A Sobolev space theory of SPDEs with constant coefficients in a half space

N. V. Krylov; S. V. Lototsky

doi:10.1137/S0036141098338843

A Sobolev space theory of SPDEs with constant coefficients in a half space

N. V. Krylov, S. V. Lototsky

Mathematics

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61 Scopus citations

Abstract

Equations of the form du = (a^{Script i signScript j sign}Script u sign_{cursive greek chiScript i signcursive greek chiScript j sign} + D_{Script i sign}Script f sign^{Script i sign}) Script d signScript t sign + Σ_{Script k sign}(σ^{Script i signScript k sign}Script u sign_{cursive greek chiScript i sign} + Script g sign^{Script k sign})Script d signScript w sign^{Script k sign}_{Script t sign} are considered for Script t sign > 0 and cursive greek chi ∈ ℝ^{Script d sign}₊. The unique solvability of these equations is proved in weighted Sobolev spaces with fractional positive or negative derivatives, summable to the power Script p sign ∈ (2, ∞).

Original language	English (US)
Pages (from-to)	19-33
Number of pages	15
Journal	SIAM Journal on Mathematical Analysis
Volume	31
Issue number	1
DOIs	https://doi.org/10.1137/S0036141098338843
State	Published - 1999

Keywords

Sobolev spaces with weights
Stochastic partial differential equations

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abstract = "Equations of the form du = (aScript i signScript j signScript u signcursive greek chiScript i signcursive greek chiScript j sign + DScript i signScript f signScript i sign) Script d signScript t sign + ΣScript k sign(σScript i signScript k signScript u signcursive greek chiScript i sign + Script g signScript k sign)Script d signScript w signScript k signScript t sign are considered for Script t sign > 0 and cursive greek chi ∈ ℝScript d sign+. The unique solvability of these equations is proved in weighted Sobolev spaces with fractional positive or negative derivatives, summable to the power Script p sign ∈ (2, ∞).",

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AB - Equations of the form du = (aScript i signScript j signScript u signcursive greek chiScript i signcursive greek chiScript j sign + DScript i signScript f signScript i sign) Script d signScript t sign + ΣScript k sign(σScript i signScript k signScript u signcursive greek chiScript i sign + Script g signScript k sign)Script d signScript w signScript k signScript t sign are considered for Script t sign > 0 and cursive greek chi ∈ ℝScript d sign+. The unique solvability of these equations is proved in weighted Sobolev spaces with fractional positive or negative derivatives, summable to the power Script p sign ∈ (2, ∞).

KW - Sobolev spaces with weights

KW - Stochastic partial differential equations

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A Sobolev space theory of SPDEs with constant coefficients in a half space

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