Let $A$ be a $2m$ order differential operator in a hermitian vector bundle $E$ over a compact riemannian manifold $\bar{\Omega }$ with boundary $\Gamma$ ; and denote by $A_B$ the realization defined by a normal differential boundary condition $B\rho u=0$ ($u\in H^{2m}\left(E\right)$, $\rho u=$ Cauchy data). We characterize, by an explicit condition on $A$ and $B$ near $\Gamma$, the realizations $A_B$ for which there exists an integro-differential sesquilinear form $a_B(u,\nu )$ on $H^m\left(E\right)$ such that $(Au,\nu )=a_B(u,\nu )$ on $D\left(A_B\right)$; moreover we show that these are exactly the realizations satisfying a weak semiboundedness...

Let $D$ be either the unit ball $B_1\left(0\right)$ or the half ball $B_1^{+}\left(0\right),$ let $f$ be a strictly positive and continuous function, and let $u$ and $\Omega \subset D$ solve the following overdetermined problem: $$\Delta u\left(x\right)={\chi }_{{}_{\Omega }}\left(x\right)f\left(x\right)\text{in}D,0\in \partial \Omega ,u=\left|\nabla u\right|=0\text{in}{\Omega }^c,$$ where ${\chi }_{{}_{\Omega }}$ denotes the characteristic function of $\Omega ,$ ${\Omega }^c$ denotes the set $D\setminus \Omega ,$ and the equation is satisfied in the sense of distributions. When $D=B_1^{+}\left(0\right),$ then we impose in addition that $$u\left(x\right)\equiv 0\text{on}\left\{(x^{\text{'}},x_n)\right|x_n=0\}.$$ We show that a fairly mild thickness assumption on ${\Omega }^c$ will ensure enough compactness on...

Let $P$ be a classical pseudodifferential operator of order $m$ on a paracompact $C^{\infty }$ manifold $X$. Let $p_m$ be the principal symbol and assume that $\Sigma =p_m^{-1}\left(0\right)$ is an involutive $C^{\infty }$ sub-manifold of $T^{*}X\setminus 0$, satisfying a certain transversality condition. We assume that $p_m$ vanishes exactly to order $M$ on $\Sigma$ and that the derivatives of order $M$ satisfy a certain condition, inspired from the Calderòn uniqueness theorem (usually empty when $M=2$). Suppose that a Levi condition is valid for the lower order symbols. If $u\in {𝒟}^{\text{'}}\left(X\right)$, $Pu\in C^{\infty }\left(X\right)$, then...

We establish a composition calculus for Fourier integral operators associated with a class of smooth canonical relations $C\subset (T^{*}X\setminus 0)\times (T^{*}Y\setminus 0)$. These canonical relations, which arise naturally in integral geometry, are such that $\pi$ : $C\to T^{*}Y$ is a Whitney fold and $\rho$ : $C\to T^{*}X$ is a blow-down mapping. If $A\in I^m\left(C\right)$, $B\in I^{m^{\text{'}}}(C^t)$, then $BA\in I^{m+m^{\text{'}},0}(\Delta ,\Lambda )$ a class of pseudodifferential operators with singular symbols. From this follows $L^2$ boundedness of $A$ with a loss of 1/4 derivative.