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

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

Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 3, pp. 667-683.

Abstract (VO)

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.

Riassunto

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.

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