Optimal Regularity Result Research Articles

Erratum to “An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs”

Erratum to “An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs”

On continuous billiard and quasigeodesic flows characterizing alcoves and isosceles tetrahedra

AbstractWe characterize fundamental domains of affine reflection groups as those polyhedral convex bodies which support a continuous billiard dynamics. We interpret this characterization in the broader context of Alexandrov geometry and prove an analogous characterization for isosceles tetrahedra in terms of continuous quasigeodesic flows. Moreover, we show an optimal regularity result for convex bodies: the billiard dynamics is continuous if the boundary is of class . In particular, billiard trajectories converge to geodesics on the boundary in this case. Our proof of the latter continuity statement is based on Alexandrov geometry methods that we discuss, respectively, establish first.

Optimal regularity results for the one-dimensional prescribed curvature equation via the strong maximum principle

A refined version of the strong maximum principle is proven for a class of second order ordinary differential equations with possibly discontinuous non-monotone nonlinearities. Then, exploiting this tool, some optimal regularity results, recently established by López-Gómez and Omari in [15–17] for the bounded variation solutions of non-autonomous quasilinear equations driven by the one-dimensional curvature operator, are substantially improved by admitting general prescribed curvatures and by incorporating general boundary conditions. The novel approach developed here yields a new, deeper, interpretation of the assumptions introduced in our previous papers, simultaneously clarifying their meaning and making fully transparent their connection with the strong maximum principle.

Transport of Gaussian measures under the flow of one-dimensional fractional nonlinear Schrödinger equations

We study the transport property of Gaussian measures on Sobolev spaces of periodic functions under the dynamics of the one-dimensional cubic fractional nonlinear Schrödinger equation. For the case of second-order dispersion or greater, we establish an optimal regularity result for the quasi-invariance of these Gaussian measures, following the approach by Debussche and Tsutsumi (2021). Moreover, we obtain an explicit formula for the Radon-Nikodym derivative and, as a corollary, a formula for the two-point function arising in wave turbulence theory. We also obtain improved regularity results in the weakly dispersive case, extending those by the first author and Trenberth (2019). Our proof combines the approach introduced by Planchon, Tzvetkov and Visciglia (2020) and that of Debussche and Tsutsumi (2021).

A study on stochastic maximal regularity for rough time-dependent problems

We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only progressively measurable in the time variable. For 2m-th order systems with VMO regularity in space, we obtain Lp(Lq) estimates for all p > 2 and q ≥ 2, leading to optimal space-time regularity results. For second order systems with continuous coefficients in space, we also include a first order linear term, under a stochastic parabolicity condition, and obtain Lp(Lp) estimates together with optimal space-time regularity. For linear second order equations in divergence form with random coefficients that are merely measurable in both space and time, we obtain estimates in the tent spaces Tp,2 of Coifman-Meyer-Stein. This is done in the deterministic case under no extra assumption, and in the stochastic case under the assumption that the coefficients are divergence free.

An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs

We show uniqueness in law for the critical SPDE dXt=AXtdt+(−A)1/2F(X(t))dt+dWt,X0=x∈H, where A:dom(A)⊂H→H is a negative definite self-adjoint operator on a separable Hilbert space H having A−1 of trace class and W is a cylindrical Wiener process on H. Here, F:H→H can be continuous with, at most, linear growth (some functions F which grow more than linearly can also be considered). This leads to new uniqueness results for generalized stochastic Burgers equations and for three-dimensional stochastic Cahn–Hilliard-type equations which have interesting applications. To get weak uniqueness, we use an infinite dimensional localization principle and also establish a new optimal regularity result for the Kolmogorov equation λu−Lu=f associated to the SPDE when F=0 (λ>0, f:H→R Borel and bounded). In particular, we prove that the first derivative Du(x) belongs to dom((−A)1/2), for any x∈H, and supx∈H|(−A)1/2Du(x)|H=‖(−A)1/2Du‖0≤C‖f‖0.

Optimal regularity for a two-phase obstacle-like problem with logarithmic singularity

We consider the semilinear problem where B 1 is the unit ball in and assume Using a monotonicity formula argument, we prove an optimal regularity result for solutions: is a log-Lipschitz function. This problem introduces two main difficulties. The first is the lack of invariance in the scaling and blow-up of the problem. The other (more serious) issue is a term in the Weiss energy which is potentially non-integrable unless one already knows the optimal regularity of the solution: this puts us in a catch-22 situation.

The Regularity of Parametrized Integer Stationary Varifolds in Two Dimensions

We establish an optimal regularity result for parametrized two‐dimensional stationary varifolds. Namely, we show that the parametrization map is a smooth minimal branched immersion and that the multiplicity function is constant.We provide some applications of this regularity result, especially in the calculus of variations for the area functional. © 2020 Wiley Periodicals LLC

Stochastic maximal regularity for rough time-dependent problems

We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only progressively measurable in the time variable. For 2m-th order systems with VMO regularity in space, we obtain L^{p}(L^{q}) estimates for all p>2 and qge 2, leading to optimal space-time regularity results. For second order systems with continuous coefficients in space, we also include a first order linear term, under a stochastic parabolicity condition, and obtain L^{p}(L^{p}) estimates together with optimal space-time regularity. For linear second order equations in divergence form with random coefficients that are merely measurable in both space and time, we obtain estimates in the tent spaces T^{p,2}_{sigma } of Coifman–Meyer–Stein. This is done in the deterministic case under no extra assumption, and in the stochastic case under the assumption that the coefficients are divergence free.

Remarks on some minimization problems associated with BV norms

The purpose of this paper is twofold. Firstly I present an optimal regularity result for minimizers of a \begin{document}$ 1D $\end{document} convex functional involving the BV-norm, under Neumann boundary condition. This functional is a simplified version of models occuring in Image Processing. Secondly I investigate the existence of minimizers for the same functional under Dirichlet boundary condition. Surprisingly, this turns out to be a delicate issue, which is still widely open.

Numerical solutions to time-fractional stochastic partial differential equations

This study is concerned with numerical approximations of time-fractional stochastic heat-type equations driven by multiplicative noise, which can be used to model the anomalous diffusion in porous media with random effects with thermal memory. A standard finite element approximation is used in space as well as a spatial-temporal discretization which is achieved by a new algorithm in time direction. The strong convergence error estimates for both semidiscrete and fully discrete schemes in a semigroup framework are proved, which are based on the error estimates for the corresponding deterministic equations, and together with the optimal regularity results of mild solutions to the time-fractional SPDEs. Numerical results are finally reported to confirm our theoretical findings.

The resolution of the Yang–Mills Plateau problem in super-critical dimensions

We study the minimization problem for the Yang–Mills energy under fixed boundary connection in supercritical dimension n≥5. We define the natural function space AG in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space AG can be also interpreted as a space of weak connections on a “real measure theoretic version” of reflexive sheaves from complex geometry.We prove the existence of weak solutions to the Yang–Mills Plateau problem in the space AG.We then prove the optimal regularity result for solutions of this Plateau problem. On the way to prove this result we establish a Coulomb gauge extraction theorem for weak curvatures with small Yang–Mills density. This generalizes to the general framework of weak L2 curvatures previous works of Meyer–Rivière and Tao–Tian in which respectively a strong approximability property and an admissibility property were assumed in addition.

NON-REGULARITY IN HÖLDER AND SOBOLEV SPACES OF SOLUTIONS TO THE SEMILINEAR HEAT AND SCHRÖDINGER EQUATIONS

In this paper, we study the Cauchy problem for the semilinear heat and Schrödinger equations, with the nonlinear term $f(u)=\unicode[STIX]{x1D706}|u|^{\unicode[STIX]{x1D6FC}}u$. We show that low regularity of $f$ (i.e., $\unicode[STIX]{x1D6FC}>0$ but small) limits the regularity of any possible solution for a certain class of smooth initial data. We employ two different methods, which yield two different types of results. On the one hand, we consider the semilinear equation as a perturbation of the ODE $w_{t}=f(w)$. This yields, in particular, an optimal regularity result for the semilinear heat equation in Hölder spaces. In addition, this approach yields ill-posedness results for the nonlinear Schrödinger equation in certain $H^{s}$-spaces, which depend on the smallness of $\unicode[STIX]{x1D6FC}$ rather than the scaling properties of the equation. Our second method is to consider the semilinear equation as a perturbation of the linear equation via Duhamel’s formula. This yields, in particular, that if $\unicode[STIX]{x1D6FC}$ is sufficiently small and $N$ is sufficiently large, then the nonlinear heat equation is ill-posed in $H^{s}(\mathbb{R}^{N})$ for all $s\geqslant 0$.

Interior partial regularity for minimal $L^p$-vector fields with integer fluxes

We use a new combinatorial technique to prove the optimal interior partial regularity result for L p-vector fields with integer fluxes that minimize the L p-energy. More precisely, we prove that the minimizing vector fields are Holder continuous outside a set that is locally finite inside the domain. The results continue the program started in [25], but this paper is self-contained.

A note on an accelerated exponential Euler method for parabolic SPDEs with additive noise

This note aims to present further results on the accelerated exponential Euler method proposed in Jentzen & Kloeden (2009). In contrast to very restrictive assumptions made there, we reformulate appropriate conditions on the drift coefficient of SPDEs to include a large class of nonlinear Nemytskii operators. In our setting, the method achieves the convergence order in time of 1 2 − ϵ for arbitrarily small ϵ > 0 in the case of space–time white noise. For the trace-class noise case, multiple spatial dimensions are allowed and an optimal convergence rate is attained based on optimal regularity results of the mild solution, which improves the corresponding convergence results in Jentzen et al. (2011).

A note on optimal regularity and regularizing effects of point mass coupling for a heat–wave system

We consider a coupled 1D heat–wave system which serves as a simplified fluid–structure interaction problem. The system is coupled in two different ways: the first, when the interface does not have mass and the second, when the interface does have mass. We prove an optimal regularity result in Sobolev spaces for both cases. The main idea behind the proof is to reduce the coupled problem to a single nonlocal equation on the interface by using Neummann to Diriclet operator. Furthermore, we show that point mass coupling regularizes the problem and quantify this regularization in the sense of Sobolev spaces.

Global attractor for a nonlocal model for biological aggregation

We investigate the long term behavior in terms of global attractors, as time goes to infinity, of solutions to a continuum model for biological aggregations in which individuals experience long-range social attraction and short range dispersal. We consider the aggregation equation with both degenerate and non-degenerate diffusion in a bounded domain subject to various boundary conditions. In the degenerate case, we prove the existence of the global attractor and derive some optimal regularity results. Furthermore, in the non-degenerate case we give a complete structural characterization of the global attractor, and also discuss the convergence of any bounded solutions to steady states. In particular, under suitable assumptions on the parameters of the problem, we establish the convergence of the bounded solution u(t) to a single steady state u(*), and the rate of convergence parallel to u(t)-u(*)parallel to(Lp(Omega)) similar to (1+t)(-rho), as t -> infinity for any p> 1, and some rho=rho(u(*),rho)is an element of(0,1). Finally, the existence of an exponential attractor is also demonstrated for sufficiently smooth kernels in the case of non-degenerate diffusion. Our analysis extends and complements the analysis from [17] and many other fundamental works.

On finite Morse index solutions of two equations with negative exponent

We consider the following equations involving negative exponent:where p > 0. Under optimal conditions on the parameters α > −2 and p > 0, we prove the non-existence of finite Morse index solution on exterior domains or near the origin. We also prove an optimal regularity result for solutions with finite Morse index and isolated rupture at 0.

Singular degenerate boundary value problems and applications

The boundary value problems for linear and nonlinear singular degenerate differential-operator equations are studied. We prove the well-posedeness of the linear problem and optimal regularity result for the nonlinear problem which occur in fluid mechanics, environmental engineering and in the atmospheric dispersion of pollutants.

The Dirichlet problem for the convex envelope

The Convex Envelope of a given function was recently characterized as the solution of a fully nonlinear Partial Differential Equation (PDE). In this article we study a modified problem: the Dirichlet problem for the underlying PDE. The main result is an optimal regularity result. Differentiability (C1,α regularity) of the boundary data implies the corresponding result for the solution in the interior, despite the fact that the solution need not be continuous up to the boundary. Secondary results are the characterization of the convex envelope as: (i) the value function of a stochastic control problem, and (ii) the optimal underestimator for a class of nonlinear elliptic PDEs.