Search results for "PDE"
showing 10 items of 558 documents
Multi-parameter analysis of the obstacle scattering problem
2022
Abstract We consider the acoustic field scattered by a bounded impenetrable obstacle and we study its dependence upon a certain set of parameters. As usual, the problem is modeled by an exterior Dirichlet problem for the Helmholtz equation Δu + k 2 u = 0. We show that the solution u and its far field pattern u ∞ depend real analytically on the shape of the obstacle, the wave number k, and the Dirichlet datum. We also prove a similar result for the corresponding Dirichlet-to-Neumann map.
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…
Quantitative Runge Approximation and Inverse Problems
2017
In this short note we provide a quantitative version of the classical Runge approximation property for second order elliptic operators. This relies on quantitative unique continuation results and duality arguments. We show that these estimates are essentially optimal. As a model application we provide a new proof of the result from \cite{F07}, \cite{AK12} on stability for the Calder\'on problem with local data.
A sharp estimate for Neumann eigenvalues of the Laplace-Beltrami operator for domains in a hemisphere
2018
Here, we prove an isoperimetric inequality for the harmonic mean of the first [Formula: see text] non-trivial Neumann eigenvalues of the Laplace–Beltrami operator for domains contained in a hemisphere of [Formula: see text].
Dažādu kaloritātes aprēķināšanas metožu un reāli uzņemtās uztura enerģijas vērtēšana apdeguma traumas slimniekiem
2018
Problēmas būtība: Apdeguma traumas slimniekiem ir raksturīgs palielināts pieprasījums pēc enerģijas, šī iemesla dēļ svarīgi ir aprēķināt slimnieku uztura enerģijas prasības un sekot slimnieku reāli uzņemtajam uztura enerģijas daudzumam. Pētījuma mērķis: Vērtēt vai pastāv atšķirības starp dažādām nepieciešamās uztura enerģijas aprēķināšanas formulām apdeguma traumas slimniekiem, kā arī noskaidrot reāli uzņemto uztura enerģijas daudzumu un tā ietekmi uz klīnisko iznākumu. Izvirzītā hipotēze: Apdeguma traumas slimnieki Valsts Apdegumu centra intensīvās terapijas un Ķirurģiskās infekcijas klīnikas palātās uzņem mazāku kaloritāti, nekā tas būtu nepieciešams atbilstoši Harisa-Benedikta, Ireton-Jo…
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.
Systematic Study of a Library of PDMAEMA-Based, Superparamagnetic Nano-Stars for the Transfection of CHO-K1 Cells.
2017
The introduction of the DNA into mammalian cells remains a challenge in gene delivery, particularly in vivo. Viral vectors are unmatched in their efficiency for gene delivery, but may trigger immune responses and cause severe side-reactions. Non-viral vectors are much less efficient. Recently, our group has suggested that a star-shaped structure improves and even transforms the gene delivery capability of synthetic polycations. In this contribution, this effect was systematically studied using a library of highly homogeneous, paramagnetic nano-star polycations with varied arm lengths and grafting densities. Gene delivery was conducted in CHO-K1 cells, using a plasmid encoding a green fluore…
An Inverse Problem for the Relativistic Boltzmann Equation
2020
We consider an inverse problem for the Boltzmann equation on a globally hyperbolic Lorentzian spacetime $(M,g)$ with an unknown metric $g$. We consider measurements done in a neighbourhood $V\subset M$ of a timelike path $\mu$ that connects a point $x^-$ to a point $x^+$. The measurements are modelled by a source-to-solution map, which maps a source supported in $V$ to the restriction of the solution to the Boltzmann equation to the set $V$. We show that the source-to-solution map uniquely determines the Lorentzian spacetime, up to an isometry, in the set $I^+(x^-)\cap I^-(x^+)\subset M$. The set $I^+(x^-)\cap I^-(x^+)$ is the intersection of the future of the point $x^-$ and the past of th…
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…
Analysis of a viscoelastic phase separation model
2020
A new model for viscoelastic phase separation is proposed, based on a systematically derived conservative two-fluid model. Dissipative effects are included by phenomenological viscoelastic terms. By construction, the model is consistent with the second law of thermodynamics, and we study well-posedness of the model, i.e., existence of weak solutions, a weak-strong uniqueness principle, and stability with respect to perturbations, which are proven by means of relative energy estimates. A good qualitative agreement with mesoscopic simulations is observed in numerical tests.