×

\(W^{1,2}_ p\) solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients. (English) Zbl 0816.35045

From the introduction: Let \(L\) be a linear parabolic operator of the form \(Lu= u_ t- a_{ij} (x) u_{x_ i^ \prime x_ j^ \prime}\) where \(x= (x', t)= (x_ 1^ \prime,\dots, x_ n^ \prime, t)\in \mathbb{R}^{n+1}\). The principal part of the operator is symmetric and uniformly elliptic. We are interested in solutions to the Cauchy- Dirichlet problem \[ Lu=f \text{ in } Q_ T, \quad u=0 \text{ on } \partial \Omega\times (0,T), \quad u(x',0) =0 \text{ in } \Omega, \tag{1} \] with \(f\in L^ p (Q_ T)\), \(1<p< +\infty\). When the coefficients \(a_{ij} (x)\) are at least uniformly continuous, existence and uniqueness results together with a priori \(W_ p^{1,2} (Q_ T)\) estimates are well known. Here we consider operators with coefficients in the Sarason’s class VMO, i.e. the closure in the (parabolic) BMO seminorm of uniformly continuous functions. Therefore we allow some discontinuities in the coefficients. We prove the same results as in the uniformly continuous case, i.e. we first prove a priori interior and boundary \(W_ p^{1,2}\) estimates and then, through the usual procedure, we obtain existence and uniqueness for a solution to (1).

MSC:

35K20 Initial-boundary value problems for second-order parabolic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
35R05 PDEs with low regular coefficients and/or low regular data
PDFBibTeX XMLCite
Full Text: DOI