Abstract:
For solutions of the Dirichlet problem for a second-order elliptic equation, we establish an analogue of the Carleson theorem on $L_p$-estimates. Under the same conditions on the coefficients for which the unique solvability of the considered problem is known, we prove this criterion for the validity of estimate of the solution norm in the space $L_p$ with a measure. We require their Dini continuity on the boundary, but we assume only their measurability and boundedness in the domain under consideration.
Citation:
A. K. Gushchin, “$L_p$-estimates for solutions of second-order elliptic equation Dirichlet problem”, TMF, 174:2 (2013), 243–255; Theoret. and Math. Phys., 174:2 (2013), 209–219
This publication is cited in the following 14 articles:
A. K. Gushchin, “On Dirichlet problem”, Theoret. and Math. Phys., 218:1 (2024), 51–67
A. K. Gushchin, “On some properties of elliptic partial differential equation solutions”, Int. J. Mod. Phys. A, 37:20 (2022), 2243002–9
A. K. Gushchin, “Extensions of the space of continuous functions and embedding theorems”, Sb. Math., 211:11 (2020), 1551–1567
A. K. Gushchin, “The boundary values of solutions of an elliptic equation”, Sb. Math., 210:12 (2019), 1724–1752
A. K. Gushchin, “On the Existence of $L_2$ Boundary Values of Solutions to an Elliptic Equation”, Proc. Steklov Inst. Math., 306 (2019), 47–65
A. K. Gushchin, “The Luzin area integral and the nontangential maximal function for solutions to a second-order elliptic equation”, Sb. Math., 209:6 (2018), 823–839
A. K. Gushchin, “A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation”, Proc. Steklov Inst. Math., 301 (2018), 44–64
A. K. Gushchin, “$L_p$-estimates for the nontangential maximal function of the solution to a second-order elliptic equation”, Sb. Math., 207:10 (2016), 1384–1409
A. K. Guschin, “O zadache Dirikhle dlya ellipticheskogo uravneniya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:1 (2015), 19–43
A. K. Gushchin, “V.A. Steklov's work on equations of mathematical physics and development of his results in this field”, Proc. Steklov Inst. Math., 289 (2015), 134–151
A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation”, Sb. Math., 206:10 (2015), 1410–1439
V. Zh. Dumanyan, “On solvability of the Dirichlet problem with the boundary function in L 2 for a second-order elliptic equation”, J. Contemp. Mathemat. Anal., 50:4 (2015), 153
V. Zh. Dumanyan, “Solvability of the Dirichlet problem for second-order elliptic equations”, Theoret. and Math. Phys., 180:2 (2014), 917–931
Citation in format AMSBIB
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