Search results for "differentiaaliyhtälö"

Optimality of Increasing Stability for an Inverse Boundary Value Problem

2021

In this work we study the optimality of increasing stability of the inverse boundary value problem (IBVP) for the Schrödinger equation. The rigorous justification of increasing stability for the IBVP for the Schrödinger equation were established by Isakov [Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), pp. 631--640] and by Isakov et al. [Inverse Problems and Applications, Contemp. Math. 615, American Math Society, Providence, RI, 2014, pp. 131--141]. In [Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), pp. 631--640] and [Inverse Problems and Applications, Contemp. Math. 615, American Math Society, Providence, RI, 2014, pp. 131--141], the authors showed that the stability of this IBVP increases …

Unique continuation results for certain generalized ray transforms of symmetric tensor fields

2022

Let $I_{m}$ denote the Euclidean ray transform acting on compactly supported symmetric $m$-tensor field distributions $f$, and $I_{m}^{*}$ be its formal $L^2$ adjoint. We study a unique continuation result for the normal operator $N_{m}=I_{m}^{*}I_{m}$. More precisely, we show that if $N_{m}$ vanishes to infinite order at a point $x_0$ and if the Saint-Venant operator $W$ acting on $f$ vanishes on an open set containing $x_0$, then $f$ is a potential tensor field. This generalizes two recent works of Ilmavirta and M\"onkk\"onen who proved such unique continuation results for the ray transform of functions and vector fields/1-forms. One of the main contributions of this work is identifying t…

On superconvergence techniques

1984

Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations

2021

We study various partial data inverse boundary value problems for the semilinear elliptic equation $\Delta u+ a(x,u)=0$ in a domain in $\mathbb R^n$ by using the higher order linearization technique introduced in [LLS 19, FO19]. We show that the Dirichlet-to-Neumann map of the above equation determines the Taylor series of $a(x,z)$ at $z=0$ under general assumptions on $a(x,z)$. The determination of the Taylor series can be done in parallel with the detection of an unknown cavity inside the domain or an unknown part of the boundary of the domain. The method relies on the solution of the linearized partial data Calder\'on problem [FKSU09], and implies the solution of partial data problems fo…

Sinin ja kosinin erilaiset määrittelytavat

2012

Local regularity estimates for general discrete dynamic programming equations

2022

We obtain an analytic proof for asymptotic H\"older estimate and Harnack's inequality for solutions to a discrete dynamic programming equation. The results also generalize to functions satisfying Pucci-type inequalities for discrete extremal operators. Thus the results cover a quite general class of equations.

Localization and dimension estimation of attractors in the Glukhovsky-Dolzhansky system

2016

The stress-strain state and stabilization of viscoelastoplastic, imperfect moving web continuum

2014

On Limits at Infinity of Weighted Sobolev Functions

2022

We study necessary and sufficient conditions for a Muckenhoupt weight $w \in L^1_{\mathrm{loc}}(\mathbb R^d)$ that yield almost sure existence of radial, and vertical, limits at infinity for Sobolev functions $u \in W^{1,p}_{\mathrm{loc}}(\mathbb R^d,w)$ with a $p$-integrable gradient $|\nabla u|\in L^p(\mathbb R^d,w)$. The question is shown to subtly depend on the sense in which the limit is taken. First, we fully characterize the existence of radial limits. Second, we give essentially sharp sufficient conditions for the existence of vertical limits. In the specific setting of product and radial weights, we give if and only if statements. These generalize and give new proofs for results of…

Recovering a variable exponent

2021

We consider an inverse problem of recovering the non-linearity in the one dimensional variable exponent $p(x)$-Laplace equation from the Dirichlet-to-Neumann map. The variable exponent can be recovered up to the natural obstruction of rearrangements. The main technique is using the properties of a moment problem after reducing the inverse problem to determining a function from its $L^p$-norms.