Geodesic
A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation
Informatics and Automation, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 53-73.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper is devoted to the study of the boundary behavior of solutions to a second-order elliptic equation. A criterion is established for the existence in $L_p$, $p>1$, of a boundary value of a solution to a homogeneous equation in the self-adjoint form without lower order terms. Under the conditions of this criterion, the solution belongs to the space of $(n-1)$-dimensionally continuous functions; thus, the boundary value is taken in a much stronger sense. Moreover, for such a solution to the Dirichlet problem, estimates for the nontangential maximal function and for an analog of the Lusin area integral hold.
Mots-clés : elliptic equation
Keywords: boundary value, Dirichlet problem, Lusin area integral. 
@article{TRSPY_2018_301_a4,
     author = {A. K. Gushchin},
     title = {A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation},
     journal = {Informatics and Automation},
     pages = {53--73},
     publisher = {mathdoc},
     volume = {301},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2018_301_a4/}
}
TY  - JOUR
AU  - A. K. Gushchin
TI  - A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation
JO  - Informatics and Automation
PY  - 2018
SP  - 53
EP  - 73
VL  - 301
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2018_301_a4/
LA  - ru
ID  - TRSPY_2018_301_a4
ER  - 
%0 Journal Article
%A A. K. Gushchin
%T A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation
%J Informatics and Automation
%D 2018
%P 53-73
%V 301
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2018_301_a4/
%G ru
%F TRSPY_2018_301_a4
A. K. Gushchin. A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation. Informatics and Automation, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 53-73. http://geodesic.mathdoc.fr/item/TRSPY_2018_301_a4/

[1] È. R. Andriyanova, F. Kh. Mukminov, “Existence and qualitative properties of a solution of the first mixed problem for a parabolic equation with non-power-law double nonlinearity”, Sb. Math., 207:1 (2016), 1–40 | DOI | DOI | MR | Zbl

[2] O. I. Bogoyavlenskii, V. S. Vladimirov, I. V. Volovich, A. K. Gushchin, Yu. N. Drozhzhinov, V. V. Zharinov, V. P. Mikhailov, “Boundary value problems of mathematical physics”, Proc. Steklov Inst. Math., 175 (1988), 65–105 | MR | Zbl | Zbl

[3] Carleson L., “An interpolation problem for bounded analytic functions”, Amer. J. Math., 80 (1958), 921–930 | DOI | MR | Zbl

[4] Carleson L., “Interpolations by bounded analytic functions and the corona problem”, Ann. Math. Ser. 2, 76 (1962), 547–559 | DOI | MR | Zbl

[5] De Giorgi E., “Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari”, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. Ser. 3, 3 (1957), 25–43 | MR | Zbl

[6] V. Zh. Dumanyan, “Solvability of the Dirichlet problem for a general second-order elliptic equation”, Sb. Math., 202:7 (2011), 1001–1020 | DOI | DOI | MR | Zbl

[7] V. Zh. Dumanyan, “Solvability of the Dirichlet problem for second-order elliptic equations”, Theor. Math. Phys., 180:2 (2014), 917–931 | DOI | DOI | MR | Zbl

[8] Dumanyan V. Zh., “On solvability of the Dirichlet problem with the boundary function in $L_2$ for a second-order elliptic equation”, J. Contemp. Math. Anal., 50:4 (2015), 153–166 | DOI | MR | Zbl

[9] Fatou P., “Séries trigonométriques et séries de Taylor”, Acta math., 30 (1906), 335–400 | DOI | MR | Zbl

[10] D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983 | MR | MR | Zbl

[11] A. K. Gushchin, “On the Dirichlet problem for a second-order elliptic equation”, Math. USSR-Sb., 65:1 (1990), 19–66 | DOI | MR | Zbl

[12] A. K. Gushchin, “On the interior smoothness of solutions to second-order elliptic equations”, Sib. Math. J., 46:5 (2005), 826–840 | DOI | MR | Zbl

[13] A. K. Gushchin, “A strengthening of the interior Hölder continuity property for solutions of the Dirichlet problem for a second-order elliptic equation”, Theor. Math. Phys., 157:3 (2008), 1655–1670 | DOI | DOI | MR | Zbl

[14] A. K. Gushchin, “Solvability of the Dirichlet problem for a second-order elliptic equation with a boundary function from $L_p$”, Dokl. Math., 83:2 (2011), 219–221 | DOI | MR | Zbl

[15] A. K. Gushchin, “The Dirichlet problem for a second-order elliptic equation with an $L_p$ boundary function”, Sb. Math., 203:1 (2012), 1–27 | DOI | DOI | MR | Zbl

[16] A. K. Guschin, “$L_p$-estimates for solutions of second-order elliptic equation Dirichlet problem”, Theor. Math. Phys., 174:2 (2013), 209–219 | DOI | DOI | MR | Zbl

[17] A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation”, Sb. Math., 206:10 (2015), 1410–1439 | DOI | DOI | MR | Zbl

[18] 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 | DOI | DOI | MR | Zbl

[19] 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 | DOI | DOI | MR | Zbl

[20] A. K. Gushchin, “The nontangential maximal function and the Lusin area integral for solutions of a second order elliptic equation”, Sb. Math., 209:6 (2018) | DOI | DOI | MR

[21] A. K. Gushchin, V. P. Mikhailov, “On boundary values in $L_p$, $p>1$, of solutions of elliptic equations”, Math. USSR-Sb., 36:1 (1980), 1–19 | DOI | DOI | MR | Zbl | Zbl

[22] A. K. Gushchin, V. P. Mikhailov, “On boundary values of solutions of elliptic equations”, Generalized Functions and Their Applications in Mathematical Physics, Proc. Int. Conf. (Moscow, Nov. 24–28, 1980), Vychisl. Tsentr Akad. Nauk SSSR, Moscow, 1981, 189–205

[23] A. K. Gushchin, V. P. Mikhailov, “On the existence of boundary values of solutions of an elliptic equation”, Math. USSR-Sb., 73:1 (1992), 171–194 | DOI | MR | Zbl

[24] A. K. Gushchin, V. P. Mikhailov, “On solvability of nonlocal problems for a second-order elliptic equation”, Russ. Acad. Sci. Sb. Math., 81:1 (1995), 101–136 | MR | Zbl

[25] Hörmander L., “$L^p$ estimates for (pluri-) subharmonic functions”, Math. Scand., 20 (1967), 65–78 | DOI | MR | Zbl

[26] M. O. Katanaev, “Lorentz invariant vacuum solutions in general relativity”, Proc. Steklov Inst. Math., 290 (2015), 138–142 | DOI | MR | Zbl

[27] M. O. Katanaev, “Killing vector fields and a homogeneous isotropic universe”, Phys. Usp., 59:7 (2016), 689–700 | DOI | DOI

[28] M. O. Katanaev, “Cosmological models with homogeneous and isotropic spatial sections”, Theor. Math. Phys., 191:2 (2017), 661–668 | DOI | DOI | MR | Zbl

[29] V. A. Kondrat'ev, I. Kopachek, O. A. Oleinik, “On the best Hölder exponents for generalized solutions of the Dirichlet problem for a second order elliptic equation”, Math. USSR-Sb., 59:1 (1988), 113–127 | DOI | MR | Zbl

[30] L. M. Kozhevnikova, A. A. Khadzhi, “Existence of solutions of anisotropic elliptic equations with nonpolynomial nonlinearities in unbounded domains”, Sb. Math., 206:8 (2015), 1123–1149 | DOI | DOI | MR | Zbl

[31] Littlewood J. E., Paley R. E. A. C., “Theorems on Fourier series and power series”, J. London Math. Soc., 6 (1931), 230–233 | DOI | MR | Zbl

[32] Littlewood J. E., Paley R. E. A. C., “Theorems on Fourier series and power series. II”, Proc. London Math. Soc. Ser. 2, 42 (1936), 52–89 | MR

[33] Littlewood J. E., Paley R. E. A. C., “Theorems on Fourier series and power series. III”, Proc. London Math. Soc. Ser. 2, 43 (1937), 105–126 | MR | Zbl

[34] Marcinkiewicz J., Zygmund A., “A theorem of Lusin”, Duke Math. J., 4 (1938), 473–485 | DOI | MR | Zbl

[35] V. G. Maz'ja, “On a degenerating problem with directional derivative”, Math. USSR-Sb., 16:3 (1972), 429–469 | DOI | MR | Zbl

[36] V. P. Mikhailov, “On boundary properties of solutions of elliptic equations”, Sov. Math. Dokl., 17:1 (1976), 274–277 | MR

[37] V. P. Mikhailov, “On the boundary values of solutions of elliptic equations in domains with a smooth boundary”, Math. USSR-Sb., 30:2 (1976), 143–166 | DOI | MR | Zbl

[38] V. P. Mikhailov, “Dirichlet's problem for a second-order elliptic equation”, Diff. Eqns., 12:10 (1977), 1320–1329 | MR | Zbl

[39] V. P. Mikhailov, “Boundary properties of solutions of elliptic equations”, Mat. Zametki, 27:1 (1980), 137–145 | MR | Zbl

[40] Yu. A. Mikhailov, “Boundary values in $L_p$, $p>1$, of solutions of second-order linear elliptic equations”, Diff. Eqns., 19:2 (1983), 243–258 | MR

[41] Nash J., “Continuity of solutions of parabolic and elliptic equations”, Amer. J. Math., 80 (1958), 931–954 | DOI | MR | Zbl

[42] I. M. Petrushko, “On boundary values of solutions of elliptic equations in domains with Lyapunov boundary”, Math. USSR-Sb., 47:1 (1984), 43–72 | DOI | MR | Zbl | Zbl

[43] I. M. Petrushko, “On boundary values in $\mathcal L_p$, $p>1$, of solutions of elliptic equations in domains with a Lyapunov boundary”, Math. USSR-Sb., 48:2 (1984), 565–585 | DOI | MR | Zbl | Zbl

[44] I. I. Privalov, Boundary Properties of Analytic Functions, Gostekhizdat, Moscow, 1950 (in Russian) | MR

[45] Riesz F., “Über die Randwerte einer analytischen Funktion”, Math. Z., 18 (1923), 87–95 | DOI | MR | Zbl

[46] F. Riesz, B. Sz.-Nagy, Functional Analysis, Dover, New York, 1990 | MR | MR | Zbl

[47] Ya. A. Roitberg, “On limiting values on surfaces, parallel to the boundary, of generalized solutions of elliptic equations”, Sov. Math. Dokl., 19 (1978), 229–233 | MR | Zbl

[48] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ, 1970 | MR | MR | Zbl

[49] V. V. Zharinov, “Conservation laws, differential identities, and constraints of partial differential equations”, Theor. Math. Phys., 185:2 (2015), 1557–1581 | DOI | DOI | MR | Zbl

[50] V. V. Zharinov, “Bäcklund transformations”, Theor. Math. Phys., 189:3 (2016), 1681–1692 | DOI | DOI | MR | Zbl

[51] V. V. Zharinov, “Lie–Poisson structures over differential algebras”, Theor. Math. Phys., 192:3 (2017), 1337–1349 | DOI | DOI | MR | Zbl

[52] A. Zygmund, Trigonometric Series, v. 2, Cambridge Univ. Press, Cambridge, 1959 | MR | Zbl