khavinson's conjecture in R4

“…Recently Kresin and Maz'ya proved the following generalization of : Here, is a bounded harmonic function in the half‐space , , and is fixed. These optimal poitwise estimates arise while proving Khavinson conjecture in the half‐space setting.…”

“…In their recent paper [, ] and in their book , Kresin and Maz'ya considered the Khavinson problem from a more general aspect including harmonic functions with ‐boundary values (). They formulated the generalized Khavinson conjecture and proved it for bounded harmonic functions in .…”

“…For some related results in two dimensional setting we refer to the papers [3], [4], [11]. Recently Kresin and Maz'ya [8] proved the following generalization of (1.1):…”

Optimal Estimates for the Gradient of Harmonic Functions in the Unit Disk

“…In the recent paper [7] Maz'ya and Kresin studied point-wise estimates of the gradient of real harmonic function u under the assumptions that the boundary values belong to L p . They obtained the following result…”