TY - GEN

T1 - Some Extremal Problems for Solutions of the Modified Helmholtz Equation in the Half-Space

AU - Kresin, Gershon

AU - Yaakov, Tehiya Ben

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

PY - 2021

Y1 - 2021

N2 - Representations for the sharp coefficients in pointwise estimates involving the gradient of the solution to the modified Helmholtz equation (Δ- c2) u= 0 in the half-space R+n are described. It is assumed that the boundary data of the Dirichlet and Neumann problems in R+n belong to the space Lp. Each of these representations includes an extremal problem with respect to a vector parameter inside of an integral over the unit sphere in Rn. Explicit formulas for solutions to the extremal problems are indicated for p∈ [ 2, ∞] and p∈ [ 2, (n+ 2 ) / 2 ] in the cases of Dirichlet and Neumann boundary data, respectively.

AB - Representations for the sharp coefficients in pointwise estimates involving the gradient of the solution to the modified Helmholtz equation (Δ- c2) u= 0 in the half-space R+n are described. It is assumed that the boundary data of the Dirichlet and Neumann problems in R+n belong to the space Lp. Each of these representations includes an extremal problem with respect to a vector parameter inside of an integral over the unit sphere in Rn. Explicit formulas for solutions to the extremal problems are indicated for p∈ [ 2, ∞] and p∈ [ 2, (n+ 2 ) / 2 ] in the cases of Dirichlet and Neumann boundary data, respectively.

KW - Extremal problems

KW - Half-space

KW - Modified Helmholtz equation

UR - http://www.scopus.com/inward/record.url?scp=85125288387&partnerID=8YFLogxK

U2 - 10.1007/978-981-16-6297-3_12

DO - 10.1007/978-981-16-6297-3_12

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AN - SCOPUS:85125288387

SN - 9789811662966

T3 - Springer Proceedings in Mathematics and Statistics

SP - 171

EP - 176

BT - Functional Differential Equations and Applications - FDEA-2019

A2 - Domoshnitsky, Alexander

A2 - Rasin, Alexander

A2 - Padhi, Seshadev

T2 - 7th International Conference on Functional Differential Equations and Applications, FDEA 2019

Y2 - 22 August 2019 through 27 August 2019

ER -