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In a recent paper we studied an equation (called the "simple equation'') introduced by one of us in 1963 for an approximate correlation function associated with the ground state of an interacting Bose gas. Solving the equation yields a relation between the density ρ of the gas and the energy per particle. Our construction of solutions gave a well-defined function ρ(e) for the density as a function of the energy e. We had conjectured that ρ(e) is a strictly monotone increasing function, so that it can be inverted to yield the strictly monotone increasing function e(ρ). We had also conjectured that ρ e(ρ) is convex as a function of ρ. We prove both conjectures here for small densities, the context in which they have the most physical relevance, and the monotonicity also for large densities. Both conjectures are grounded in the underlying physics, and their proof provides further mathematical evidence for the validity of the assumptions underlying the derivation of the simple equation, at least for low or high densities, if not intermediate densities, although the equation gives surprisingly good predictions for all densities ρ. Another problem left open in our previous paper was whether the simple equation could be used to compute accurate predictions of observables other than the energy. Here, we provide a recipe for computing predictions for any one- or two-particle observables for the ground state of the Bose gas. We focus on the condensate fraction and the momentum distribution, and show that they have the same low density asymptotic behavior as that predicted for the Bose gas. Along with the computation of the low density energy of the simple equation in our previous paper, this shows that the simple equation reproduces the known and conjectured properties of the Bose gas at low densities. [ABSTRACT FROM AUTHOR]

The bitemperature Euler model describes a crucial step of inertial confinement fusion (ICF) when the plasma is quasineutral while ionic and electronic temperatures remain distinct. The model is written as a first-order hyperbolic system in nonconservative form with partially dissipative source terms. We consider the polytropic case for both ions and electrons with different γ-law pressures. The system does not fulfill the Shizuta-Kawashima condition and, the physical entropy, which is a strictly convex function, does not provide a symmetrizer of the system. In this paper we exhibit a symmetrizer to apply the result on the local existence of smooth solutions in several space dimensions. In the one-dimensional case we establish energy and dissipation estimates leading to global existence for small perturbations of equilibrium states. [ABSTRACT FROM AUTHOR]

We present an improvement to the Jensen inequality for certain convex functions φ on (0 ∞). This implies finding a nonzero lower bound for the Jensen gap. In particular, just as the Jensen inequality becomes an equality when φ is linear, for the new inequality we obtain an equality also when φ is quadratic. At the heart of the paper is an apparent new technique for improving classical inequalities based on nonlinear functions. [ABSTRACT FROM AUTHOR]

We consider the convergence of the Douglas{Rachford splitting method (DRSM) for minimizing the sum of a strongly convex function and a weakly convex function; this setting has various applications, especially in some sparsity-driven scenarios with the purpose of avoiding biased estimates which usually occur when convex penalties are used. Though the convergence of the DRSM has been well studied for the case where both functions are convex, its results for some nonconvexfunction- involved cases, including the "strongly + weakly" convex case, are still in their infancy. In this paper, we prove the convergence of the DRSM for the "strongly + weakly" convex setting under relatively mild assumptions compared with some existing work in the literature. Moreover, we establish the rate of asymptotic regularity and the local linear convergence rate in the asymptotical sense under some regularity conditions. [ABSTRACT FROM AUTHOR]

We study the well-posedness of a hyperbolic quasi-linear version of hydrostatic Navier--Stokes system in I x TT and prove the global well-posedness of the system with initial data which are small and analytic in both variables. We also prove the convergence of such analytic solutions to that of the classical hydrostatic Navier--Stokes system when the delay time converges to zero. Furthermore, we obtain a local well-posedness result in Gevrey class 2 when the initial datum is a small perturbation of some convex function. [ABSTRACT FROM AUTHOR]

Abstract. In this paper we compute explicit solutions of the eigenvalue problem -div(Du/|Du|) = u in R2, in particular explicit solutions whose truncatures are in Wloc1,1 (R2),and piecewise constant ones which are sums of characteristic functions of convex sets. The solutions of the above eigenvalue problem describe the asymptotic behavior of solutions of the minimizing total variation flow. As an application,we also construct explicit solutions of the denoising problem in image processing. [ABSTRACT FROM AUTHOR]

This paper describes a variational approach for estimating a discontinuous optical flow from a sequence of images. Defined as the apparent motion of the image brightness pattern, the optical flow is very important in the computer vision community, where its accurate estimation is strongly needed. After a short overview of existing methods, we present a new variational method that we study in the space of bounded variations. We first present an integral representation of the optical flow problem which appears to be not lower semicontinuous. The relaxed functional is then calculated. We conclude by challenging questions about the possible numerical analysis of the abstract results. [ABSTRACT FROM AUTHOR]

We consider, on a trivial vector bundle over a Riemannian manifold with boundary, the inverse problem of uniquely recovering time- and space-dependent coefficients of the dynamic, vector-valued Schrödinger equation from the knowledge of the Dirichlet-to-Neumann (D-to-N) map. We show that the D-to-N map uniquely determines both the connection form and the potential appearing in the Schrödinger equation, under the assumption that the manifold is either (a) twodimensional and simple or (b) of higher dimension with strictly convex boundary and admits a smooth, strictly convex function. [ABSTRACT FROM AUTHOR]

We use the distances introduced in a previous joint paper to exhibit the gradient flow structure of some drift-diffusion equations for a wide class of entropy functionals. Functional inequalities obtained by the comparison of the entropy with the entropy production functional reflect the contraction properties of the flow. Our approach provides a unified framework for the study of the Kolmogorov--Fokker--Planck equation. [ABSTRACT FROM AUTHOR]

Abstract. In this paper we compute explicit solutions of the eigenvalue problem -div(Du/|Du|) = u in R2, in particular explicit solutions whose truncatures are in Wloc1,1 (R2),and piecewise constant ones which are sums of characteristic functions of convex sets. The solutions of the above eigenvalue problem describe the asymptotic behavior of solutions of the minimizing total variation flow. As an application,we also construct explicit solutions of the denoising problem in image processing. [ABSTRACT FROM AUTHOR]

We propose an extension of the classical variational theory of evolution equations that accounts for dynamics also in possibly nonreflexive and nonseparable spaces. The pivoting point is to establish a novel variational structure, based on abstract modular spaces associated to a given convex function. First, we show that the new variational triple is suited for framing the evolution, in the sense that a novel duality pairing can be introduced and a generalized computational chain rule holds. Second, we prove well-posedness in an extended variational sense for evolution equations, without relying on any reflexivity assumption and any polynomial requirement on the nonlinearity. Finally, we discuss several important applications that can be addressed in this framework: these cover, but are not limited to, equations in Musielak--Orlicz--Sobolev spaces, such as variable exponent, Orlicz, weighted Lebesgue, and double-phase spaces. [ABSTRACT FROM AUTHOR]

We study the properties of the Hopf-Lax formula restricted to convex functions and provide a characterization of the optimal transfer plan for weak transport problems on the real line. On n dimensional real space, we also provide a sufficient condition on the potential function such that the optimal plan of the classical Monge-Kantorovich problem does not depend on the cost function. As a byproduct, we establish a link between the combinatorial object (the permutation polytope) and the Hamilton-Jacobi equation. [ABSTRACT FROM AUTHOR]

In the absence of the entropy condition, we construct an L∞ solution to the Cauchy problem of a scalar conservation law in one space dimension that exhibits fine-scale oscillations in the interior of its support when the initial function is nonconstant, continuous, and compactly supported. As a result, such a solution turns out to be nowhere continuous in the interior of the support. The method of proof is to convert the main equation into a suitable partial differential inclusion and to rely on the convex integration method of Müller and Šverák [Ann. of Math. (2), 157 (2003), pp. 715-742]. In doing so, we find an appropriate subsolution by solving certain ordinary differential equations and make use of it to tailor an in-approximation scheme that reflects persistence of oscillations. [ABSTRACT FROM AUTHOR]

We rephrase Monge's optimal transportation (OT) problem with quadratic cost--via a Monge-Ampère equation--as an infinite-dimensional optimization problem, which is in fact a convex problem when the target is a log-concave measure with convex support. We define a natural finite-dimensional discretization to the problem and associate a piecewise affine convex function to the solution of this discrete problem. The discrete problems always admit a solution, which can be obtained by standard convex optimization algorithms whenever the target is a log-concave measure with convex support. We show that under suitable regularity conditions the convex functions retrieved from the discrete problems converge to the convex solution of the original OT problem furnished by Brenier's theorem. We demonstrate numerical solutions of these discrete problems and then explain (at the expense of provable convergence) how to modify our method to make it more efficient and more accurate, as well as how to remove the restriction on the target measure via a fixed point method that only involves solving OT problems with constant target densities. [ABSTRACT FROM AUTHOR]

We prove that for all smooth nonconstant initial data the initial-Neumann boundary value problem for the Perona-Malik equation in image processing possesses infinitely many Lipschitz weak solutions on smooth bounded convex domains in all dimensions. Such existence results have not been known except for the one-dimensional problems. Our approach is motivated by reformulating the Perona-Malik equation as a nonhomogeneous partial differential inclusion with linear constraint but some uncontrollable components of gradient. We establish a general existence result by a suitable Baire category method under a pivotal density hypothesis. We finally fulfill this density hypothesis by convex integration based on certain approximations from an explicit formula of lamination convex hull of some matrix set involved. [ABSTRACT FROM AUTHOR]

We consider weak stationary solutions to the incompressible Euler equations and show that the analogue of the h-principle obtained by the second author in joint work with C. De Lellis for time-dependent weak solutions in L∞ continues to hold. The key difference arises in dimension d = 2, where it turns out that the relaxation set is strictly smaller than what one obtains in the time-dependent case. [ABSTRACT FROM AUTHOR]

We present a variational reformulation of a class of doubly nonlinear parabolic equations as (limits of) constrained convex minimization problems. In particular, an ε-dependent family of weighted energy-dissipation (WED) functionals on entire trajectories is introduced and proved to admit minimizers. These minimizers converge to solutions of the original doubly nonlinear equation as ε → 0. The argument relies on the suitable dualization of the former analysis of [G. Akagi and U. Stefanelli, J. Funct. Anal., 260 (2011), pp. 2541-2578] and results in a considerable extension of the possible application range of the WED functional approach to nonlinear diffusion phenomena, including the Stefan problem and the porous media equation. [ABSTRACT FROM AUTHOR]

We study an abstract second order nonlinear evolution inclusion in a framework of evolution triple of spaces. We consider time-dependent possibly nonconvex nonsmooth functions and their Clarke subdifferentials operating on the unknown function. First we prove the existence of a weak solution. Then we study the asymptotic behavior of a sequence of solutions when a small parameter in the inertial term tends to zero. We prove that the limit function is a solution of a parabolic hemivariational inequality. Finally, we give an application of the abstract theorem to a quasi-static viscoelastic contact problem. [ABSTRACT FROM AUTHOR]

We give a class of optimally stable solutions to Tietze's extension problem in general, metrically convex compact metric spaces. The extensions which solve this problem are the unique stationary states of nonlinear processes of regularization. [ABSTRACT FROM AUTHOR]