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Register a new userAleksandrovs estimates for elliptic equations with drift in Morrey spaces containing $L_{d}$
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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 $p<d$.
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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.
We present new conditions on the drift of the Morrey type with mixed norms allowing us to obtain Aleksandrov type estimates of potentials of time inhomogeneous diffusion processes in spaces with mixed norms and, for instance, in $L_{d_{0}+1}$ with $d_{0}<d$.
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.
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.
We consider Dirichlet problems for linear elliptic equations of second order in divergence form on a bounded or exterior smooth domain $Omega$ in $mathbb{R}^n$, $n ge 3$, with drifts $mathbf{b}$ in the critical weak $L^n$-space $L^{n,infty}(Omega ; mathbb{R}^n )$. First, assuming that the drift $mathbf{b}$ has nonnegative weak divergence in $L^{n/2, infty }(Omega )$, we establish existence and uniqueness of weak solutions in $W^{1,p}(Omega )$ or $D^{1,p}(Omega )$ for any $p$ with $n = n/(n-1)< p < n$. By duality, a similar result also holds for the dual problem. Next, we prove $W^{1,n+varepsilon}$ or $W^{2, n/2+delta}$-regularity of weak solutions of the dual problem for some $varepsilon, delta >0$ when the domain $Omega$ is bounded. By duality, these results enable us to obtain a quite general uniqueness result as well as an existence result for weak solutions belonging to $bigcap_{p< n }W^{1,p}(Omega )$. Finally, we prove a uniqueness result for exterior problems, which implies in particular that (very weak) solutions are unique in both $L^{n/(n-2),infty}(Omega )$ and $L^{n,infty}(Omega )$.
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