Geodesic
Remarks on the Uniqueness of a Solution of the Dirichlet Problem for Second-Order Elliptic Equations with Lower-Order Terms
Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 3, pp. 15-28.

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

We give an example of an incompressible diffusion equation whose solution is nonunique. It is shown that this equation has an approximation solution as well as another solution that cannot be obtained by approximation. We give sufficient conditions for the uniqueness of a solution as well as for the uniqueness of an approximation solution.
Keywords: approximation solution, nonuniqueness, energy identity, maximal function, higher integrability. 
@article{FAA_2004_38_3_a1,
     author = {V. V. Zhikov},
     title = {Remarks on the {Uniqueness} of a {Solution} of the {Dirichlet} {Problem} for {Second-Order} {Elliptic} {Equations} with {Lower-Order} {Terms}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {15--28},
     publisher = {mathdoc},
     volume = {38},
     number = {3},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2004_38_3_a1/}
}
TY  - JOUR
AU  - V. V. Zhikov
TI  - Remarks on the Uniqueness of a Solution of the Dirichlet Problem for Second-Order Elliptic Equations with Lower-Order Terms
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2004
SP  - 15
EP  - 28
VL  - 38
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2004_38_3_a1/
LA  - ru
ID  - FAA_2004_38_3_a1
ER  - 
%0 Journal Article
%A V. V. Zhikov
%T Remarks on the Uniqueness of a Solution of the Dirichlet Problem for Second-Order Elliptic Equations with Lower-Order Terms
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2004
%P 15-28
%V 38
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2004_38_3_a1/
%G ru
%F FAA_2004_38_3_a1
V. V. Zhikov. Remarks on the Uniqueness of a Solution of the Dirichlet Problem for Second-Order Elliptic Equations with Lower-Order Terms. Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 3, pp. 15-28. http://geodesic.mathdoc.fr/item/FAA_2004_38_3_a1/

[1] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[2] Kinderlerer D., Stampakkya G., Vvedenie v variatsionnye neravenstva i ikh prilozheniya, Mir, M., 1983 | MR

[3] Zhikov V. V., “Diffuziya v neszhimaemom sluchainom potoke”, Funkts. analiz i ego pril., 31:3 (1997), 10–22 | DOI | MR | Zbl

[4] Fannjiang M. A., Papanicolaou G. C., “Diffusion in turbulence”, Probab. Theory Related Fields, 105 (1996), 279–334 | DOI | MR | Zbl

[5] Nadirashvili N. S., “Nonuniqueness in the martingale problem and the Dirichlet problem for uniformly elliptic operators”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 537–550 | MR | Zbl

[6] Gilbarg D., Serrin J., “On isolated singularities of solutions of second order elliptic differential equations”, J. Anal. Math., 4 (1955/56), 309–340 | DOI | MR

[7] Gilbarg D., Trudinger N., Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989 | MR | Zbl

[8] Safonov M., “Nonuniqueness for second order elliptic equations with measurable coefficients”, SIAM J. Math. Anal., 30 (1999), 879–895 | DOI | MR | Zbl

[9] Stein E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, 1993 | MR | Zbl

[10] Giaquinta and Modica G., “Regularity results for some classes of higher order nonlinear elliptic systems”, J. Reine Angew. Math., 311/312 (1979), 145–169 | MR | Zbl

[11] Gehring F. W., “The $L^p$-integrability of the partial derivatives of a quasiconformal mapping”, Acta Math., 130 (1973), 265–277 | DOI | MR | Zbl

[12] Evans L. C., Gariepy R. F., Measure theory and fine properties of functions, CRC Press, Boca Ration, Ann Arbor, London, 1992 | MR | Zbl