Do you want to publish a course? Click here

A generalization of an integrability theorem of Darboux

126   0   0.0 ( 0 )
 Added by Irina Kogan A
 Publication date 2018
  fields
and research's language is English




Ask ChatGPT about the research

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 $n$.



rate research

Read More

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.
We show that among sets of finite perimeter balls are the only volume-constrained critical points of the perimeter functional.
83 - Siran Li 2019
We give a soft proof of Albertis Luzin-type theorem in [1] (G. Alberti, A Lusintype theorem for gradients, J. Funct. Anal. 100 (1991)), using elementary geometric measure theory and topology. Applications to the $C^2$-rectifiability problem are also discussed.
We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem $$ Delta^2 u=|u|^{p-1}u {in} R^n,$$ where $ p>1$ and $nge1$. We give a complete classification of stable and finite Morse index solutions (whether positive or sign changing), in the full exponent range. We also compute an upper bound of the Hausdorff dimension of the singular set of extremal solutions. Our approach is motivated by Flemings tangent cone analysis technique for minimal surfaces and Federers dimension reduction principle in partial regularity theory. A key tool is the monotonicity formula for biharmonic equations.
Caratheodory showed that $n$ complex numbers $c_1,...,c_n$ can uniquely be written in the form $c_p=sum_{j=1}^m rho_j {epsilon_j}^p$ with $p=1,...,n$, where the $epsilon_j$s are different unimodular complex numbers, the $rho_j$s are strictly positive numbers and integer $m$ never exceeds $n$. We give the conditions to be obeyed for the former property to hold true if the $rho_j$s are simply required to be real and different from zero. It turns out that the number of the possible choices of the signs of the $rho_j$s are {at most} equal to the number of the different eigenvalues of the Hermitian Toeplitz matrix whose $i,j$-th entry is $c_{j-i}$, where $c_{-p}$ is equal to the complex conjugate of $c_{p}$ and $c_{0}=0$. This generalization is relevant for neutron scattering. Its proof is made possible by a lemma - which is an interesting side result - that establishes a necessary and sufficient condition for the unimodularity of the roots of a polynomial based only on the polynomial coefficients. Keywords: Toeplitz matrix factorization, unimodular roots, neutron scattering, signal theory, inverse problems. PACS: 61.12.Bt, 02.30.Zz, 89.70.+c, 02.10.Yn, 02.50.Ga
comments
Fetching comments Fetching comments
Sign in to be able to follow your search criteria
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا