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: 0392-4041

Abstract

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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

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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

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