An $L^2$-estimate of the finite element error is proved for a Dirichlet and a Neumann boundary value problem on a three-dimensional, prismatic and non-convex domain that is discretized by an anisotropic tetrahedral mesh. To this end, an approximation error estimate for an interpolation operator that is preserving the Dirichlet boundary conditions is given. The challenge for the Neumann problem is the proof of a local interpolation error estimate for functions from a weighted Sobolev space.

We prove endpoint estimates for operators given by oscillating spectral multipliers on Riemannian manifolds with $C^{\infty }$-bounded geometry and nonnegative Ricci curvature.

We consider the Schrödinger operators $-\Delta +V\left(x\right)$ in ${ℝ}^n$ where the nonnegative potential $V\left(x\right)$ belongs to the reverse Hölder class $B_q$ for some $q\ge n/2$. We obtain the optimal $L^p$ estimates for the operators $(-\Delta +V{)}^{i\gamma },{\nabla }^2(-\Delta +V{)}^{-1},\nabla (-\Delta +V{)}^{-1/2}$ and $\nabla (-\Delta +V{)}^{-1}$ where $\gamma \in ℝ$. In particular we show that $(-\Delta +V{)}^{i\gamma }$ is a Calderón-Zygmund operator if $V\in B_{n/2}$ and $\nabla (-\Delta +V{)}^{-1/2},\nabla (-\Delta +V{)}^{-1}\nabla$ are Calderón-Zygmund operators if $V\in B_n$.

We prove an inequality of the type$$\parallel |x{|}^{r_f}{\parallel }_{L^p({\mathbf{R}}^n)}\le c(n,p,q,r)\parallel |x{|}^{\tau +\mu }\Delta f{\parallel }_{L^q({\mathbf{R}}^n)}.$$This is then used to derive the unique continuation property for the differential inequality $\left|\Delta f\right(x\left)\right|\le \left|v\right(x\left)\right|\left|f\right(x\left)\right|$ under suitable local integrability assumptions on the function $v$.

Let Ω be an open subset of ${ℝ}^d$ with 0 ∈ Ω. Furthermore, let $H_{\Omega }=-{\sum }_{i,j=1}^d{\partial }_i{c}_{ij}{\partial }_j$ be a second-order partial differential operator with domain $C_c^{\infty }\left(\Omega \right)$ where the coefficients $c_{ij}\in W_{loc}^{1,\infty }\left(\Omega ̅\right)$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=\left(c_{ij}\right)$ satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ∈ Ω. If ${\int }_0^{\infty }ds{s}^{d/2}{e}^{-\lambda \mu \left(s\right)²}<\infty$ for some λ > 0 where $\mu \left(s\right)={\int }_0^{s}dtc{\left(t\right)}^{-1/2}$ then we establish that $H_{\Omega }$ is L₁-unique, i.e. it has a unique L₁-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique L₂-extension which generates a submarkovian semigroup. Moreover...

We solve the L²-data Dirichlet boundary problem for a weighted form Laplacian in the unit Euclidean ball. The solution is given explicitly as a sum of four series.