Relevant bibliographies by topics / Geometric PDEs

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Geometric PDEs'

Author: Grafiati
Published: 14 March 2023

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Journal articles on the topic "Geometric PDEs"

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Shardlow, Tony. "Geometric ergodicity for stochastic pdes." Stochastic Analysis and Applications 17, no. 5 (January 1999): 857–69. http://dx.doi.org/10.1080/07362999908809639.

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Krantz, Steven G., and Vicentiu D. Radulescu. "Perspectives of Geometric Analysis in PDEs." Journal of Geometric Analysis 30, no. 2 (November 1, 2019): 1411. http://dx.doi.org/10.1007/s12220-019-00303-2.

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Vitagliano, Luca. "Characteristics, bicharacteristics and geometric singularities of solutions of PDEs." International Journal of Geometric Methods in Modern Physics 11, no. 09 (October 2014): 1460039. http://dx.doi.org/10.1142/s0219887814600391.

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Many physical systems are described by partial differential equations (PDEs). Determinism then requires the Cauchy problem to be well-posed. Even when the Cauchy problem is well-posed for generic Cauchy data, there may exist characteristic Cauchy data. Characteristics of PDEs play an important role both in Mathematics and in Physics. I will review the theory of characteristics and bicharacteristics of PDEs, with a special emphasis on intrinsic aspects, i.e. those aspects which are invariant under general changes of coordinates. After a basically analytic introduction, I will pass to a modern, geometric point of view, presenting characteristics within the jet space approach to PDEs. In particular, I will discuss the relationship between characteristics and singularities of solutions and observe that: "wave-fronts are characteristic surfaces and propagate along bicharacteristics". This remark may be understood as a mathematical formulation of the wave/particle duality in optics and/or quantum mechanics. The content of the paper reflects the three-hour mini-course that I gave at the XXII International Fall Workshop on Geometry and Physics, September 2–5, 2013, Évora, Portugal.
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Bezerra Júnior, Elzon C., João Vitor da Silva, and Gleydson C. Ricarte. "Geometric estimates for doubly nonlinear parabolic PDEs." Nonlinearity 35, no. 5 (April 21, 2022): 2334–62. http://dx.doi.org/10.1088/1361-6544/ac636e.

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Abstract In this manuscript, we establish C loc α , α θ regularity estimates for bounded weak solutions of a certain class of doubly degenerate evolution PDEs, whose simplest model case is given by ∂ u ∂ t − d i v ( m | u | m − 1 | ∇ u | p − 2 ∇ u ) = f ( x , t ) in Ω T ≔ Ω × ( 0 , T ) , where m ⩾ 1, p ⩾ 2 and f belongs to a suitable anisotropic Lebesgue space. Employing intrinsic scaling techniques and geometric tangential methods, we derive sharp regularity estimates for such models, which depend only on universal and compatibility parameters of the problem. In this scenario, our results are natural improvements for former ones in the context of nonlinear evolution PDEs with degenerate structure via a unified approach. As a consequence of our findings and approach, we address a Liouville type result for entire weak solutions of a related homogeneous problem with frozen coefficients and asymptotic estimates under a certain approximating regime, which may have their own mathematical interest. We also present examples of degenerate PDEs where our results can be applied.
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Tehseen, Naghmana, and Geoff Prince. "Integration of PDEs by differential geometric means." Journal of Physics A: Mathematical and Theoretical 46, no. 10 (February 21, 2013): 105201. http://dx.doi.org/10.1088/1751-8113/46/10/105201.

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NODA, TAKAHIRO, and KAZUHIRO SHIBUYA. "ON IMPLICIT SECOND-ORDER PDE OF A SCALAR FUNCTION ON A PLANE VIA DIFFERENTIAL SYSTEMS." International Journal of Mathematics 22, no. 07 (July 2011): 907–24. http://dx.doi.org/10.1142/s0129167x11007069.

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In the present paper, we study implicit second-order PDEs (i.e. partial differential equations) of single type for one unknown function of two variables. In particular, by using the theory of differential systems, we give a geometric characterization of PDEs which have a certain singularity. Moreover, we provide a new invariant of PDEs under contact transformations.
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Udriste, Constantin, and Ionel Tevy. "Geometric Dynamics on Riemannian Manifolds." Mathematics 8, no. 1 (January 3, 2020): 79. http://dx.doi.org/10.3390/math8010079.

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The purpose of this paper is threefold: (i) to highlight the second order ordinary differential equations (ODEs) as generated by flows and Riemannian metrics (decomposable single-time dynamics); (ii) to analyze the second order partial differential equations (PDEs) as generated by multi-time flows and pairs of Riemannian metrics (decomposable multi-time dynamics); (iii) to emphasise second order PDEs as generated by m-distributions and pairs of Riemannian metrics (decomposable multi-time dynamics). We detail five significant decomposed dynamics: (i) the motion of the four outer planets relative to the sun fixed by a Hamiltonian, (ii) the motion in a closed Newmann economical system fixed by a Hamiltonian, (iii) electromagnetic geometric dynamics, (iv) Bessel motion generated by a flow together with an Euclidean metric (created motion), (v) sinh-Gordon bi-time motion generated by a bi-flow and two Euclidean metrics (created motion). Our analysis is based on some least squares Lagrangians and shows that there are dynamics that can be split into flows and motions transversal to the flows.
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Boyer, A. L., C. Cardenas, F. Gibou, and D. Levy. "Segmentation for radiotherapy treatment planning using geometric PDEs." International Journal of Radiation Oncology*Biology*Physics 54, no. 2 (October 2002): 82–83. http://dx.doi.org/10.1016/s0360-3016(02)03200-5.

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SURI, JASJIT, DEE WU, LAURA REDEN, JIANBO GAO, SAMEER SINGH, and SWAMY LAXMINARAYAN. "MODELING SEGMENTATION VIA GEOMETRIC DEFORMABLE REGULARIZERS, PDE AND LEVEL SETS IN STILL AND MOTION IMAGERY: A REVISIT." International Journal of Image and Graphics 01, no. 04 (October 2001): 681–734. http://dx.doi.org/10.1142/s0219467801000402.

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Partial Differential Equations (PDEs) have dominated image processing research recently. The three main reasons for their success are: first, their ability to transform a segmentation modeling problem into a partial differential equation framework and their ability to embed and integrate different regularizers into these models; second, their ability to solve PDEs in the level set framework using finite difference methods; and third, their easy extension to a higher dimensional space. This paper is an attempt to survey and understand the power of PDEs to incorporate into geometric deformable models for segmentation of objects in 2D and 3D in still and motion imagery. The paper first presents PDEs and their solutions applied to image diffusion. The main concentration of this paper is to demonstrate the usage of regularizers in PDEs and level set framework to achieve the image segmentation in still and motion imagery. Lastly, we cover miscellaneous applications such as: mathematical morphology, computation of missing boundaries for shape recovery and low pass filtering, all under the PDE framework. The paper concludes with the merits and the demerits of PDEs and level set-based framework for segmentation modeling. The paper presents a variety of examples covering both synthetic and real world images.
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Hirica, Iulia, Constantin Udriste, Gabriel Pripoae, and Ionel Tevy. "Least Squares Approximation of Flatness on Riemannian Manifolds." Mathematics 8, no. 10 (October 13, 2020): 1757. http://dx.doi.org/10.3390/math8101757.

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The purpose of this paper is fourfold: (i) to introduce and study the Euler–Lagrange prolongations of flatness PDEs solutions (best approximation of flatness) via associated least squares Lagrangian densities and integral functionals on Riemannian manifolds; (ii) to analyze some decomposable multivariate dynamics represented by Euler–Lagrange PDEs of least squares Lagrangians generated by flatness PDEs and Riemannian metrics; (iii) to give examples of explicit flat extremals and non-flat approximations; (iv) to find some relations between geometric least squares Lagrangian densities.
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Dissertations / Theses on the topic "Geometric PDEs"

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Ndiaye, Cheikh Birahim. "Geometric PDEs on compact Riemannian manifolds." Doctoral thesis, SISSA, 2007. http://hdl.handle.net/20.500.11767/4088.

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In this thesis, some nonlinear problems coming from conformal geometry and physics, namely the prescription of Q-curvature, T-curvature ones and the generalized 2×2 Toda system are studied. We study also the existence of extremal functions of two Moser-Trudinger type inequalities (which is a common feature of those problems) due to Fontana[40] and Chang-Yang[23].
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Burovskiy, Pavel Andreevich. "Second order quasilinear PDEs in 3D : integrability, classification and geometric aspects." Thesis, Loughborough University, 2009. https://dspace.lboro.ac.uk/2134/26691.

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CHERMISI, MILENA. "Crystalline flow of planar partitions and a geometric approach for systems of PDEs." Doctoral thesis, Università degli Studi di Roma "Tor Vergata", 2006. http://hdl.handle.net/2108/202647.

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La presente tesi tratta due argomenti distinti. Il Capitolo 1 e il Capitolo 2 riguardano problemi di evoluzione di interfacce nel piano. Nel Capitolo 1 viene considerata l’evoluzione di un materiale policristallino con tre (o più) fasi, in presenza di un’anisotropia cristallina (pari) ϕo la cui linea di livello 1, Fϕ :={ϕo ≤1} (Frank diagram), è un poligono regolare di n lati. La funzione duale ϕ : R2 →R definita da ϕ(ξ) := sup{ξ·η : ϕo(η)≤1}´e anch’essa un’anisotropia cristallina e Wϕ := {ϕ ≤ 1} è detta Wulff shape. In particolare, viene studiato il moto per curvatura cristallina di triodi elementari, ossia speciali reti piane di curve che sono frontiere regolari di insiemi rappresentanti tre fasi distinte di un materiale. Un triodo elementare è formato dall’unione di tre curve Lipschitziane, le interfacce, che si intersecano in un unico punto detto giunzione tripla. Ogni interfaccia è l’unione di un segmento di lunghezza finita e di una semiretta che riproduce due lati consecutivi della Wulff shape Wϕ. Viene analizzata l’esitenza locale e globale e la stabilità del flusso. Si dimostra l’esistenza locale di un unico flusso regolare stabile a partire da un dato iniziale regolare stabile: se n, il numero dei lati della Wulff shapeWϕ, è un multiplo di 6 allora il flusso è globale e converge a un flusso omotetico per t →+∞. L’analisi del comportamento del flusso per tempi grandi richiede lo studio della stabilità. La stabilità è l’ingrediente che assicura che nessun segmento si sviluppa dalla giunzione tripla durante il flusso. In generale, il flusso può diventare instabile in un tempo finito: se ciò accade e tutte le lunghezze dei segmenti finiti sono strettamente positive per tale tempo,è possibile costruire un flusso regolare per tempi successivi aggiungendo in corrispondenza della giunzione tripla in una delle tre interfacce un segmento infinitesimo opportuno (o addirittura un arco di curva a curvatura cristallina nulla). ´E anche possibile che durante il flusso uno dei tre segmenti scompaia in un tempo finito. In tal caso, in tale tempo il campo vettoriale di Cahn-Hoffman ha un salto di discontinuità e ai tempi successivi la giunzione tripla si muove traslando lungo la semiretta adiacente. Ognuno di questi flussi ha la proprietà che tutte le curvature cristalline rimangono limitate (persino se un segmento appare o scompare). ´E importante sottolineare che Taylor aveva già predetto la nascita di nuovi segmenti dalla giunzione tripla (senza però dimostrarlo). Viene inoltre considerato il flusso per curvatura cristalina di una partizione regolare stabile formata da due triodi elementari adiacenti. Vengono discussi alcuni esempi di situazioni di colasso che portano a cambi di topologia, come ad esempio la collisione di due giunzioni triple. Questi esempi (come anche il risultato di esistenza per tempi piccoli) mostrano uno dei vantaggi del flusso per curvatura cristallino rispetto, ad esempio, all’usuale moto per curvatura: calcoli espliciti possono essere fatti, e nel caso di non unicità, è possibile confrontare le energie delle diverse evoluzioni (difficile nel caso euclideo). Nel Capitolo 2 viene introdotta, usando la teoria delle funzioni a variazione limitata a valori in S1, la sfera diR2, una nuova classe di funzionali energia definiti su partizioni. Attraverso la variazione prima del funzionale energia, viene fornito un nuovo modello per l’evoluzione di interfacce che parzialmente estende quello introdotto nel Capitolo 1 e che consiste in un problema di frontiera libera definito sulle funzioni a variazione limitata a valori in S1. Questo modello è legato all’evoluzione di materiali policristallini dove è consentito alla Wulff shape di ruotare. Assumendo l’esitenza locale del flusso, si dimostra che durante il flusso curve chiuse convesse rimangono convesse e curve chiuse embedded rimangono embedded. Il secondo argomento della tesi è trattato nel Capitolo 3: l’obiettivo è quello di estendere il metodo delle linee di livello a sistemi di equazioni differenziali alle derivate parziali. Il metodo che viene proposto è consistente con la precedente ricerca portata avanti da Evans per l’equazione del calore e da Giga e Sato per equazioni di Hamilton-Jacobi. Il nostro approccio segue una costruzione geometrica che è legate alla nozione di barriera introdotta da De Giorgi. L’idea principale è quella di forzare un principio di confronto tra varietà di diversa codimensione e richiedere che ogni sottolivello di una soluzione dell’equazione per le linee di livello, detta level set equation, sia una barriera per i grafici di soluzioni del corrispondente sistema. Tale metodo ben si applica a una classe di sistemi di equazioni quasi-lineari del primo ordine. Viene fornita la level set equation associata ad opportuni sitemi di leggi di conservazione del primo ordine, al flusso per curvatura media di una varietà di codimensione arbitraria e a sitemi di equazioni di reazione-diffusione. Infine, viene calcolata la level set equation associata al sistema soddisfatto dalle parametrizzazioni di curve piane che si muovono per curvatura.
The present thesis deals with two different subjects. Chapter 1 and Chapter 2 concern interfaces evolution problems in the plane. In Chapter 1 I consider the evolution of a polycrystalline material with three (or more) phases, in presence of for an even crystalline anisotropy ϕo whose one-sublevel set Fϕ := {ϕo ≤ 1} (the Frank diagram) is a regular polygon of n sides. The dual function ϕ : R2 → R defined by ϕ(ξ) := sup{ξ ·η : ϕo(η) ≤ 1} is crystalline too and Wϕ := {ϕ ≤ 1} is called the Wulff shape. I am particularly interested in the motion by crystalline curvature of special planar networks called elementary triods, namely a regular three-phase boundary given by the union of three Lipschitz curves, the interfaces, intersecting at a point called triple junction. Each interface is the union of a segment of finite length and a half-line, reproducing two consecutive sides of Wϕ. I analyze local and global existence and stability of the flow. I prove that there exists, locally in time, a unique stable regular flow starting from a stable regular initial datum. I show that if n, the number of sides of Wϕ, is a multiple of 6 then the flow is global and converge to a homothetic flow as t → +∞. The analysis of the long time behavior requires the study of the stability. Stability is the ingredient that ensures that no additional segments develop at the triple junction during the flow. In general, the flow may become unstable at a finite time: if this occurs and none of the segments desappears, it is possible to construct a regular flow at subsequent times by adding an infinitesimal segment (or even an arc with zero crystalline curvature) at the triple junction. I also show that a segment may desappear. In such a case, the Cahn-Hoffman vector field Nmin has a jump discontinuity and the triple junction translates along the remaining adjacent half-line at subsequent times. Each of these flows has the property that all crystalline curvatures remain bounded (even if a segment appears or disappears). I want to stress that Taylor already predicted the appearance of new edges from a triple junction. I also consider the crystalline curvature flow starting from a stable ϕ-regular partition formed by two adjacent elementary triods. I discuss some examples of collapsing situations that lead to changes of topology, such as for instance the collision of two triple junctions. These examples (as well as the local in time existence result) show one of the advantages of crystalline flows with respect, for instance, to the usual mean curvature flow: explicit computations can be performed to some extent, and in case of nonuniqueness, a comparison between the energies of different evolutions (difficult in the euclidean case) can be made. In Chapter 2 we introduce, using the theory of S1-valued functions of bounded variations, a class of energy functionals defined on partitions and we produce, through the first variation, a new model for the evolution of interfaces which partially extends the one in Chapter 1 and which consists of a free boundary problem defined on S1-valued functions of bounded variation. This model is related to the evolution of polycrystals where the Wulff shape is allowed to rotate. Assuming the local existence of the flow, we show convexity preserving and embeddedness preserving properties. The second subject of the thesis is considered in Chapter 3 where we aim to extend the level set method to systems of PDEs. The method we propose is consistent with the previous research pursued by Evans for the heat equation and by Giga and Sato for Hamilton-Jacobi equations. Our approach follows a geometric construction related to the notion of barriers introduced by De Giorgi. The main idea is to force a comparison principle between manifolds of different codimension and require each sub-level of a solution of the level set equation to be a barrier for the graph of a solution of the corresponding system. We apply the method for a class of systems of first order quasi-linear equations. We compute the level set equation associated with suitable first order systems of conservation laws, with the mean curvature flow of a manifold of arbitrary codimension and with systems of reaction-diffusion equations. Finally, we provide a level set equation associated with the parametric curvature flow of planar curves.
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Benatti, Luca. "Monotonicity Formulas in Nonlinear Potential Theory and their geometric applications." Doctoral thesis, Università degli studi di Trento, 2022. http://hdl.handle.net/11572/346959.

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In the setting of Riemannian manifolds with nonnegative Ricci curvature, we provide geometric inequalities as consequences of the Monotonicity Formulas holding along the flow of the level sets of the p-capacitary potential. The work is divided into three parts. (1) In the first part, we describe the asymptotic behaviour of the p-capactitary potential in a natural class of Riemannian manifolds. (2) The second part is devoted to the proof of our Monotonicity-Rigidity Theorems. (3) In the last part, we apply the Monotonicity Theorems to obtain geometric inequalities, focusing on the Extended Minkowski Inequality.
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Mascellani, Giovanni. "Fourth-order geometric flows on manifolds with boundary." Doctoral thesis, Scuola Normale Superiore, 2017. http://hdl.handle.net/11384/85715.

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Nakauchi, Gene. "Analytical and numerical results for a curvature-driven geometric flow rule." Thesis, Queensland University of Technology, 2019. https://eprints.qut.edu.au/127335/1/Gene_Nakauchi_Thesis.pdf.

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This research studies a particular curvature-driven geometric flow rule in the plane using techniques from differential geometry, computational mathematics, and formal asymptotics. The flow rule is a combination of the well-studied curve shortening flow, which is governed by a parabolic system of partial differential equations, and the Eikonal equation, which is governed by a hyperbolic system. The physical motivations for considering our model include propagating fire fronts and phase separation. The focus is on a variety of mathematical problems related to the flow rule, such as the explicit form of travelling wave solutions, linear stability, self-intersection, singularity formation, and the extinction problem for convex curves.
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Junca, Stéphane. "Oscillating waves for nonlinear conservation laws." Habilitation à diriger des recherches, Université Nice Sophia Antipolis, 2013. http://tel.archives-ouvertes.fr/tel-00845827.

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The manuscript presents my research on hyperbolic Partial Differential Equations (PDE), especially on conservation laws. My works began with this thought in my mind: ''Existence and uniqueness of solutions is not the end but merely the beginning of a theory of differential equations. The really interesting questions concern the behavior of solutions.'' (P.D. Lax, The formation and decay of shock waves 1974). To study or highlight some behaviors, I started by working on geometric optics expansions (WKB) for hyperbolic PDEs. For conservation laws, existence of solutions is still a problem (for large data, $L^\infty$ data), so I early learned method of characteristics, Riemann problem, $BV$ spaces, Glimm and Godunov schemes, \ldots In this report I emphasize my last works since 2006 when I became assistant professor. I use geometric optics method to investigate a conjecture of Lions-Perthame-Tadmor on the maximal smoothing effect for scalar multidimensional conservation laws. With Christian Bourdarias and Marguerite Gisclon from the LAMA (Laboratoire de \\ Mathématiques de l'Université de Savoie), we have obtained the first mathematical results on a $2\times2$ system of conservation laws arising in gas chromatography. Of course, I tried to put high oscillations in this system. We have obtained a propagation result exhibiting a stratified structure of the velocity, and we have shown that a blow up occurs when there are too high oscillations on the hyperbolic boundary. I finish this subject with some works on kinetic équations. In particular, a kinetic formulation of the gas chromatography system, some averaging lemmas for Vlasov equation, and a recent model of a continuous rating system with large interactions are discussed. Bernard Rousselet (Laboratoire JAD Université de Nice Sophia-Antipolis) introduced me to some periodic solutions related to crak problems and the so called nonlinear normal modes (NNM). Then I became a member of the European GDR: ''Wave Propagation in Complex Media for Quantitative and non Destructive Evaluation.'' In 2008, I started a collaboration with Bruno Lombard, LMA (Laboratoire de Mécanique et d'Acoustique, Marseille). We details mathematical results and challenges we have identified for a linear elasticity model with nonlinear interfaces. It leads to consider original neutral delay differential systems.
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Jevnikar, Aleks. "Variational aspects of Liouville equations and systems." Doctoral thesis, SISSA, 2015. http://hdl.handle.net/20.500.11767/4847.

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Cekić, Mihajlo. "The Calderón problem for connections." Thesis, University of Cambridge, 2017. https://www.repository.cam.ac.uk/handle/1810/267829.

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This thesis is concerned with the inverse problem of determining a unitary connection $A$ on a Hermitian vector bundle $E$ of rank $m$ over a compact Riemannian manifold $(M, g)$ from the Dirichlet-to-Neumann (DN) map $\Lambda_A$ of the associated connection Laplacian $d_A^*d_A$. The connection is to be determined up to a unitary gauge equivalence equal to the identity at the boundary. In our first approach to the problem, we restrict our attention to conformally transversally anisotropic (cylindrical) manifolds $M \Subset \mathbb{R}\times M_0$. Our strategy can be described as follows: we construct the special Complex Geometric Optics solutions oscillating in the vertical direction, that concentrate near geodesics and use their density in an integral identity to reduce the problem to a suitable $X$-ray transform on $M_0$. The construction is based on our proof of existence of Gaussian Beams on $M_0$, which are a family of smooth approximate solutions to $d_A^*d_Au = 0$ depending on a parameter $\tau \in \mathbb{R}$, bounded in $L^2$ norm and concentrating in measure along geodesics when $\tau \to \infty$, whereas the small remainder (that makes the solution exact) can be shown to exist by using suitable Carleman estimates. In the case $m = 1$, we prove the recovery of the connection given the injectivity of the $X$-ray transform on $0$ and $1$-forms on $M_0$. For $m > 1$ and $M_0$ simple we reduce the problem to a certain two dimensional $\textit{new non-abelian ray transform}$. In our second approach, we assume that the connection $A$ is a $\textit{Yang-Mills connection}$ and no additional assumption on $M$. We construct a global gauge for $A$ (possibly singular at some points) that ties well with the DN map and in which the Yang-Mills equations become elliptic. By using the unique continuation property for elliptic systems and the fact that the singular set is suitably small, we are able to propagate the gauges globally. For the case $m = 1$ we are able to reconstruct the connection, whereas for $m > 1$ we are forced to make the technical assumption that $(M, g)$ is analytic in order to prove the recovery. Finally, in both approaches we are using the vital fact that is proved in this work: $\Lambda_A$ is a pseudodifferential operator of order $1$ acting on sections of $E|_{\partial M}$, whose full symbol determines the full Taylor expansion of $A$ at the boundary.
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Ugail, Hassan, M. I. G. Bloor, and M. J. Wilson. "Manipulation of PDE surfaces using an interactively defined parameterisation." Elsevier, 1999. http://hdl.handle.net/10454/2669.

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No
Manipulation of PDE surfaces using a set of interactively defined parameters is considered. The PDE method treats surface design as a boundary-value problem and ensures that surfaces can be defined using an appropriately chosen set of boundary conditions and design parameters. Here we show how the data input to the system, from a user interface such as the mouse of a computer terminal, can be efficiently used to define a set of parameters with which to manipulate the surface interactively in real time.
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Books on the topic "Geometric PDEs"

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Gursky, Matthew J., Ermanno Lanconelli, Andrea Malchiodi, Gabriella Tarantello, Xu-Jia Wang, and Paul C. Yang. Geometric Analysis and PDEs. Edited by Sun-Yung Alice Chang, Antonio Ambrosetti, and Andrea Malchiodi. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-01674-5.

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Ponce, Augusto C. Elliptic PDEs, measures and capacities: From the Poisson equation to nonlinear Thomas-Fermi problems. Zürich: European Mathematical Society, 2016.

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C.I.M.E. Summer School (2007 : Cetraro, Italy), ed. Geometric analysis and PDEs: Lectures given at the C.I.M.E. summer school held in Cetraro, Italy June 11-16, 2007. Berlin: Springer-Verlag, 2009.

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Hyperbolic partial differential equations and geometric optics. Providence, R.I: American Mathematical Society, 2012.

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Citti, Giovanna, Maria Manfredini, Daniele Morbidelli, Sergio Polidoro, and Francesco Uguzzoni, eds. Geometric Methods in PDE’s. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-02666-4.

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Geometric asymptotics for nonlinear PDE. Providence, R.I: American Mathematical Society, 2001.

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Geometry of PDEs and mechanics. Singapore: World Scientific, 1996.

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Ferone, Vincenzo, Tatsuki Kawakami, Paolo Salani, and Futoshi Takahashi, eds. Geometric Properties for Parabolic and Elliptic PDE's. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-73363-6.

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Gazzola, Filippo, Kazuhiro Ishige, Carlo Nitsch, and Paolo Salani, eds. Geometric Properties for Parabolic and Elliptic PDE's. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-41538-3.

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Magnanini, Rolando, Shigeru Sakaguchi, and Angelo Alvino, eds. Geometric Properties for Parabolic and Elliptic PDE's. Milano: Springer Milan, 2013. http://dx.doi.org/10.1007/978-88-470-2841-8.

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Book chapters on the topic "Geometric PDEs"

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Gursky, Matthew J. "PDEs in Conformal Geometry." In Geometric Analysis and PDEs, 1–33. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-01674-5_1.

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Lanconelli, Ermanno. "Heat Kernels in Sub-Riemannian Settings." In Geometric Analysis and PDEs, 35–61. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-01674-5_2.

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Malchiodi, Andrea. "Concentration of Solutions for Some Singularly Perturbed Neumann Problems." In Geometric Analysis and PDEs, 63–115. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-01674-5_3.

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Tarantello, Gabriella. "On Some Elliptic Problems in the Study of Selfdual Chern-Simons Vortices." In Geometric Analysis and PDEs, 117–75. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-01674-5_4.

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Wang, Xu-Jia. "The k-Hessian Equation." In Geometric Analysis and PDEs, 177–252. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-01674-5_5.

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Yang, Paul. "Minimal Surfaces in CR Geometry." In Geometric Analysis and PDEs, 253–73. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-01674-5_6.

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Ugail, Hassan. "Elliptic PDEs for Geometric Design." In Partial Differential Equations for Geometric Design, 31–45. London: Springer London, 2011. http://dx.doi.org/10.1007/978-0-85729-784-6_4.

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Calogero, F. "Universal Integrable Nonlinear PDEs." In Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, 109–14. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-011-2082-1_11.

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Macià, Fabricio. "Geometric Control of Eigenfunctions of Schrödinger Operators." In Research in PDEs and Related Fields, 151–68. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-14268-0_5.

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Chan, Tony F., and Jianhong Shen. "Bayesian Inpainting Based on Geometric Image Models." In Recent Progress in Computational and Applied PDES, 73–99. Boston, MA: Springer US, 2002. http://dx.doi.org/10.1007/978-1-4615-0113-8_5.

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Conference papers on the topic "Geometric PDEs"

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Toutain, Matthieu, Abderrahim Elmoataz, and Olivier Lezoray. "Geometric PDEs on Weighted Graphs for Semi-supervised Classification." In 2014 13th International Conference on Machine Learning and Applications (ICMLA). IEEE, 2014. http://dx.doi.org/10.1109/icmla.2014.43.

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Baniamerian, A., and K. Khorasani. "Fault detection and isolation of dissipative parabolic PDEs: Finite-dimensional geometric approach." In 2012 American Control Conference - ACC 2012. IEEE, 2012. http://dx.doi.org/10.1109/acc.2012.6315006.

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Zhao, Yiming, and Jason D. Dykstra. "Vibrations of Curved and Twisted Beam." In ASME 2015 Dynamic Systems and Control Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/dscc2015-9880.

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This paper studies the vibration of beams in 3D space with arbitrary shape. Based on results from differential geometry of curves, a set of beam vibration dynamics equations is developed, comprising six partial differential equations (PDE). The beam dynamics equations account for both the in-plane and out-of-plane beam vibrations simultaneously. In addition, the equations explicitly capture the coupling between different vibration mode types, which occur when the beam exhibits geometric irregularities such as bending, torsion, and twisting. The proposed beam dynamics equations are solved numerically. Comparison between experimental results and numerical results obtained by solving the PDEs proposed in this paper shows a good match for in-plane and out-of-plane curved beam vibrations.
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Marquez, Ricardo, and Michael Modest. "Implementation of the PN-Approximation for Radiative Heat Transfer on OpenFOAM." In ASME 2013 Heat Transfer Summer Conference collocated with the ASME 2013 7th International Conference on Energy Sustainability and the ASME 2013 11th International Conference on Fuel Cell Science, Engineering and Technology. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/ht2013-17556.

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This work presents an OpenFOAM implementation of the PN approximation for radiative heat transfer, including higher orders P3, P5, and P7. Also described is a procedure which enables the sequential numerical computations of the coupled partial differential equations (PDEs) by re-expressing the boundary conditions in matrix form so that individual boundary conditions can be associated with each PDE. The implementation of the software programs are verified with derived analytical solutions for 1-D slabs with constant and variable properties, and are also tested with various orientations in order to demonstrate the geometric invariance properties of the 3-dimensional PN formulation. A few examples taken from the literature are also considered in this work and could be taken as benchmark solutions for the PN approximations.
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Banerjee, Abhishek, and Ameeya Kumar Nayak. "Assessment and Prediction of EOF Mixing in Binary Electrolytes." In ASME 2017 Fluids Engineering Division Summer Meeting. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/fedsm2017-69524.

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A two dimensional simulation is made to analyse the mixing enhancement due to surface roughness and geometric modulation in a sufficiently long rectangular nano-channel filled with electrolyte solutions of different concentrations. Geometric modulation is made by mounting non-conducting rectangular blocks on the bottom wall of the channel. An overpotential patch is placed on the upper wall of each block to create surface heterogeneity. Based on a finite volume staggered grid approach, the flow characteristics and mixing efficiency are discussed by a complete numerical solution of coupled nonlinear set of PDEs involving Nernst-Planck equation for ion distribution, Navier-Stokes equation for velocity components and Maxwells equation for potential distribution. A linear pressure drop is observed above the overpotential region which creates a recirculating zone. Mixing efficiency is improved with increasing vortex strength which is enhanced by decreasing EDL (electric double layer) thickness and increasing overpotential patch strength.
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Li, H. Z. "Variational problems and PDEs in affine differential geometry." In PDEs, Submanifolds and Affine Differential Geometry. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2005. http://dx.doi.org/10.4064/bc69-0-1.

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Belkhelfa, Mohamed, Franki Dillen, and Jun-ichi Inoguchi. "Surfaces with parallel second fundamental form in Bianchi-Cartan-Vranceanu spaces." In PDEs, Submanifolds and Affine Differential Geometry. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2002. http://dx.doi.org/10.4064/bc57-0-5.

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Djorić, Mirjana, and Masafumi Okumura. "CR submanifolds of maximal CR dimension in complex manifolds." In PDEs, Submanifolds and Affine Differential Geometry. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2002. http://dx.doi.org/10.4064/bc57-0-6.

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Gálvez, J. A., and A. Martínez. "Hypersurfaces with constant curvature in Rn+1." In PDEs, Submanifolds and Affine Differential Geometry. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2002. http://dx.doi.org/10.4064/bc57-0-7.

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Gollek, Hubert. "Natural algebraic representation formulas for curves in C3." In PDEs, Submanifolds and Affine Differential Geometry. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2002. http://dx.doi.org/10.4064/bc57-0-8.

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Reports on the topic "Geometric PDEs"

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Moreno, Giovanni. A Natural Geometric Framework for the Space of Initial Data of Nonlinear PDEs. GIQ, 2012. http://dx.doi.org/10.7546/giq-13-2012-245-257.

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Yau, Stephen S. PDE, Differential Geometric and Algebraic Methods in Nonlinear Filtering. Fort Belvoir, VA: Defense Technical Information Center, January 1993. http://dx.doi.org/10.21236/ada260967.

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Yau, Stephen S. PDE, Differential Geometric and Algebraic Methods for Nonlinear Filtering. Fort Belvoir, VA: Defense Technical Information Center, February 1996. http://dx.doi.org/10.21236/ada310330.

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Tannenbaum, Allen R. Geometric PDE's and Invariants for Problems in Visual Control Tracking and Optimization. Fort Belvoir, VA: Defense Technical Information Center, January 2005. http://dx.doi.org/10.21236/ada428955.

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Yau, Stephen S. T. PDE, Differential Geometric, Algebraic, Wavelet and Parallel Computation Methods in Nonlinear Filtering. Fort Belvoir, VA: Defense Technical Information Center, June 2003. http://dx.doi.org/10.21236/ada415460.

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