Palagachev, Dian K. and Ragusa, Maria A. and Softova, Lubomira G.:
Cauchy-Dirichlet problem in Morrey spaces for parabolic equations with discontinuous coefficients
Bollettino dell'Unione Matematica Italiana Serie 8 6-B (2003), fasc. n.3, p. 667-683, Unione Matematica Italiana (English)
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Sunto
Siano $Q_{T}$ un cilindro in $\mathbb{R}^{n+1}$ ed $x=(x',t)\in \mathbb{R}^{n}\times \mathbb{R}$. Si studia il problema di Cauchy-Dirichlet per l'operatore uniformemente parabolico $$ \begin{cases} u_{t}-\sum_{i,j=1}^{n}a^{ij}(x) D_{ij}u=f(x) & \text{q.o. in } Q_{T}, \\ u(x)=0 & \text{su } \partial Q_{T}, \end{cases} $$ nell'ambito degli spazi di Morrey $W^{2,1}_{p,\lambda}(Q_{T})$, $p\in (1, \infty)$, $\lambda\in (0, n+2)$ supponendo che i coefficienti della parte principale appartengano alla classe delle funzioni con oscillazione media infinitesima. Si ottengono inoltre delle stime a priori nei suddetti spazi, e regolarità Hölderiana della soluzione e della sua derivata spaziale.
Referenze Bibliografiche
[1]
P. ACQUISTAPACE,
On BMO regularity for linear elliptic systems,
Ann. Mat. Pura Appl.,
161 (
1992), 231-269. |
MR 1174819 |
Zbl 0802.35015[2]
M. BRAMANTI,
Commutators of integral operators with positive kernels,
Le Matematiche,
49 (
1994), 149-168. |
MR 1386370 |
Zbl 0840.42009[3]
M. BRAMANTI-
M. C. CERUTTI,
$W_p^{1, 2}$ solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients,
Comm. in Partial Diff. Equations,
18 (
1993), 1735-1763. |
MR 1239929 |
Zbl 0816.35045[4]
A. P. CALDERÓN-
A. ZYGMUND,
On the existence of certain singular integrals,
Acta. Math.,
88 (
1952), 85-139. |
MR 52553 |
Zbl 0047.10201[5]
A. P. CALDERÓN-
A. ZYGMUND,
Singular integral operators and differential equations,
Amer. J. Math.,
79 (
1957), 901-921. |
MR 100768 |
Zbl 0081.33502[6]
P. CANNARSA,
Second order nonvariational parabolic systems,
Boll. Unione Mat. Ital.,
18-C (
1981), 291-315. |
MR 631584 |
Zbl 0473.35043[7]
F. CHIARENZA-
M. FRASCA,
Morrey spaces and Hardy-Littlewood maximal functions,
Rend. Mat. Appl.,
7 (
1987), 273-279. |
MR 985999 |
Zbl 0717.42023[8]
F. CHIARENZA-
M. FRASCA-
P. LONGO,
Interior $W^{2, p}$ estimates for nondivergence elliptic equations with discontinuous coefficients,
Ric. Mat.,
40 (
1991), 149-168. |
MR 1191890 |
Zbl 0772.35017[9]
F. CHIARENZA-
M. FRASCA-
P. LONGO,
$W^{2,p}$ solvability of the Dirichlet problem for nondivergence form elliptic equations with VMO coefficients,
Trans. Amer. Math. Soc.,
336 (
1993), 841-853. |
MR 1088476 |
Zbl 0818.35023[10]
G. DA PRATO,
Spazi $\mathcal{L}^{p,\theta}(\Omega, \delta)$ e loro proprietà,
Ann. Mat. Pura Appl.,
69 (
1965), 383-392. |
MR 192330 |
Zbl 0145.16207[11]
G. DI FAZIO-
D. K. PALAGACHEV-
M. A. RAGUSA,
Global Morrey regularity of strong solutions to Dirichlet problem for elliptic equations with discontinuous coefficients,
J. Funct. Anal.,
166 (
1999), 179-196. |
MR 1707751 |
Zbl 0942.35059[13]
D. GILBARG-
N. S. TRUDINGER,
Elliptic Partial Differential Equations of Second Order, 2nd ed.,
Springer-Verlag, Berlin,
1983. |
MR 737190 |
Zbl 0562.35001[14]
F. GUGLIELMINO,
Sulle equazioni paraboliche del secondo ordine di tipo non variazionale,
Ann. Mat. Pura Appl.,65 (
1964), 127-151. |
MR 186940 |
Zbl 0141.29202[15]
F. JOHN-
L. NIRENBERG,
On functions of bounded mean oscillation,
Comm. Pure Appl. Math.,
14 (
1961), 415-426. |
MR 131498 |
Zbl 0102.04302[16]
O. A. LADYZHENSKAYA-
V. A. SOLONNIKOV-
N. N. URAL'TSEVA,
Linear and Quasilinear Equations of Parabolic Type,
Transl. Math. Monographs, Vol.
23,
Amer. Math. Soc., Providence, R.I.
1968. |
Zbl 0174.15403[17]
A. MAUGERI-
D. K. PALAGACHEV-
L. G. SOFTOVA,
Elliptic and Parabolic Equations with Discontinuous Coefficients,
Wiley-VCH, Berlin,
2000. |
MR 2260015 |
Zbl 0958.35002[18]
D. K. PALAGACHEV-
M. A. RAGUSA-
L. G. SOFTOVA,
Regular oblique derivative problem in Morrey spaces,
Electr. J. Diff. Equations,
2000 (
2000), No. 39, 1-17. |
MR 1764709 |
Zbl 1002.35033[19]
D.K. PALAGACHEV-
L.G. SOFTOVA,
Singular integral operators with mixed homogeneity in Morrey spaces C. R. Acad. Bulgare Sci.,
54 (
2001), No. 11, 11-16. |
MR 1878039 |
Zbl 0983.42008[20]
D. SARASON,
Functions of vanishing mean oscillation,
Trans. Amer. Math. Soc.,
207 (
1975), 391-405. |
MR 377518 |
Zbl 0319.42006[21]
L. G. SOFTOVA,
Regular oblique derivative problem for linear parabolic equations with VMO principal coefficients,
Manuscr. Math.,
103, No. 2 (
2000), 203-220. |
MR 1796316 |
Zbl 0963.35032[22]
L. G. SOFTOVA,
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. |
MR 1868228 |
Zbl 0990.35035[23]
L. G. SOFTOVA,
Parabolic equations with VMO coefficients in Morrey spaces,
Electr. J. Diff. Equations,
2001, No. 51 (
2001), 1-25. |
MR 1846667 |
Zbl 1068.35517