Geodesic
The boundary values of solutions of an elliptic equation
Sbornik. Mathematics, Tome 210 (2019) no. 12, pp. 1724-1752 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The paper is devoted to the study of the boundary behaviour of solutions of a second-order elliptic equation. Criteria are established for the existence of a boundary value of a solution of the homogeneous equation under the same conditions on the coefficients of the equation as were used to establish that the Dirichlet problem with a boundary function in $L_p$, $p>1$, has a unique solution. In particular, an analogue of Riesz's well-known theorem (on the boundary values of an analytic function) is proved: if a family of norms in the space $L_p$ of the traces of a solution on surfaces ‘parallel’ to the boundary is bounded, then this family of traces converges in $L_p$. This means that the solution of the equation under consideration is a solution of the Dirichlet problem with a certain boundary value in $L_p$. Estimates of the nontangential maximal function and of an analogue of the Luzin area integral hold for such a solution, which make it possible to claim that the boundary value is taken in a substantially stronger sense. Bibliography: 57 titles.
Keywords: boundary value, Dirichlet problem.
Mots-clés : elliptic equation 
@article{SM_2019_210_12_a3,
     author = {A. K. Gushchin},
     title = {The boundary values of solutions of an elliptic equation},
     journal = {Sbornik. Mathematics},
     pages = {1724--1752},
     year = {2019},
     volume = {210},
     number = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2019_210_12_a3/}
}
TY  - JOUR
AU  - A. K. Gushchin
TI  - The boundary values of solutions of an elliptic equation
JO  - Sbornik. Mathematics
PY  - 2019
SP  - 1724
EP  - 1752
VL  - 210
IS  - 12
UR  - http://geodesic.mathdoc.fr/item/SM_2019_210_12_a3/
LA  - en
ID  - SM_2019_210_12_a3
ER  - 
%0 Journal Article
%A A. K. Gushchin
%T The boundary values of solutions of an elliptic equation
%J Sbornik. Mathematics
%D 2019
%P 1724-1752
%V 210
%N 12
%U http://geodesic.mathdoc.fr/item/SM_2019_210_12_a3/
%G en
%F SM_2019_210_12_a3
A. K. Gushchin. The boundary values of solutions of an elliptic equation. Sbornik. Mathematics, Tome 210 (2019) no. 12, pp. 1724-1752. http://geodesic.mathdoc.fr/item/SM_2019_210_12_a3/

[1] V. P. Mikhaĭlov, “Dirichlet's problem for a second-order elliptic equation”, Differ. Equ., 12:10 (1977), 1320–1329 | MR | Zbl

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

[3] 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

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

[5] 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

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

[7] A. K. Gushchin, “Estimates of the nontangential maximal function for solutions of a second-order elliptic equation”, Dokl. Math., 86:2 (2012), 667–669 | DOI | MR | Zbl

[8] 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

[9] 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 | DOI | DOI | MR | Zbl

[10] L. Carleson, “An interpolation problem for bounded analytic functions”, Amer. J. Math., 80:4 (1958), 921–930 | DOI | MR | Zbl

[11] L. Carleson, “Interpolations by bounded analytic functions and the corona problem”, Ann. of Math. (2), 76:3 (1962), 547–559 | DOI | MR | Zbl

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

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

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

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

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

[17] J. Nash, “Continuity of solutions of parabolic and elliptic equations”, Amer. J. Math., 80:4 (1958), 931–954 | DOI | MR | Zbl

[18] 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

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

[20] V. Zh. Dumanyan, “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

[21] L. M. Kozhevnikova, “Entropy and renormalized solutions of anisotropic elliptic equations with variable nonlinearity exponents”, Sb. Math., 210:3 (2019), 417–446 | DOI | DOI | MR | Zbl

[22] F. Kh. Mukminov, “Existence of a renormalized solution to an anisotropic parabolic problem with variable nonlinearity exponents”, Sb. Math., 209:5 (2018), 714–738 | DOI | DOI | MR | Zbl

[23] M. O. Katanaev, “Chern–Simons action and disclinations”, Proc. Steklov Inst. Math., 301 (2018), 114–133 | DOI | DOI | MR | Zbl

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

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

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

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

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

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

[30] Yu. A. Mikhaĭlov, “Boundary values in $L_p$, $p>1$, of solutions of a second-order linear elliptic equation”, Differ. Equ., 19:2 (1983), 243–258 | MR | Zbl

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

[32] 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

[33] 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

[34] Yu. A. Alkhutov, V. A. Kondrat'ev, “Solvability of the Dirichlet problem for second-order elliptic equations in a convex domain”, Differ. Equ., 28:5 (1992), 650–662 | MR | Zbl

[35] Yu. A. Alkhutov, “$L_p$-estimates of the solution of the Dirichlet problem for second-order elliptic equations”, Sb. Math., 189:1 (1998), 1–17 | DOI | DOI | MR | Zbl

[36] P. Fatou, “Séries trigonometriques et séries de Taylor”, Acta Math., 30:1 (1906), 335–400 | DOI | MR | Zbl

[37] F. Riesz, “Über die Randwerte einer analytischen Funktion”, Math. Z., 18:1 (1923), 87–95 | DOI | MR | Zbl

[38] J. E. Littlewood, R. E. A.Ċ. Paley, “Theorems on Fourier series and power series”, J. London Math. Soc., 6:3 (1931), 230–233 | DOI | MR | Zbl

[39] J. E. Littlewood, R. E. A.Ċ. Paley, “Theorems on Fourier series and power series. II”, Proc. London Math. Soc. (2), 42:1 (1936), 52–89 | DOI | MR | Zbl

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

[41] J. Marcinkiewicz, A. Zygmund, “A theorem of Lusin. Part I”, Duke Math. J., 4:3 (1938), 473–485 | DOI | MR | Zbl

[42] A. Zygmund, Trigonometric series, v. II, 2nd ed., Cambridge Univ. Press, New York, 1959, vii+354 pp. | MR | MR | Zbl | Zbl

[43] I. I. Priwalow, Randeigenschaften analytischer Funktionen, Hochschulbücher für Math., 25, 2. Aufl., VEB Deutscher Verlag Wissensch., Berlin, 1956, viii+247 pp. | MR | MR | Zbl | Zbl

[44] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Ser., 30, Princeton Univ. Press, Princeton, NJ, 1970, xiv+290 pp. | MR | MR | Zbl | Zbl

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

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

[47] V. P. Mikhaĭlov, “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

[48] V. P. Mikhailov, “O granichnykh svoistvakh reshenii ellipticheskikh uravnenii”, Matem. zametki, 27:1 (1980), 137–145 | MR | Zbl

[49] 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

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

[51] A. K. Guschin, V. P. Mikhailov, “O granichnykh znacheniyakh reshenii ellipticheskikh uravnenii”, Obobschennye funktsii i ikh primeneniya v matematicheskoi fizike (Moskva, 1980), VTs AN SSSR, M., 1981, 189–205 | MR | Zbl

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

[53] V. A. Kondrat'ev, J. Kopáček, 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

[54] 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 | DOI | DOI | MR | Zbl

[55] A. K. Guschin, O suschestvovanii granichnykh znachenii v $L_2$ reshenii ellipticheskogo uravneniya, Tr. MIAN, 306, MAIK «Nauka/Interperiodika», M., 2019 (to appear)

[56] A. K. Guschin, “$L_p$-otsenki nekasatelnoi maksimalnoi funktsii dlya reshenii ellipticheskogo uravneniya vtorogo poryadka”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(30) (2013), 53–69 | DOI

[57] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren Math. Wiss., 224, 2nd ed., Springer-Verlag, Berlin, 1983, xiii+513 pp. | MR | MR | Zbl | Zbl