Cauchy-Dirichlet problem in Morrey spaces for parabolic equations with discontinuous coefficients

Dian K. Palagachev; Maria A. Ragusa; Lubomira G. Softova

Bollettino dell'Unione Matematica Italiana (2003)

  • Volume: 6-B, Issue: 3, page 667-683
  • ISSN: 0392-4033

Abstract

top
Let Q T be a cylinder in R n + 1 and x = x , t R n × R . It is studied the Cauchy-Dirichlet problem for the uniformly parabolic operator u t - i , j = 1 n a i j x D i j u = f x q.o. in  Q T , u x = 0 su  Q T , in the Morrey spaces W p , λ 2 , 1 Q T , p 1 , , λ 0 , n + 2 , supposing the coefficients to belong to the class of functions with vanishing mean oscillation. There are obtained a priori estimates in Morrey spaces and Hölder regularity for the solution and its spatial derivatives.

How to cite

top

Palagachev, Dian K., Ragusa, Maria A., and Softova, Lubomira G.. "Cauchy-Dirichlet problem in Morrey spaces for parabolic equations with discontinuous coefficients." Bollettino dell'Unione Matematica Italiana 6-B.3 (2003): 667-683. <http://eudml.org/doc/195716>.

@article{Palagachev2003,
abstract = {Let $Q_\{T\}$ be a cylinder in $\mathbb\{R\}^\{n+1\}$ and $x=(x',t)\in \mathbb\{R\}^\{n\}\times \mathbb\{R\}$. It is studied the Cauchy-Dirichlet problem for the uniformly parabolic operator $$ \begin\{cases\} u\_\{t\}-\sum\_\{i,j=1\}^\{n\}a^\{ij\}(x) D\_\{ij\}u=f(x) & \text\{q.o. in \} Q\_\{T\}, \\ u(x)=0 & \text\{su \} \partial Q\_\{T\}, \end\{cases\} $$ in the Morrey spaces $W^\{2,1\}_\{p,\lambda\}(Q_\{T\})$, $p\in (1, \infty)$, $\lambda\in (0, n+2)$, supposing the coefficients to belong to the class of functions with vanishing mean oscillation. There are obtained a priori estimates in Morrey spaces and Hölder regularity for the solution and its spatial derivatives.},
author = {Palagachev, Dian K., Ragusa, Maria A., Softova, Lubomira G.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {667-683},
publisher = {Unione Matematica Italiana},
title = {Cauchy-Dirichlet problem in Morrey spaces for parabolic equations with discontinuous coefficients},
url = {http://eudml.org/doc/195716},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Palagachev, Dian K.
AU - Ragusa, Maria A.
AU - Softova, Lubomira G.
TI - Cauchy-Dirichlet problem in Morrey spaces for parabolic equations with discontinuous coefficients
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/10//
PB - Unione Matematica Italiana
VL - 6-B
IS - 3
SP - 667
EP - 683
AB - Let $Q_{T}$ be a cylinder in $\mathbb{R}^{n+1}$ and $x=(x',t)\in \mathbb{R}^{n}\times \mathbb{R}$. It is studied the Cauchy-Dirichlet problem for the uniformly parabolic operator $$ \begin{cases} u_{t}-\sum_{i,j=1}^{n}a^{ij}(x) D_{ij}u=f(x) & \text{q.o. in } Q_{T}, \\ u(x)=0 & \text{su } \partial Q_{T}, \end{cases} $$ in the Morrey spaces $W^{2,1}_{p,\lambda}(Q_{T})$, $p\in (1, \infty)$, $\lambda\in (0, n+2)$, supposing the coefficients to belong to the class of functions with vanishing mean oscillation. There are obtained a priori estimates in Morrey spaces and Hölder regularity for the solution and its spatial derivatives.
LA - eng
UR - http://eudml.org/doc/195716
ER -

References

top
  1. ACQUISTAPACE, P., On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl., 161 (1992), 231-269. Zbl0802.35015MR1174819
  2. BRAMANTI, M., Commutators of integral operators with positive kernels, Le Matematiche, 49 (1994), 149-168. Zbl0840.42009MR1386370
  3. BRAMANTI, M.- CERUTTI, M. C., W p 1 , 2 solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients, Comm. in Partial Diff. Equations, 18 (1993), 1735-1763. Zbl0816.35045MR1239929
  4. CALDERÓN, A. P.- ZYGMUND, A., On the existence of certain singular integrals, Acta. Math., 88 (1952), 85-139. Zbl0047.10201MR52553
  5. CALDERÓN, A. P.- ZYGMUND, A., Singular integral operators and differential equations, Amer. J. Math., 79 (1957), 901-921. Zbl0081.33502MR100768
  6. CANNARSA, P., Second order nonvariational parabolic systems, Boll. Unione Mat. Ital., 18-C (1981), 291-315. Zbl0473.35043MR631584
  7. CHIARENZA, F.- FRASCA, M., Morrey spaces and Hardy-Littlewood maximal functions, Rend. Mat. Appl., 7 (1987), 273-279. Zbl0717.42023MR985999
  8. CHIARENZA, F.- FRASCA, M.- LONGO, P., Interior W 2 , p estimates for nondivergence elliptic equations with discontinuous coefficients, Ric. Mat., 40 (1991), 149-168. Zbl0772.35017MR1191890
  9. CHIARENZA, F.- FRASCA, M.- LONGO, P., W 2 , p solvability of the Dirichlet problem for nondivergence form elliptic equations with VMO coefficients, Trans. Amer. Math. Soc., 336 (1993), 841-853. Zbl0818.35023MR1088476
  10. DA PRATO, G., Spazi L p , θ Ω , δ e loro proprietà, Ann. Mat. Pura Appl., 69 (1965), 383-392. Zbl0145.16207MR192330
  11. DI FAZIO, G.- PALAGACHEV, D. K.- RAGUSA, M. A., Global Morrey regularity of strong solutions to Dirichlet problem for elliptic equations with discontinuous coefficients, J. Funct. Anal., 166 (1999), 179-196. Zbl0942.35059MR1707751
  12. FABES, E. B.- RIVIÈRE, N., Singular integrals with mixed homogeneity, Studia Math., 27 (1966), 19-38. Zbl0161.32403MR209787
  13. GILBARG, D.- TRUDINGER, N. S., Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, 1983. Zbl0562.35001MR737190
  14. GUGLIELMINO, F., Sulle equazioni paraboliche del secondo ordine di tipo non variazionale, Ann. Mat. Pura Appl.,65 (1964), 127-151. Zbl0141.29202MR186940
  15. JOHN, F.- NIRENBERG, L., On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415-426. Zbl0102.04302MR131498
  16. LADYZHENSKAYA, O. A.- SOLONNIKOV, V. A.- URAL'TSEVA, N. N., Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monographs, Vol. 23, Amer. Math. Soc., Providence, R.I.1968. Zbl0174.15403
  17. MAUGERI, A.- PALAGACHEV, D. K.- SOFTOVA, L. G., Elliptic and Parabolic Equations with Discontinuous Coefficients, Wiley-VCH, Berlin, 2000. Zbl0958.35002MR2260015
  18. PALAGACHEV, D. K.- RAGUSA, M. A.- SOFTOVA, L. G., Regular oblique derivative problem in Morrey spaces, Electr. J. Diff. Equations, 2000 (2000), No. 39, 1-17. Zbl1002.35033MR1764709
  19. PALAGACHEV, D.K.- SOFTOVA, L.G., Singular integral operators with mixed homogeneity in Morrey spacesC. R. Acad. Bulgare Sci., 54 (2001), No. 11, 11-16. Zbl0983.42008MR1878039
  20. SARASON, D., Functions of vanishing mean oscillation, Trans. Amer. Math. Soc., 207 (1975), 391-405. Zbl0319.42006MR377518
  21. SOFTOVA, L. G., Regular oblique derivative problem for linear parabolic equations with VMO principal coefficients, Manuscr. Math., 103, No. 2 (2000), 203-220. Zbl0963.35032MR1796316
  22. SOFTOVA, L. G., Morrey regularity of strong solutions to parabolic equations with VMO coefficients, C. R. Acad. Sci., Paris, Ser. I, Math., 333, No. 7 (2001), 635-640. Zbl0990.35035MR1868228
  23. SOFTOVA, L. G., Parabolic equations with VMO coefficients in Morrey spaces, Electr. J. Diff. Equations, 2001, No. 51 (2001), 1-25. Zbl1068.35517MR1846667

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.