7 problems

Let $\Omega\subset\mathbb{R}^3$ be a bounded domain diffeomorphic to the ball with smooth boundary, and consider the stability estimate asserting that sufficiently small…

Let $\rho(k)$ denote the stability constant for the Helmholtz problem at wavenumber $k$, and let $C$ and $k_0$ be positive constants. The best known general upper bound is…

Let $n$ be the dimension, let $f_0\colon\mathbb{R}^n\to\mathbb{R}_+$ be even log-concave, and suppose that for sufficiently small $\varepsilon>0$ its functional Blaschke–Santaló de…

Let $Omega y$ be a bounded Lipschitz domain in $R^d$, let $a=1$, and let $cin L^{infty}(Omega)$ satisfy … For $omegageq omega00$, consider the heterogeneous Helmholtz solution…

Logarithmic stability conjecture. For small $\varepsilon$ and large $N$, the error satisfies

Consider the inverse boundary value problem for the Helmholtz equation in the continuous setting, with stability constants depending on the parameter $N$. The full data case uses t…

Let $u$ solve … where $Omega$ is a bounded domain in $\mathbb{R}^N$ with boundary $Gamma$ of class $C^{2,tau}$, $0<tau<1$, and $f$ is locally Lipschitz continuous. Let $r_e$ and…