Search results for "Mathematical analysis"
showing 10 items of 2409 documents
Fixed angle inverse scattering in the presence of a Riemannian metric
2020
We consider a fixed angle inverse scattering problem in the presence of a known Riemannian metric. First, assuming a no caustics condition, we study the direct problem by utilizing the progressing wave expansion. Under a symmetry assumption on the metric, we obtain uniqueness and stability results in the inverse scattering problem for a potential with data generated by two incident waves from opposite directions. Further, similar results are given using one measurement provided the potential also satisfies a symmetry assumption. This work extends the results of [23,24] from the Euclidean case to certain Riemannian metrics.
Discontinuous solutions of linear, degenerate elliptic equations
2008
Abstract We give examples of discontinuous solutions of linear, degenerate elliptic equations with divergence structure. These solve positively conjectures of De Giorgi.
Dimensional reduction for energies with linear growth involving the bending moment
2008
A $\Gamma$-convergence analysis is used to perform a 3D-2D dimension reduction of variational problems with linear growth. The adopted scaling gives rise to a nonlinear membrane model which, because of the presence of higher order external loadings inducing a bending moment, may depend on the average in the transverse direction of a Cosserat vector field, as well as on the deformation of the mid-plane. The assumption of linear growth on the energy leads to an asymptotic analysis in the spaces of measures and of functions with bounded variation.
Mappings of finite distortion: the degree of regularity
2005
This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K(x)⩾1 be a measurable function defined on a domain Ω⊂Rn,n⩾2, and such that exp(βK(x))∈Lloc1(Ω), β>0. We show that there exist two universal constants c1(n),c2(n) with the following property: Let f be a mapping in Wloc1,1(Ω,Rn) with |Df(x)|n⩽K(x)J(x,f) for a.e. x∈Ω and such that the Jacobian determinant J(x,f) is locally in L1log−c1(n)βL. Then automatically J(x,f) is locally in L1logc2(n)βL(Ω). This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite disto…
Fokker–Planck equation with respect to heat measures on loop groups
2011
Abstract The Dirichlet form on the loop group L e ( G ) with respect to the heat measure defines a Laplacian Δ DM on L e ( G ) . In this note, we will use Wasserstein distance variational method to solve the associated heat equation for a given data of finite entropy.
Analysis on free Riemannian path spaces
2005
Abstract The gradient operator is defined on the free path space with reference measure P μ , the law of the Brownian motion on the base manifold with initial distribution μ, where μ has strictly positive density w.r.t. the volume measure. The formula of integration by parts is established for the underlying directional derivatives, which implies the closability of the gradient operator so that it induces a conservative Dirichlet form on the free path space. The log-Sobolev inequality for this Dirichlet form is established and, consequently, the transportation cost inequality is obtained for the associated intrinsic distance.
Dimension gap under conformal mappings
2012
Abstract We give an estimate for the Hausdorff gauge dimension of the boundary of a simply connected planar domain under p -integrability of the hyperbolic metric, p > 1 . This estimate does not degenerate when p tends to one; for p = 1 the boundary can even have positive area. The same phenomenon is extended to general planar domains in terms of the quasihyperbolic metric. We also give an example which shows that our estimates are essentially sharp.
Stochastic anticipative calculus on the path space over a compact Riemannian manifold
1998
Abstract In this paper, we shall first give another expression for Cruzeiro-Malliavin structure equation, by means of the Skorohod integral. The torsion tensor with respect to the Markovian connection used in [CF] is computed. This is the key step to establish a Stroock-like formula of commutation on the derivative of the Skorohod integral, which enables us to prove an Ito formula. As an application, we shall give a maximal inequality for Skorohod integrals following [AN2].
Transport equations and quasi-invariant flows on the Wiener space
2010
Abstract We shall investigate on vector fields of low regularity on the Wiener space, with divergence having low exponential integrability. We prove that the vector field generates a flow of quasi-invariant measurable maps with density belonging to the space L log L . An explicit expression for the density is also given.
Geometry and analysis of Dirichlet forms
2012
Let $ \mathscr E $ be a regular, strongly local Dirichlet form on $L^2(X, m)$ and $d$ the associated intrinsic distance. Assume that the topology induced by $d$ coincides with the original topology on $ X$, and that $X$ is compact, satisfies a doubling property and supports a weak $(1, 2)$-Poincar\'e inequality. We first discuss the (non-)coincidence of the intrinsic length structure and the gradient structure. Under the further assumption that the Ricci curvature of $X$ is bounded from below in the sense of Lott-Sturm-Villani, the following are shown to be equivalent: (i) the heat flow of $\mathscr E$ gives the unique gradient flow of $\mathscr U_\infty$, (ii) $\mathscr E$ satisfies the Ne…