arXiv subjects
Publications and source records attributed to M. A. Perelmuter.
We construct, for every $d\ge 5$, a nonnegative radial potential $V\in L^\infty_{\mathrm{loc}}(\mathbb{R}^d\setminus\{0\})\cap L^1(\mathbb{R}^d)$ for which $V(-Δ\dotplus V)^{-1}$ is not bounded on $L^2(\mathbb{R}^d)$.
We consider weak solutions of the second-order elliptic equation $\operatorname{div} (\mathbf{M}\nabla u) = \operatorname{div}\mathbf{F}$ in $Ω\subset {\mathbb R}^d$ with Dirichlet boundary conditions, where $\mathbf{M}$ is a uniformly elliptic real-valued symmetric matrix $\mathbf{M}:Ω\rightarrow \mathbb{R}^{d\times d}$ such that $\frac{1}{K} |ξ|^2 \leq\langle \mathbf{M}ξ,ξ\rangle \leq K |ξ|^2, \forall ξ\in \mathbb{R}^d, K > 1$. We prove that $\|\nabla u\|_p \leq C\|\mathbf{F}\|_p$ for any $p\in \left(\frac{26K-16}{13K-3}, \frac{26K-16}{13K-13}\right)$.
Let $A$ be a dissipative operator on a Banach space with a dense domain. It is proved that $A$ has a quasi-dissipative extension (possibly in an enlarged Banach space) which generates a quasi-contractive $C_0$-semigroup. \par This gives a positive answer to the question posed by P.R.Chernoff and H.F.Trotter.
We give $L^p$ estimates for the second derivatives of weak solutions to the Dirichlet problem for equation $\Div(\mathbf{A}\nabla u) = f$ in $Ω\subset \mathbb{R}^d$ with Sobolev coefficients. In particular, for $f\in L^2(Ω) \bigcap L^s(Ω)$ $$\|Δu\|_{2} \leq \begin{cases} c_1\|f\|_2 + c_2 \|\nabla \mathbf{A}\|_q^2\|f\|_s, & \text{if } 1 < s < d/2, \frac{1}{2}=\frac{2}{q}+ \frac{1}{s} - \frac{2}{d}\\ c_1\|f\|_2 + c_2 \|\nabla \mathbf{A}\|_4^2\|f\|_s, & \text{if } s > d/2 \end{cases}.$$