Strong unique continuation for general elliptic equations in 2D

We prove that solutions to elliptic equations in two variables in divergence form, possibly non-selfadjoint and with lower order terms, satisfy the strong unique continuation property.Comment: 10 page

Unique continuation for solutions to the induced Cauchy-Riemann equations

AbstractLet M be a real infinitely differentiable closed hypersurface in X, a complex manifold of dimension n ⩾ 2, and let \̄t6M denote the induced Cauchy-Riemann operator on M. The problem considered in this paper is unique continuation for distribution solutions to the equation \̄t6Mu = 0 (these solutions are called CR distributions). In a local version of the problem it is shown that a CR distribution u in an open set U ⊂M which vanishes on one side of a C1 hypersurface S ⊂U which is noncharacteristic at a point p ϵ S necessarily vanishes in a neighborhood of p. If the CR distribution u is a continuous function on U, then it is only necessary to assume that u vanishes on S in order to prove that u vanishes in a neighborhood of p in M. It is also proved that if u is a CR distribution on M, then the boundary of the support of u is foliated by complex hypersurfaces. Thus a global unique continuation theorem is obtained by assuming that such a set is not contained in M

Wave equation with Robin condition, quantitative estimates of strong unique continuation at the boundary

The main result of the present paper consists in a quantitative estimate of unique continuation at the boundary for solutions to the wave equation. Such estimate is the sharp quantitative counterpart of the following strong unique continuation property: let $u$ be a solution to the wave equation that satisfies an homogeneous Robin condition on a portion $S$ of the boundary and the restriction of $u_{\mid S}$ on $S$ is flat on a segment $\{0\}\times J$ with $0\in S$ then $u_{\mid S}$ vanishes in a neighborhood of $\{0\}\times J$

Dual Variational Methods for a nonlinear Helmholtz system

This paper considers a pair of coupled nonlinear Helmholtz equations \begin{align*} -\Delta u - \mu u = a(x) \left( |u|^\frac{p}{2} + b(x) |v|^\frac{p}{2} \right)|u|^{\frac{p}{2} - 2}u, \end{align*} \begin{align*} -\Delta v - \nu v = a(x) \left( |v|^\frac{p}{2} + b(x) |u|^\frac{p}{2} \right)|v|^{\frac{p}{2} - 2}v \end{align*} on $\mathbb{R}^N$ where $\frac{2(N+1)}{N-1} < p < 2^\ast$. The existence of nontrivial strong solutions in $W^{2, p}(\mathbb{R}^N)$ is established using dual variational methods. The focus lies on necessary and sufficient conditions on the parameters deciding whether or not both components of such solutions are nontrivial.Comment: Published version. Contains minor revisions: Quote added, explanations on p.12 concerning F_{\mu\nu} = \infty, correction of exponent on p.1