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Register a new userA generalization of pdes from a Krylov point of view
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We introduce and investigate the notion of a `generalized equation of the form $f(D^2 u)=0$, based on the notions of subequations and Dirichlet duality. Precisely, a subset ${{mathbb H}}subset {rm Sym}^2({mathbb R}^n)$ is a generalized equation if it is an intersection ${{mathbb H}} = {{mathbb E}}cap (-widetilde{{{mathbb G}}})$ where ${{mathbb E}}$ and ${{mathbb G}}$ are subequations and $widetilde{{{mathbb G}}}$ is the subequation dual to ${{mathbb G}}$. We utilize a viscosity definition of `solution to ${{mathbb H}}$. The mirror of ${{mathbb H}}$ is defined by ${{mathbb H}}^* equiv {{mathbb G}}cap (-widetilde {{mathbb E}})$. One of the main results here concerns the Dirichlet problem on arbitrary bounded domains $Omegasubset {mathbb R}^n$ for solutions to ${{mathbb H}}$ with prescribed boundary function $varphi in C(partial Omega)$. We prove that: (A) Uniqueness holds $iff$ ${{mathbb H}}$ has no interior, and (B) Existence holds $iff$ ${{mathbb H}}^*$ has no interior. For (B) the appropriate boundary convexity of $partial Omega$ must be assumed. Many examples of generalized equations are discussed, including the constrained Laplacian, the twisted Monge-Amp`ere equation, and the $C^{1,1}$-equation. The closed sets ${{mathbb H}}$ which can be written as generalized equations are intrinsically characterized. For such an ${{mathbb H}}$ the set of subequation pairs with ${{mathbb H}} = {{mathbb E}}cap (-widetilde{{{mathbb G}}})$ is partially ordered, and there is a canonical least element, contained in all others. Harmonics for the canonical equation are harmonic for all others giving ${{mathbb H}}$. A general form of the main theorem, which holds on any manifold, is also established.
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In his monograph Lec{c}ons sur les syst`emes orthogonaux et les coordonnees curvilignes. Principes de geometrie analytique, 1910, Darboux stated three theorems providing local existence and uniqueness of solutions to first order systems of the type [partial_{x_i} u_alpha(x)=f^alpha_i(x,u(x)),quad iin I_alphasubseteq{1,dots,n}.] For a given point $bar xin mathbb{R}^n$ it is assumed that the values of the unknown $u_alpha$ are given locally near $bar x$ along ${x,|, x_i=bar x_i , text{for each}, iin I_alpha}$. The more general of the theorems, Theor`eme III, was proved by Darboux only for the cases $n=2$ and $3$. In this work we formulate and prove a generalization of Darbouxs Theor`eme III which applies to systems of the form [{mathbf r}_i(u_alpha)big|_x = f_i^alpha (x, u(x)), quad iin I_alphasubseteq{1,dots,n}] where $mathcal R={{mathbf r}_i}_{i=1}^n$ is a fixed local frame of vector fields near $bar x$. The data for $u_alpha$ are prescribed along a manifold $Xi_alpha$ containing $bar x$ and transverse to the vector fields ${{mathbf r}_i,|, iin I_alpha}$. We identify a certain Stable Configuration Condition (SCC). This is a geometric condition that depends on both the frame $mathcal R$ and on the manifolds $Xi_alpha$; it is automatically met in the case considered by Darboux. Assuming the SCC and the relevant integrability conditions are satisfied, we establish local existence and uniqueness of a $C^1$-solution via Picard iteration for any number of independent variables $n$.
The energy dependence of the cross sections for electromagnetic diffractive processes can be well described by a single power, $W^delta$. For $J/psi$ photoproduction this holds in the range from 20 GeV to 2 TeV. This feature is most easily explained by a single pole in the angular momentum plane which depends on the scale of the process, at least in a certain range of values of the momentum transfer. It is shown that this assumption allows a unified description of all electromagnetic elastic diffractive processes. We also discuss an alternative model with an energy dependent dipole cross section, which is compatible with the data up to 2 TeV and which shows an energy behaviour typical for a cut in the angular momentum plane.
We give an exposition of the Horn inequalities and their triple role characterizing tensor product invariants, eigenvalues of sums of Hermitian matrices, and intersections of Schubert varieties. We follow Belkales geometric method, but assume only basic representation theory and algebraic geometry, aiming for self-contained, concrete proofs. In particular, we do not assume the Littlewood-Richardson rule nor an a priori relation between intersections of Schubert cells and tensor product invariants. Our motivation is largely pedagogical, but the desire for concrete approaches is also motivated by current research in computational complexity theory and effective algorithms.
We consider stability of non-rotating gaseous stars modeled by the Euler-Poisson system. Under general assumptions on the equation of states, we proved a turning point principle (TPP) that the stability of the stars is entirely determined by the mass-radius curve parameterized by the center density. In particular, the stability can only change at extrema (i.e. local maximum or minimum points) of the total mass. For very general equation of states, TPP implies that for increasing center density the stars are stable up to the first mass maximum and unstable beyond this point until next mass extremum (a minimum). Moreover, we get a precise counting of unstable modes and exponential trichotomy estimates for the linearized Euler-Poisson system. To prove these results, we develop a general framework of separable Hamiltonian PDEs. The general approach is flexible and can be used for many other problems including stability of rotating and magnetic stars, relativistic stars and galaxies.
In this work, we propose a method for solving Kolmogorov hypoelliptic equations based on Fourier transform and Feynman-Kac formula. We first explain how the Feynman-Kac formula can be used to compute the fundamental solution to parabolic equations with linear or quadratic potential. Then applying these results after a Fourier transform we deduce the computation of the solution to a a first class of Kolmogorov hypoelliptic equations. Then we solve partial differential equations obtained via Feynman-Kac formula from the Ornstein-Uhlenbeck generator. Also, a new small time approximation of the solution to Kolmogorov hypoelliptic equations is provided. We finally present the results of numerical experiments to check the practical efficiency of this approximation.
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