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Register a new userElliptic equations with VMO a, b$,in L_{d}$, and c$,in L_{d/2}$
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Authors
N.V. Krylov
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We consider elliptic equations with operators $L=a^{ij}D_{ij}+b^{i}D_{i}-c$ with $a$ being almost in VMO, $bin L_{d}$ and $cin L_{q}$, $cgeq0$, $d>qgeq d/2$. We prove the solvability of $Lu=fin L_{p}$ in bounded $C^{1,1}$-domains, $1<pleq q$, and of $lambda u-Lu=f$ in the whole space for any $lambda>0$. Weak uniqueness of the martingale problem associated with such operators is also obtained.
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In subdomains of $mathbb{R}^{d}$ we consider uniformly elliptic equations $Hbig(v( x),D v( x),D^{2}v( x), xbig)=0$ with the growth of $H$ with respect to $|Dv|$ controlled by the product of a function from $L_{d}$ times $|Dv|$. The dependence of $H$ on $x$ is assumed to be of BMO type. Among other things we prove that there exists $d_{0}in(d/2,d)$ such that for any $pin(d_{0},d)$ the equation with prescribed continuous boundary data has a solution in class $W^{2}_{p,text{loc}}$. Our results are new even if $H$ is linear.
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