Dissertations / Theses on the topic 'Fully nonlinear equation'
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Terrone, Gabriele. "Singular Perturbation and Homogenization Problems in Control Theory, Differential Games and fully nonlinear Partial Differential Equations." Doctoral thesis, Università degli studi di Padova, 2008. http://hdl.handle.net/11577/3426271.
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In this thesis we address different topics related to homogenization of first and second order fully nonlinear PDEs, essentially of Hamilton--Jacobi type, and more generally to singular perturbation in optimal control problems and differential games, in the light of the viscosity solution theory.
We take into account a singularly perturbed control systems (i.e. a system where the state variables evolve with two different time scales), both in the deterministic and in the stochastic setting, and the related first and second order Hamilton-Jacobi equations. A first part of the work is devoted to order reduction procedures: the goal of such procedures is to obtain, as the perturbation parameter tends to zero, a system where only the slow variables appear. The construction of the limit dynamics relies on the asymptotic behavior of the fast variables of the original system.
We use limiting relaxed controls, i.e. suitably defined Radon probability measures to average the fast part of the controlled dynamics. We give - both in the deterministic and in the stochastic framework - representation formulae for the effective Hamiltonian in terms of limiting relaxed controls. This allow a control interpretation of the limiting dynamics. As an application of these reduction procedures, we study the propagation of fronts moving with normal velocity depending on the position and undergoing fast oscillations.
In the second part of the work we study asymptotic controllability properties of a deterministic singularly perturbed systems and of the limit system. We prove first that, under suitable assumptions, the weak lower semilimit of Lyapunov functions of a singularly perturbed system is a lower semicontinuous Lyapunov function for the limiting system.
Furthermore, we also prove that the asymptotic controllability to the origin of the (smaller) limit system is enough to infer asymptotic controllability of the slow part of the (larger) perturbed system. More precisely, perturbing a Lyapunov pair for the limit dynamics, we construct a Lyapunov pair for the original system.
The third and last part of the thesis concerns homogenization of non-coercive Hamilton-Jacobi equations with oscillating Hamiltonian and initial data. We take into account a rather general class of Hamiltonians convex in some gradient variables and concave with respect to the others. In particular it is shown that for some of these equations homogenization does not take place, in contrast with the usual coercive case. Sufficient conditions for homogenization are provided involving the structure of the running cost and the initial data.
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ALESSANDRONI, ROBERTA. "Evolution of hypersurfaces by curvature functions." Doctoral thesis, Università degli Studi di Roma "Tor Vergata", 2008. http://hdl.handle.net/2108/661.
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Consideriamo un'ipersuperficie liscia di ℝⁿ⁺¹, con n≥2, e la sua evoluzione secondo una classe di flussi geometrici. La velocità di questi flussi ha direzione normale alla superficie e il modulo è una funzione simmetrica delle curvature principali. Inizialmente mostriamo alcune proprietà generali di questi flussi e calcoliamo l'equazione di evoluzione per una generica funzione omogenea delle curvature principali.
In particolare applichiamo il flusso con velocità S=(H/(logH)), dove H è la curvatura media a meno di una costante, ad una superficie con curvatura media positiva per ottenere delle stime di convessità. Usando solamente il principio del massimo dimostriamo che, su un limite di riscalamenti delle superfici che si evolvono vicino alla singolarità, la parte negativa della curvatura scalare tende a zero.
La parte successiva è dedicata allo studio di un'ipersuperficie convessa che si evolve secondo potenze della curvatura scalare: S=R^{p}, con p>1/2. Si dimostra che se la superficie iniziale soddisfa delle stime di "pinching" sulle curvature principali allora si contrae ad un punto in tempo finito e la forma delle superfici che si evolvono approssima sempre più quella di una sfera. In questo caso il grado di omogeneità, strettamente maggiore di uno, permette di concludere la dimostrazione della convergenza ad un "punto rotondo" tramite il solo principio del massimo, evitando l'uso di stime integrali. Viene anche costruito un esempio di superficie convessa che forma una singolarità di tipo "neck pinching".
Infine studiamo il caso di un grafico intero su ℝⁿ con crescita al più lineare all'infinito e mostriamo che un grafico che si evolve secondo un qualsiasi flusso nella classe considerata rimane un grafico. Inoltre dimostriamo un risultato di esistenza per tempi lunghi per i flussi con velocità S=R^{p} con p≥1/2 e descriviamo delle soluzioni esplicite per grafici a simmetria di rotazione.We consider a smooth n-dimensional hypersurface of ℝⁿ⁺¹, with n≥2, and its evolution by a class of geometric flows. The speed of these flows has normal direction with respect to the surface and its modulus S is a symmetric function of the principal curvatures. We show some general properties of these flows and compute the evolution equation for any homogeneous function of principal curvatures. Then we apply the flow with speed S=(H/(logH)), where H is the mean curvature plus a constant, to a mean convex surface to prove some convexity estimates. Using only the maximum principle we prove that the negative part of the scalar curvature tends to zero on a limit of rescalings of the evolving surfaces near a singularity. The following part is dedicated to the study of a convex initial manifold moving by powers of scalar curvature: S=R^{p}, with p>1/2. We show that if the initial surface satisfies a pinching estimate on the principal curvatures then it shrinks to a point in finite time and the shape of the evolving surfaces approaches the one of a sphere. Since the homogeneity degree of this speed is strictly greater than one, the convergence to a "round point" can be proved using just the maximum principle, avoiding the integral estimates. Then we also construct an example of a non convex surface forming a neck pinching singularity. Finally we study the case of an entire graph over ℝⁿ with at most linear growth at infinity. We show that a graph evolving by any flow in the considered class remains a graph. Moreover we prove a long time existence result for flows where the speed is S=R^{p} with p≥1/2 and describe some explicit solutions in the rotationally symmetric case.
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Chen, Huyuan. "Fully nonlinear elliptic equations and semilinear fractional equations." Tesis, Universidad de Chile, 2014. http://www.repositorio.uchile.cl/handle/2250/115532.
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Doctor en Ciencias de la Ingeniería, Mención Modelación MatemáticaEsta tesis esta dividida en seis partes. La primera parte está dedicada a probar propiedades de Hadamard y teoremas del tipo de Liouville para soluciones viscosas de ecuaciones diferenciales parciales elípticas completamente no lineales con término gradiente \begin{equation}\label{eq06-10-13 1} \mathcal{M}^{-}(|x|,D^2u)+\sigma(|x|)|Du|+f(x,u)\leq 0,\quad \ x\in\Omega, \end{equation} donde $\Omega=\mathbb{R}^N$ o un dominio exterior, las funciones $\sigma:[0,\infty)\to\mathbb{R}$ y $f:\Omega\times (0,\infty)\to (0,\infty)$ son continuas las cuales satisfacen algunas condiciones extras. En la segunda parte se estudia la existencia de soluciones que explotan en la frontera para ecuaciones elípticas fraccionarias semilineales \begin{equation}\label{eq06-10-13 2} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^{\alpha} u(x)+|u|^{p-1}u(x)=h(x),\quad & x\in\Omega,\\[2mm] \phantom{ (-\Delta)^{\alpha} u(x)+|u|^{p-1}} u(x)=0,\quad & x\in\bar\Omega^c,\\[2mm] \phantom{ (-\Delta)^{\alpha} \ } \lim_{x\in\Omega, x\to\partial\Omega}u(x)=+\infty, \end{array} \end{equation} donde $p>1$, $\Omega$ es un dominio abierto acotado $C^2$ de $\mathbb{R}^N(N\geq2)$, el operador $(-\Delta)^{\alpha}$ con $\alpha\in(0,1)$ es el Laplaciano fraccionario y $h:\Omega\to\R$ es una función continua la cual satisface algunas condiciones extras. Por otra parte, analizamos la unicidad y el comportamiento asimptótico de soluciones al problema (\ref{eq06-10-13 2}). El objetivo principal de la tercera parte es investigar soluciones positivas para ecuaciones elípticas fraccionarias \begin{equation}\label{eq06-10-13 3} \arraycolsep=1pt \begin{array}{lll} (-\Delta)^{\alpha} u(x)+|u|^{p-1}u(x)=0,\quad & x\in\Omega\setminus\mathcal{C},\\[2mm] \phantom{ (-\Delta)^{\alpha} u(x)+|u|^{p-1}} u(x)=0,\quad & x\in\Omega^c,\\[2mm] \phantom{ (-\Delta) \ } \lim_{x\in\Omega\setminus\mathcal{C}, \ x\to\mathcal{C}}u(x)=+\infty, \end{array} \end{equation} donde $p>1$ y $\Omega$ es un dominio abierto acotado $C^2$ de $\mathbb{R}^N(N\geq2)$, $\mathcal{C}\subset \Omega$ es el frontera de dominio $G$ que es $C^2$ y satisface $\bar G\subset\Omega$. Consideramos la existencia de soluciones positivas para el problema (\ref{eq06-10-13 3}). Mas aún, analizamos la unicidad, el comportamiento asimptótico y la no existencia al problema (\ref{eq06-10-13 3}). En la cuarta parte, estudiamos la existencia de soluciones débiles de (F) $ (-\Delta)^\alpha u+g(u)=\nu $ en un dominio $\Omega$ abierto acotado $C^2$ de $\R^N (N\ge2)$ el cual se desvanece en $\Omega^c$, donde $\alpha\in(0,1)$, $\nu$ es una medida de Radon y $g$ es una función no decreciente satisfaciendo algunas hipótesis extras. Cuando $g$ satisface una condición de integrabilidad subcrítica, probamos la existencia y unicidad de una solución débil para el problema (F) para cualquier medida. En el caso donde $\nu$ es una masa de Dirac, caracterizamos el comportamiento asimptótico de soluciones a (F). Asimismo, cuando $g(r)=|r|^{k-1}r$ con $k$ supercrítico, mostramos que una condición de absoluta continuidad de la medida con respecto a alguna capacidad de Bessel es una condición necesaria y suficiente para que (F) sea resuelta. El propósito de la quinta parte es investigar soluciones singulares débiles y fuertes de ecuaciones elípticas fraccionarias semilineales. Sean $p\in(0,\frac{N}{N-2\alpha})$, $\alpha\in(0,1)$, $k>0$ y $\Omega\subset \R^N(N\geq2)$ un dominio abierto acotado $C^2$ conteniendo a $0$ y $\delta_0$ la masa de Dirac en $0$, estudiamos que la solución débil de $(E)_k$ $ (-\Delta)^\alpha u+u^p=k\delta_0 $ en $\Omega$ la cual se desvanece en $\Omega^c$ es una solución débil singular de $(E^*)$ $ (-\Delta)^\alpha u+u^p=0 $ en $\Omega\setminus\{0\}$ con el mismo dato externo. Por otra parte, estudiamos el límite de soluciones débiles de $(E)_k$ cuando $k\to\infty$. Para $p\in(0, 1+\frac{2\alpha}{N}]$, el límite es infinito en $\Omega$. Para $p\in(1+\frac{2\alpha}N,\frac{N}{N-2\alpha})$, el límite es una solución fuertemente singular de $(E^*)$. Finalmente, en la sexta parte estudiamos la ecuación elíptica fraccionaria semilineal (E1) $(-\Delta)^\alpha u+\epsilon g(|\nabla u|)=\nu $ en un dominio $\Omega$ abierto acotado $C^2$ de $\R^N (N\ge2)$, el cual se desvanece en $\Omega^c$, donde $\epsilon=\pm1$, $\alpha\in(1/2,1)$, $\nu$ es una medida de Radon y $g:\R_+\mapsto\R_+$ es una funci\'on continua. Probamos la existencia de soluciones débiles para el problema (E1) cuando $g$ es subcrítico. Además, el comportamiento asimptótico y la unicidad de soluciones son descritas cuando $\epsilon=1$, $\nu$ es una masa de Dirac y $g(s)=s^p$ con $p\in(0,\frac)$.
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Sui, Zhenan. "On Some Classes of Fully Nonlinear Partial Differential Equations." The Ohio State University, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=osu1429640709.
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Liu, Weian, Yin Yang, and Gang Lu. "Viscosity solutions of fully nonlinear parabolic systems." Universität Potsdam, 2002. http://opus.kobv.de/ubp/volltexte/2008/2621/.
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In this paper, we discuss the viscosity solutions of the weakly coupled systems of fully nonlinear second order degenerate parabolic equations and their Cauchy-Dirichlet problem. We prove the existence, uniqueness and continuity of viscosity solution by combining Perron's method with the technique of coupled solutions. The results here generalize those in [2] and [3].
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Rang, Marcus [Verfasser]. "Regularity results for nonlocal fully nonlinear elliptic equations / Marcus Rang." Bielefeld : Universitätsbibliothek Bielefeld, 2013. http://d-nb.info/103805026X/34.
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Lai, Mijia. "Fully nonlinear flows and Hessian equations on compact Kahler manifolds." Diss., University of Iowa, 2011. https://ir.uiowa.edu/etd/1010.
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In this thesis, we will study a class of fully nonlinear flows on Kähler manifolds. This family of flows generalizes the previously studied J-flow. We use the quotients of elementary symmetric polynomials or log of them to construct the flow. We obtain a necessary and sufficient condition in terms of positivity of certain cohomology class to guarantee the convergence of the flow. The corresponding limit metric gives rise to a critical metric satisfying a Hessian type equation on the manifold. We shall also discuss several geometric applications of our main result.
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Sotoudeh, Zahra. "Nonlinear static and dynamic analysis of beam structures using fully intrinsic equations." Diss., Georgia Institute of Technology, 2011. http://hdl.handle.net/1853/41179.
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Beams are structural members with one dimension much larger than the other two. Examples of beams include propeller blades, helicopter rotor blades, and high aspect-ratio aircraft wings in aerospace engineering; shafts and wind turbine blades in mechanical engineering; towers, highways and bridges in civil engineering; and DNA modeling in biomedical engineering. Beam analysis includes two sets of equations: a generally linear two-dimensional problem over the cross-sectional plane and a nonlinear, global one-dimensional analysis.
This research work deals with a relatively new set of equations for one-dimensional beam analysis, namely the so-called fully intrinsic equations. Fully intrinsic equations comprise a set of geometrically exact, nonlinear, first-order partial differential equations that is suitable for analyzing initially curved and twisted anisotropic beams. A fully intrinsic formulation is devoid of displacement and rotation variables, making it especially attractive because of the absence of singularities, infinite-degree nonlinearities, and other undesirable features associated with finite rotation variables.
In spite of the advantages of these equations, using them with certain boundary conditions presents significant challenges. This research work will take a broad look at these challenges of modeling various boundary conditions when using the fully intrinsic equations. Hopefully it will clear the path for wider and easier use of the fully intrinsic equations in future research.
This work also includes application of fully intrinsic equations in structural analysis of joined-wing aircraft, different rotor blade configuration and LCO analysis of HALE aircraft.
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Zhang, Wei [Verfasser]. "Asymptotics for subcritical fully nonlinear equations with isolated singularities / Wei Zhang." Hannover : Gottfried Wilhelm Leibniz Universität Hannover, 2018. http://d-nb.info/1172414165/34.
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Coutinho, Francisco Edson Gama. "Universal moduli of continuity for solutions to fully nonlinear elliptic equations." Universidade Federal do CearÃ, 2013. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=11427.
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CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel SuperiorIn this paper we provide a universal solution for continuity module in the direction of the viscosity of fully nonlinear elliptic equations considering properties of the function f integrable in different situations. Established inner estimate for the solutions of these equations based on some conditions the norm of the function f. To obtain regularity in solutions of these inhomogeneous equations and coefficients of variables we use a method of compactness, which consists essentially of approximating solutions of inhomogeneous equations for a solution of a homogeneous equation in order to "inherit" the regularity that those equations possess.
Neste trabalho fornecemos mÃdulo de continuidade universal para soluÃÃes, no sentido da viscosidade,de equaÃÃes elÃpticas totalmente nÃo lineares, considerando propriedades de integrabilidade da funÃÃo f em diferentes situaÃÃes. Estabelecemos estimativa interior para as soluÃÃes dessas equaÃÃes baseadas em algumas condiÃÃes da norma da funÃÃo f. Para se obter regularidade nas soluÃÃes dessas equacÃes nÃo homogÃneas e de coeficientes variÃveis usamos um mÃtodo de compacidade, o qual consiste, essencialmente, em aproximar soluÃÃes de equaÃÃes nÃo homogÃneas por uma soluÃÃo de uma equaÃÃo homogÃnea com o objetivo de âherdarâ a regularidade que essas equaÃÃes possuem.
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Guo, Sheng. "On Neumann Problems for Fully Nonlinear Elliptic and Parabolic Equations on Manifolds." The Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1571696906482925.
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Galise, Giulio. "Maximum principles, entire solutions and removable singularities of fully nonlinear second order equations." Doctoral thesis, Universita degli studi di Salerno, 2013. http://hdl.handle.net/10556/928.
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2011 - 2012This PhD thesis is devoted to some qualitative aspect of viscosity solutions of nonlinear second order elliptic partial di erential equations of the form F(x; u(x);Du(x);D2u(x)) = f(x)... [edited by author]
XI n.s.
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Prazeres, Disson Soares dos. "Improved regularity estimates in nonlinear elliptic equations." Universidade Federal do CearÃ, 2014. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=13536.
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CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel SuperiorConselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico
In this work we establish local regularity estimates for at solutions to non-convex fully nonlinear elliptic equations and we study cavitation type equations modeled within coef- icients bounded and measurable.
Neste trabalho estabelecemos estimativas de regularidade local para soluÃÃes "flat" de equaÃÃes elÃpticas totalmente nÃo-lineares nÃo-convexas e estudamos equations do tipo cavidade com coeficientes meramente mensurÃveis.
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Chen, Huyuan. "Fully linear elliptic equations and semilinear fractionnal elliptic equations." Thesis, Tours, 2014. http://www.theses.fr/2014TOUR4001/document.
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Cette thèse est divisée en six parties. La première partie est consacrée à l'étude de propriétés de Hadamard et à l'obtention de théorèmes de Liouville pour des solutions de viscosité d'équations aux dérivées partielles elliptiques complètement non-linéaires avec des termes de gradient,This thesis is divided into six parts. The first part is devoted to prove Hadamard properties and Liouville type theorems for viscosity solutions of fully nonlinear elliptic partial differential equations with gradient term
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Neji, Ali. "Existence unicité et régularité de solutions de problèmes non linéaires et complètement non linéaires elliptiques singuliers." Thesis, Cergy-Pontoise, 2019. http://www.theses.fr/2019CERG1017.
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Dans cette thèse on s'intéresse à l'existence, et la régularité pour des équations aux dérivées partielles non linéaires relatives au p-Laplacien , avec des termes d'ordre critiques ou sous critique, utilisant dans un cas le lemme du col d'Ambrozetti Rabinowitz, dans l'autre la concentration compacité de P L Lions. On considère ensuite un problème qui présente un terme d'ordre zéro qui "explose " près du bord, sur le modèle d'un article de Lazer mackenna, la différence essentielle étant ici que l'on a aussi un terme d'ordre 0 linéaire, qui demande donc l'utilisation de certaines fonctions propres. Une généralisation de ce problème à des cas complètement non linéaires et donc à des solutions de viscosité est étudiée dans la dernière partie de la thèseWe studied in this thesis the properties of existence and regularity for various nonlinear partial differential equations of elliptic type. We proved the existence of weak solutions to certain problems involving the p-Laplacian operator using critical point theory and the mountain pass theorem . We have also showed the existence of viscosity solutions for singular equations involving fully nonlinear operators
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Girard, Réjean. "Relativistic nonlinear wave equations with groups of internal symmetry." Thesis, McGill University, 1988. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=75688.
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A nonlinear wave equation invariant with respect to unitary representations of the Lorentz group is considered in an attempt to describe extended particles with spin and positive definite energy by means of a self-confined classical field. The wave function has an infinite number of components and, in the specific representations used, the corresponding internal degree of freedom is identified with the spin. A fractional power of the scalar bilinear invariant appears as an appropriate choice for the nonlinearity in order that all the stationary states be localized. Two approximation methods are proposed and both lead to results that bear a resemblance to the results of the MIT bag model.
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von, Nessi Gregory Thomas, and greg vonnessi@maths anu edu au. "Regularity Results for Potential Functions of the Optimal Transportation Problem on Spheres and Related Hessian Equations." The Australian National University. Mathematical Sciences Institute, 2008. http://thesis.anu.edu.au./public/adt-ANU20081215.120059.
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In this thesis, results will be presented that pertain to the global regularity of solutions to boundary value problems having the general form \begin{align} F\left[D^2u-A(\,\cdot\,,u,Du)\right] &= B(\,\cdot\,,u,Du),\quad\text{in}\ \Omega^-,\notag\\ T_u(\Omega^-) &= \Omega^+, \end{align} where $A$, $B$, $T_u$ are all prescribed; and $\Omega^-$ along with $\Omega^+$ are bounded in $\mathbb{R}^n$, smooth and satisfying notions of c-convexity and c^*-convexity relative to one another (see [MTW05] for definitions). In particular, the case where $F$ is a quotient of symmetric functions of the eigenvalues of its argument matrix will be investigated. Ultimately, analogies to the global regularity result presented in [TW06] for the Optimal Transportation Problem to this new fully-nonlinear elliptic boundary value problem will be presented and proven. It will also be shown that the A3w condition (first presented in [MTW05]) is also necessary for global regularity in the case of (1). The core part of this research lies in proving various a priori estimates so that a method of continuity argument can be applied to get the existence of globally smooth solutions. The a priori estimates vary from those presented in [TW06], due to the structure of F, introducing some complications that are not present in the Optimal Transportation case.¶
In the final chapter of this thesis, the A3 condition will be reformulated and analysed on round spheres. The example cost-functions subsequently analysed have already been studied in the Euclidean case within [MTW05] and [TW06]. In this research, a stereographic projection is utilised to reformulate the A3 condition on round spheres for a general class of cost-functions, which are general functions of the geodesic distance as defined relative to the underlying round sphere. With this general expression, the A3 condition can be readily verified for a large class of cost-functions that depend on the metrics of round spheres, which is tantamount (combined with some geometric assumptions on the source and target domains) to the classical regularity for solutions of the Optimal Transportation Problem on round spheres.
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Michelis, Katina. "A sequential eigenfunction expansion approach for certain nonlinear integral equations /." Thesis, McGill University, 1997. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=20588.
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In the numerical solution of nonlinear integral equations, classic finite difference and series methods lead to systems of nonlinear algebraic or transcendental equations which are solved by iterative schemes such as the Newton method. The present work develops a sequential eigenfunction expansion for the numerical solution of certain nonlinear integral equations. The nonlinear term provides constraints for the amplitudes of the eigenfunctions and a subsequent iteration is used to refine these coefficients. A comparative study of the present method with the Broyden method is conducted. It is shown that the expansion procedure provides an early indication about the multiplicity of solutions which is not present when using the classic methods of solution. Numerical examples are presented which demonstrate the robustness of the expansion method in determining multiple solutions.
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Samurkas, Tony. "Nonlinear viscoelastic behaviour of linear polyethylene : molecular weight effects and constitutive equation evaluations." Thesis, McGill University, 1993. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=41766.
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The nonlinear viscoelastic properties of a series of blends of linear polyethylene were studied using the McGill sliding plate rheometer. A more reliable and sensitive shear stress transducer for this rheometer was designed, built and used in this work. The molecular weight dependence of a variety of nonlinear viscoelastic properties was investigated. It was determined that, as with steady state properties, the sensitivity of such properties to molecular weight diminishes with increasing shear rate. The behavior of these materials in large amplitude oscillatory shear (LAOS) was also studied. By using harmonic analysis, the frequency content of the nonlinear stress response to the sinusoidal strain was studied as a function of molecular weight, strain amplitude and frequency. The predictive abilities of the Wagner model in LAOS, exponential shear, start-up and cessation of steady shear and interrupted shear, at high shear rates were evaluated. Qualitative trends were well predicted by the model for a variety of sigmoidal and exponential damping function forms. For the first time, it has been shown that Wagner model predictions for molten thermoplastics are insensitive to the damping function form. The damping functions in simple shear and planar extension were obtained for a branched low density polyethylene (LDPE). Simple shear is similar to planar extension in a rotated reference frame and thus the two flows should have similar damping functions. It was found that the damping functions that fitted these two flows are, in fact, quite different. Thus we have shown that the contribution of kinematics cannot be simply described.
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Bradley, Aoibhinn Maire. "Analysis of nonlinear spatio-temporal partial differential equations : applications to host-parasite systems and bubble growth." Thesis, University of Strathclyde, 2014. http://oleg.lib.strath.ac.uk:80/R/?func=dbin-jump-full&object_id=24405.
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The mountain hare population currently appears to be under threat in Scotland. The natural population cycles exhibited by this species are thought to be, at least in part, due to its infestation by a parasitic worm. We seek to gain an understanding of these population dynamics through a mathematical model of this system and so determine whether low population levels observed in the field are a natural trough associated with this cycling, or whether they point to a more serious decline in overall population densities. A generic result, that can be used to predict the presence of periodic travelling waves (PTWs) in a spatially heterogeneous system, is reported. This result is applicable to any two population host-parasite system with a supercritical Hopf bifurcation in the reaction kinetics. Application of this result to two examples of well studied host-parasite systems, namely the mountain hare and the red grouse systems, predicts and illustrates, for the first time, the existence of PTWs as solutions for these reaction advection diffusion schemes. One method for designing bone scaffolds involves the acoustic irradiation of a reacting polymer foam resulting in a final sample with graded porosity. The work in this thesis represents the first attempt to derive a mathematical model, for this empirical method, in order to inform the experimental design and tailor the porosity profile of samples. We isolate and study the direct effect of the acoustic pressure amplitude as well as its indirect effect on the reaction rate. We demonstrate that the direct effect of the acoustic pressure amplitude is negligible due to a high degree of attenuation by the sample. The indirect effect, on reaction rate, is significant and the standing wave is shown to produce a heterogeneous bubble size distribution. Several suggestions for further work are made.
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Filippini, Andrea Gilberto. "Free surface flow simulation in estuarine and coastal environments : numerical development and application on unstructured meshes." Thesis, Bordeaux, 2016. http://www.theses.fr/2016BORD0404/document.
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Over the last decades, there has been considerable attention in the accurate mathematical modeling and numerical simulations of free surface wave propagation in near-shore environments. A physical correct description of the large scale phenomena, which take place in the shallow water region, must account for strong nonlinear and dispersive effects, along with the interaction with complex topographies. First, a study on the behavior in nonlinear regime of different Boussinesq-type models is proposed, showing the advantage of using fully-nonlinear models with respect to weakly-nonlinear and weakly dispersive models (commonly employed). Secondly, a new flexible strategy for solving the fully-nonlinear and weakly-dispersive Green-Naghdi equations is presented, which allows to enhance an existing shallow water code by simply adding an algebraic term to the momentum balance and is particularly adapted for the use of hybrid techniques for wave breaking. Moreover, the first discretization of the Green-Naghdi equations on unstructured meshes is proposed via hybrid finite volume/ finite element schemes. Finally, the models and the methods developed in the thesis are deployed to study the physical problem of bore formation in convergent alluvial estuary, providing the first characterization of natural estuaries in terms of bore inceptionCes dernières décennies, une attention particulière a été portée sur la modélisation mathématique et la simulation numérique de la propagation de vagues en environnements côtiers. Une description physiquement correcte des phénomènes à grande échelle, qui apparaissent dans les régions d'eau peu profonde, doit prendre en compte de forts effets non-linéaires et dispersifs, ainsi que l'interaction avec des bathymétries complexes. Dans un premier temps, une étude du comportement en régime non linéaire de différents modèles de type Boussinesq est proposée, démontrant l'avantage d'utiliser des modèles fortement non-linéaires par rapport à des modèles faiblement non-linéaires et faiblement dispersifs (couramment utilisés). Ensuite, une nouvelle approche flexible pour résoudre les équations fortement non-linéaires et faiblement dispersives de Green-Naghdi est présentée. Cette stratégie permet d'améliorer un code "shallow water" existant par le simple ajout d'un terme algébrique dans l'équation du moment et est particulièrement adapté à l'utilisation de techniques hybrides pour le déferlement des vagues. De plus, la première discrétisation des équations de Green-Naghdi sur maillage non structuré est proposée via des schémas hybrides Volume Fini/Élément Fini. Finalement, les modèles et méthodes développés dans la thèse sont appliqués à l'étude du problème physique de la formation du mascaret dans des estuaires convergents et alluviaux. Cela a amené à la première caractérisation d'estuaire naturel en terme d'apparition de mascaret
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Labutin, Denis. "Potential theory for fully nonlinear elliptic equations." Phd thesis, 1999. http://hdl.handle.net/1885/147899.
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Holtby, Derek William. "Higher order estimates for fully nonlinear difference equations." Phd thesis, 1996. http://hdl.handle.net/1885/145341.
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Reye, Stephen James. "Fully non-linear parabolic differential equations of second order." Phd thesis, 1985. http://hdl.handle.net/1885/138487.
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Neilan, Michael Joseph. "Numerical methods for fully nonlinear second order partial differential equations." 2009. http://etd.utk.edu/2009/Spring2009Dissertations/NeilanMichaelJoseph.pdf.
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Chang, Lara Hector Andres. "Regularity for solutions of nonlocal fully nonlinear parabolic equations and free boundaries on two dimensional cones." 2013. http://hdl.handle.net/2152/21668.
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On the first part, we consider nonlinear operators I depending on a family of nonlocal linear operators [mathematical equations]. We study the solutions of the Dirichlet initial and boundary value problems [mathematical equations]. We do not assume even symmetry for the kernels. The odd part bring some sort of nonlocal drift term, which in principle competes against the regularization of the solution. Existence and uniqueness is established for viscosity solutions. Several Hölder estimates are established for u and its derivatives under special assumptions. Moreover, the estimates remain uniform as the order of the equation approaches the second order case. This allows to consider our results as an extension of the classical theory of second order fully nonlinear equations. On the second part, we study two phase problems posed over a two dimensional cone generated by a smooth curve [mathematical symbol] on the unit sphere. We show that when [mathematical equation] the free boundary avoids the vertex of the cone. When [mathematical equation]we provide examples of minimizers such that the vertex belongs to the free boundary.text
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LIN, PO-AN, and 林柏安. "A Survey on The Book ”Fully Nonlinear Elliptic Equations” By Luis.A. Caarelli and Xavier Cabr´e." Thesis, 2014. http://ndltd.ncl.edu.tw/handle/5449up.
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von, Nessi Gregory Thomas. "Regularity Results for Potential Functions of the Optimal Transportation Problem on Spheres and Related Hessian Equations." Phd thesis, 2008. http://hdl.handle.net/1885/49370.
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In this thesis, results will be presented that pertain to the global regularity of solutions to a class of boundary value problems closely related to the Optimal Transportation Equation. Ultimately, analogies to the global regularity result presented in [TW06] for the Optimal Transportation Problem to this new fully-nonlinear elliptic boundary value problem will be presented and proven. It will also be shown that the A3w condition (first presented in [MTW05]) is also necessary for global regularity for this class of problems. The core part of this research lies in proving various a priori estimates so that a method of continuity argument can be applied to get the existence of globally smooth solutions. The a priori estimates vary from those presented in [TW06], due to the structure of these new equations, introducing some complications that are not present in the Optimal Transportation case.
In the final chapter of this thesis, the A3 condition will be reformulated and analysed on round spheres. The example cost-functions subsequently analysed have already been studied in the Euclidean case within [MTW05] and [TW06]. In this research, a stereographic projection is utilised to reformulate the A3 condition on round spheres for a general class of cost-functions, which are general functions of the geodesic distance as defined relative to the underlying round sphere. With this general expression, the A3 condition can be readily verified for a large class of cost-functions that depend on the metrics of round spheres, which is tantamount (combined with some geometric assumptions on the source and target domains) to the classical regularity for solutions of the Optimal Transportation Problem on round spheres.