### A global continuation theorem for obtaining eigenvalues and bifurcation points

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Let $A$ be a closed set of $M\simeq {\mathbb{R}}^{n}$, whose conormai cones $x+{y}_{x}^{*}\left(A\right)$, $x\in A$, have locally empty intersection. We first show in §1 that $\text{dist}\left(x,A\right)$, $x\in M\setminus A$ is a ${C}^{1}$ function. We then represent the n microfunctions of ${\mathcal{C}}_{A|X}$, $X\simeq {\mathbb{C}}^{n}$, using cohomology groups of ${\mathcal{O}}_{X}$ of degree 1. By the results of § 1-3, we are able to prove in §4 that the sections of ${\left.{\mathcal{C}}_{A|X}\right|}_{{\dot{\pi}}^{-1}\left({x}_{0}\right)}$, ${x}_{0}\in \partial A$, satisfy the principle of the analytic continuation in the complex integral manifolds of ${\left\{H\left({\varphi}_{i}^{C}\right)\right\}}_{i=1,\mathrm{\dots},m}$, $\left\{{\varphi}_{i}\right\}$ being a base for the linear hull of ${\gamma}_{{x}_{0}}^{*}\left(A\right)$ in ${T}_{{x}_{0}}^{*}M$; in particular we get ${\left.{\mathrm{\Gamma}}_{A\times {}_{M}T{}^{*}{}_{M}X}\left({\mathcal{C}}_{A|X}\right)\right|}_{\partial A\times {}_{M}\dot{T}{}^{*}{}_{M}X}=0$. When $A$is a half space with ${C}^{\omega}$-boundary,...

If $P$ is a polynomial in ${\mathbf{R}}^{n}$ such that $1/P$ integrable, then the inverse Fourier transform of $1/P$ is a fundamental solution ${E}_{P}$ to the differential operator $P\left(D\right)$. The purpose of the article is to study the dependence of this fundamental solution on the polynomial $P$. For $n=1$ it is shown that ${E}_{P}$ can be analytically continued to a Riemann space over the set of all polynomials of the same degree as $P$. The singularities of this extension are studied.

We consider a linear model of interaction between a viscous incompressible fluid and a thin elastic structure located on a part of the fluid domain boundary, the other part being rigid. After having given an existence and uniqueness result for the direct problem, we study the question of approximate controllability for this system when the control acts as a normal force applied to the structure. The case of an analytic boundary has been studied by Lions and Zuazua in [9] where, in particular,...

Consider the nonlinear heat equation (E): ${u}_{t}-\Delta u={\left|u\right|}^{p-1}u+b{\left|\nabla u\right|}^{q}$. We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates $C\u2081{(T-t)}^{-1/(p-1)}{\le \left|\right|u\left(t\right)\left|\right|}_{\infty}\le C\u2082{(T-t)}^{-1/(p-1)}$. Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality ${u}_{t}-{u}_{xx}\ge {u}^{p}$. More general inequalities of the form ${u}_{t}-{u}_{xx}\ge f\left(u\right)$ with, for instance, $f\left(u\right)=(1+u)lo{g}^{p}(1+u)$ are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions of the ordinary...

We establish a Carleman type inequality for the subelliptic operator $\mathcal{L}={\Delta}_{z}+|x{|}^{2}{\partial}_{t}^{2}$ in ${\mathbb{R}}^{n+1}$, $n\ge 2$, where $z\in {\mathbb{R}}^{n}$, $t\in \mathbb{R}$. As a consequence, we show that $-\mathcal{L}+V$ has the strong unique continuation property at points of the degeneracy manifold $\left\{\right(0,t)\in {\mathbb{R}}^{n+1}|t\in \mathbb{R}\}$ if the potential $V$ is locally in certain ${L}^{p}$ spaces.

In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lamé system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lamé coefficients by a single measurement of solution over $(0,T)\times \omega $, where $T\>0$ is a sufficiently large time interval and a subdomain $\omega $ satisfies a non-trapping condition.

In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lamé system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lamé coefficients by a single measurement of solution over (0,T) x ω, where T > 0 is a sufficiently large time interval and a subdomain ω satisfies a non-trapping condition.

We derive Carleman type estimates with two large parameters for a general partial differential operator of second order. The weight function is assumed to be pseudo-convex with respect to the operator. We give applications to uniqueness and stability of the continuation of solutions and identification of coefficients for the Lamé system of dynamical elasticity with residual stress. This system is anisotropic and cannot be principally diagonalized, but it can be transformed into an "upper triangular"...

A necessary and sufficient condition for the continuous extendibility of a solution of the Neumann problem for the Laplace equation is given.

We consider a controllability problem for a beam, clamped at one boundary and free at the other boundary, with an attached piezoelectric actuator. By Hilbert Uniqueness Method (HUM) and new results on diophantine approximations, we prove that the space of exactly initial controllable data depends on the location of the actuator. We also illustrate these results with numerical simulations.