Search results for "PDE"
showing 8 items of 558 documents
Muckenhoupt $A_p$-properties of distance functions and applications to Hardy-Sobolev -type inequalities
2017
Let $X$ be a metric space equipped with a doubling measure. We consider weights $w(x)=\operatorname{dist}(x,E)^{-\alpha}$, where $E$ is a closed set in $X$ and $\alpha\in\mathbb R$. We establish sharp conditions, based on the Assouad (co)dimension of $E$, for the inclusion of $w$ in Muckenhoupt's $A_p$ classes of weights, $1\le p<\infty$. With the help of general $A_p$-weighted embedding results, we then prove (global) Hardy-Sobolev inequalities and also fractional versions of such inequalities in the setting of metric spaces.
Convergence of dynamic programming principles for the $p$-Laplacian
2018
We provide a unified strategy to show that solutions of dynamic programming principles associated to the $p$-Laplacian converge to the solution of the corresponding Dirichlet problem. Our approach includes all previously known cases for continuous and discrete dynamic programming principles, provides new results, and gives a convergence proof free of probability arguments.
Duality theory for multi-marginal optimal transport with repulsive costs in metric spaces
2018
In this paper we extend the duality theory of the multi-marginal optimal transport problem for cost functions depending on a decreasing function of the distance (not necessarily bounded). This class of cost functions appears in the context of SCE Density Functional Theory introduced in "Strong-interaction limit of density-functional theory" by M. Seidl.
Game-Theoretic Approach to Hölder Regularity for PDEs Involving Eigenvalues of the Hessian
2021
AbstractWe prove a local Hölder estimate for any exponent $0<\delta <\frac {1}{2}$ 0 < δ < 1 2 for solutions of the dynamic programming principle $$ \begin{array}{@{}rcl@{}} u^{\varepsilon} (x) = \sum\limits_{j=1}^{n} \alpha_{j} \underset{\dim(S)=j}{\inf} \underset{|v|=1}{\underset{v\in S}{\sup}} \frac{u^{\varepsilon} (x + \varepsilon v) + u^{\varepsilon} (x - \varepsilon v)}{2} \end{array} $$ u ε ( x ) = ∑ j = 1 n α j inf dim ( S ) = j sup v ∈ S | v | = 1 u ε ( x + ε v ) + u ε ( x − ε v ) 2 with α1,αn > 0 and α2,⋯ ,αn− 1 ≥ 0. The proof is based on a new coupling idea from game theory. As an application, we get the same regularity estimate for viscosity solutions of the PDE $…
Fixed angle inverse scattering for sound speeds close to constant
2021
We study the fixed angle inverse scattering problem of determining a sound speed from scattering measurements corresponding to a single incident wave. The main result shows that a sound speed close to constant can be stably determined by just one measurement. Our method is based on studying the linearized problem, which turns out to be related to the acoustic problem in photoacoustic imaging. We adapt the modified time-reversal method from [P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems 25 (2009), 075011] to solve the linearized problem in a stable way, and use this to give a local uniqueness result for the nonlinear inverse problem.
Akūtas nieru mazspējas riska faktoru analīze smagas termiskas traumas slimniekiem
2022
Par spīti arvien plašākām diagnostikas, prevencijas un ārstēšanas iespējām, akūts nieru bojājums joprojām ir nozīmīga smagas termiskas traumas komplikācija ar augstu incidenci un mirstību. Darba mērķis: Identificēt riska faktorus, kas izraisa akūtu nieru mazspēju smagas apdeguma traumas pacientiem (TBSA ≥ 20%). Metodes un materiāli: Pētījums veikts RAKUS stacionāra “Biķernieki” Valsts Apdegumu centrā. Dati iegūti retrospektīvi analizējot 1997 pacientu medicīniskās slimības vēstures, laika posmā no 2011. – 2021. gadam, kas atlasītas, izmantojot SSK-10 klasifikāciju pēc koda T29 – vairāku un neprecizētu ķermeņa apvidu termiski un ķīmiski apdegumi, pētījumā iekļaujot 248 pacientus, vecumā no 1…
Free boundary methods and non-scattering phenomena
2021
We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of the obstacle has to be very thin near the point. These facts are proved by invoking results from t…
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…