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Register a new userLinear and fully nonlinear elliptic equations with $L_{d}$-drift
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Authors
N.V. Krylov
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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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We present some results concerning the solvability of linear elliptic equations in bounded domains with the main coefficients almost in VMO, the drift and the free terms in Morrey classes containing $L_{d}$, and bounded zeroth order coefficient. We prove that the second-order derivatives of solutions are in a local Morrey class containing $W^{2}_{p,loc}$. Actually, the exposition is given for fully nonlinear equations and encompasses the above mentioned results, which are new even if the main part of the equation is just the Laplacian.
In this note, we obtain a version of Aleksandrovs maximum principle when the drift coefficients are in Morrey spaces, which contains $L_d$, and when the free term is in $L_p$ for some $p<d$.
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.
In this paper we deal with the heat equation with drift in $L_{d+1}$. Basically, we prove that, if the free term is in $L_{q}$ with high enough $q$, then the equation is uniquely solvable in a rather unusual class of functions such that $partial_{t}u, D^{2}uin L_{p}$ with $p<d+1$ and $Duin L_{q}$.
Under structural conditions which are almost optimal, we derive a quantitative version of boundary estimate then prove existence of solutions to Dirichlet problem for a class of fully nonlinear elliptic equations on Hermitian manifolds.
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