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Register a new userPoint electrode problems in piecewise smooth plane domains
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Authors
Otto Seiskari
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Conductivity equation is studied in piecewise smooth plane domains and with measure-valued current patterns (Neumann boundary values). This allows one to extend the recently introduced concept of bisweep data to piecewise smooth domains, which yields a new partial data result for Calderon inverse conductivity problem. It is also shown that bisweep data are (up to a constant scaling factor) the Schwartz kernel of the relative Neumann-to-Dirichlet map. A numerical method for reconstructing the supports of inclusions from discrete bisweep data is also presented.
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Given any $f$ a locally finitely piecewise affine homeomorphism of $Omega subset rn$ onto $Delta subset rn$ in $W^{1,p}$, $1leq p < infty$ and any $epsilon >0$ we construct a smooth injective map $tilde{f}$ such that $|f-tilde{f}|_{W^{1,p}(Omega,rn)} < epsilon$.
Let $c$ be a piecewise smooth wave speed on $mathbb R^n$, unknown inside a domain $Omega$. We are given the solution operator for the scalar wave equation $(partial_t^2-c^2Delta)u=0$, but only outside $Omega$ and only for initial data supported outside $Omega$. Using our recently developed scattering control method, we prove that piecewise smooth wave speeds are uniquely determined by this map, and provide a reconstruction formula. In other words, the wave imaging problem is solvable in the piecewise smooth setting under mild conditions. We also illustrate a separate method, likewise constructive, for recovering the locations of interfaces in broken geodesic normal coordinates using scattering control.
Two frameworks that have been used to characterize reflected diffusions include stochastic differential equations with reflection and the so-called submartingale problem. We introduce a general formulation of the submartingale problem for (obliquely) reflected diffusions in domains with piecewise C^2 boundaries and piecewise continuous reflection vector fields. Under suitable assumptions, we show that well-posedness of the submartingale problem is equivalent to existence and uniqueness in law of weak solutions to the corresponding stochastic differential equation with reflection. Our result generalizes to the case of reflecting diffusions a classical result due to Stroock and Varadhan on the equivalence of well-posedness of martingale problems and well-posedness of weak solutions of stochastic differential equations in d-dimensional Euclidean space. The analysis in the case of reflected diffusions in domains with non-smooth boundaries is considerably more subtle and requires a careful analysis of the behavior of the reflected diffusion on the boundary of the domain. In particular, the equivalence can fail to hold when our assumptions are not satisfied. The equivalence we establish allows one to transfer results on reflected diffusions characterized by one approach to reflected diffusions analyzed by the other approach. As an application, we provide a characterization of stationary distributions of a large class of reflected diffusions in convex polyhedral domains.
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In this paper, we prove the existence of nontrivial unbounded domains $Omegasubsetmathbb{R}^{n+1},ngeq1$, bifurcating from the straight cylinder $Btimesmathbb{R}$ (where $B$ is the unit ball of $mathbb{R}^n$), such that the overdetermined elliptic problem begin{equation*} begin{cases} Delta u +f(u)=0 &mbox{in $Omega$, } u=0 &mbox{on $partialOmega$, } partial_{ u} u=mbox{constant} &mbox{on $partialOmega$, } end{cases} end{equation*} has a positive bounded solution. We will prove such result for a very general class of functions $f: [0, +infty) to mathbb{R}$. Roughly speaking, we only ask that the Dirichlet problem in $B$ admits a nondegenerate solution. The proof uses a local bifurcation argument.
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