Parabolic oblique derivative problem with discontinuous coefficients in generalized Morrey spaces

We study the global Morrey-type regularity of the solution of the regular oblique derivative problem for linear uniformly parabolic operators with $VMO$ coefficients.     

Regularity results for fully nonlinear parabolic integro-differential operators

In this paper, we consider the regularity theory for fully nonlinear parabolic integro-differential equations with symmetric kernels. We are able to find parabolic versions of Alexandrov–Backelman–Pucci estimate with $0<\sigma <2$ . And we show a Harnack inequality, Hölder regularity, and $C^{1,\alpha }$ -regularity of the solutions by obtaining decay estimates of their level sets.     

The Laplacian on Cylindrical Domains

The aim of this paper is to prove elliptic regularity and parabolic maximal regularity of the Laplacian with mixed boundary conditions on domains Ω carrying a cylindrical structure. More precisely, we consider Ω to be given as the Cartesian product of whole or half spaces, a cube ${\mathcal{Q}}$ , and a standard domain V having compact boundary. Taking advantage of this structure we apply operator-valued Fourier multiplier results to transfer ${\mathcal{H}^{\infty}}$ -calculus results known for the Laplacian in L ^p (V) to the Laplacian in L ^p (Ω). This approach turns out to inherit elliptic regularity, i.e. the domain of the Dirichlet Laplacian equals ${W^{2,p}(\Omega) \cap W_0^{1,p}(\Omega)}$ , for instance. This is surprising since Ω may be unbounded and non-convex with boundary neither compact nor of class C ^1,1 at the same time. More generally, we consider the following mixture of boundary conditions: on every smooth part of the boundary Dirichlet or Neumann boundary conditions are imposed and on parts related to ${\mathcal{Q}}$ generalized periodic boundary conditions are included. Via ${\mathcal{R}}$ -sectoriality we deduce maximal regularity in the parabolic sense which seems to be new for this general class of boundary conditions. Parabolic equations with such a mixture of boundary conditions on such type of domains appear for example in models describing growth of biological cells.     

Continuous maximal regularity on uniformly regular Riemannian manifolds

We establish continuous maximal regularity results for parabolic differential operators acting on sections of tensor bundles on uniformly regular Riemannian manifolds M. As an application, we show that solutions to the Yamabe flow on M instantaneously regularize and become real analytic in space and time. The regularity result is obtained by introducing a family of parameter-dependent diffeomorphisms acting on functions on M in conjunction with maximal regularity and the implicit function theorem.     

Everywhere regularity of certain nonlinear diffusion systems

In this paper I discuss nonlinear parabolic systems that are generalizations of scalar diffusion equations. More precisely, I consider systems of the form $$\mathbf{u}_t -\Delta\left[ \mathbf{\nabla}\Phi(\mathbf{u})\right] = 0,$$ where ${\Phi(z)}$ is a strictly convex function. I show that when ${\Phi}$ is a function only of the norm of u, then bounded weak solutions of these parabolic systems are everywhere Hölder continuous and thus everywhere smooth. I also show that the method used to prove this result can be easily adopted to simplify the proof of the result due to Wiegner (Math Ann 292(4):711–727, 1992) on everywhere regularity of bounded weak solutions of strongly coupled parabolic systems.     

Open subgroups of locally compact Kac–Moody groups

Let $G$ be a complete Kac–Moody group over a finite field. It is known that $G$ possesses a BN-pair structure, all of whose parabolic subgroups are open in $G$ . We show that, conversely, every open subgroup of $G$ is contained with finite index in some parabolic subgroup; moreover there are only finitely many such parabolic subgroups. The proof uses some new results on parabolic closures in Coxeter groups. In particular, we give conditions ensuring that the parabolic closure of the product of two elements in a Coxeter group contains the respective parabolic closures of those elements.     

Lower large deviations for the maximal flow through a domain of {\mathbb{R}^d} in first passage percolation

We consider the standard first passage percolation model in the rescaled graph ${\mathbb{Z}^d/n}$ for d??? 2, and a domain ?? of boundary ?? in ${\mathbb{R}^d}$ . Let ??¹ and ??² be two disjoint open subsets of ??, representing the parts of ?? through which some water can enter and escape from ??. We investigate the asymptotic behaviour of the flow ${\phi_n}$ through a discrete version ?? _n of ?? between the corresponding discrete sets ${\Gamma^{1}_{n}}$ and ${\Gamma^{2}_{n}}$ . We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the lower large deviations of ${\phi_n/ n^{d-1}}$ below a certain constant are of surface order.     

A uniformly stable Fortin operator for the Taylor–Hood element

We construct a new Fortin operator for the lowest order Taylor–Hood element, which is uniformly stable both in $L^2$ and $H^1$ . The construction, which is restricted to two space dimensions, is based on a tight connection between a subspace of the Taylor–Hood velocity space and the lowest order Nedelec edge element. General shape regular triangulations are allowed for the $H^1$ -stability, while some mesh restrictions are imposed to obtain the $L^2$ -stability. As a consequence of this construction, a uniform inf–sup condition associated the corresponding discretizations of a parameter dependent Stokes problem is obtained, and we are able to verify uniform bounds for a family of preconditioners for such problems, without relying on any extra regularity ensured by convexity of the domain.     

A general regularity theory for weak mean curvature flow

We give a new proof of Brakke’s partial regularity theorem up to $C^{1,\varsigma }$ for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and any given background flow field in a dimensionally sharp integrability class. It is a natural parabolic generalization of Allard’s regularity theorem in the sense that the special time-independent case reduces to Allard’s theorem.     

On some free boundary problem for a compressible barotropic viscous fluid flow

In this paper, we prove a local in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the \(L_p\) in time and \(L_q\) in space framework with \(2 < p < \infty \) and \(N < q < \infty \) under the assumption that the initial domain is a uniform \(W^{2-1/q}_q\) one in \({\mathbb {R}}^{N}\, (N \ge 2\) ). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve problem by the Banach contraction mapping principle based on the maximal \(L_p\) – \(L_q\) regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key issue for the linear theorem is the existence of \({\mathcal {R}}\) -bounded solution operator in a sector, which combined with Weis’s operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal \(L_p\) – \(L_q\) regularity theorem. The nonlinear problem we studied here was already investigated by several authors (Denisova and Solonnikov, St. Petersburg Math J 14:1–22, 2003; J Math Sci 115:2753–2765, 2003; Secchi, Commun PDE 1:185–204, 1990; Math Method Appl Sci 13:391–404, 1990; Secchi and Valli, J Reine Angew Math 341:1–31, 1983; Solonnikov and Tani, Constantin carathéodory: an international tribute, vols 1, 2, pp 1270–1303, World Scientific Publishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin, 1992; Tani, J Math Kyoto Univ 21:839–859, 1981; Zajaczkowski, SIAM J Math Anal 25:1–84, 1994) in the \(L_2\) framework and Hölder spaces, but our approach is different from them.     

On Metric Regularity and the Boundary of the Feasible Set in Linear Optimization

This paper deals with semi-infinite linear inequality systems in ? ⁿ and studies the stability of the boundary of their feasible sets. We analyze the equivalence between the metric regularity of the inverse of the boundary set mapping, $\mathcal{N}$ , and the stability of the feasible set mapping in the sense of the maintenance of the consistency. In doing this we provide operational formulae for distances from points to some useful sets. We also include relationships between the regularity moduli corresponding to the mappings $\mathcal{N}$ and the inverse, $\mathcal{M}$ , of the feasible set mapping, and prove their equality for finite systems and some special cases in the semi-infinite framework. Moreover, we provide conditions to assure that the metric regularity of $\mathcal{N}$ is equivalent to the lower semi-continuity of the boundary set mapping, which is important because the latter property has many characterizations. Since the boundary of a feasible set may not be convex, we cannot make use of the general theory for mappings with convex graph, as for example, the Robinson–Ursescu theorem.     

Sobolev regularity of the \overline{\partial }-equation on the Hartogs triangle

The regularity of the $\overline{\partial }$ -problem on the domain $\{\left|{z_1}\right|\!<\!\left|{z_2}\right|\!<\!1\}$ in $\mathbb C ^2$ is studied using $L^2$ -methods. Estimates are obtained for the canonical solution in weighted $L^2$ -Sobolev spaces with a weight that is singular at the point $(0,0)$ . In particular, the singularity of the Bergman projection for the Hartogs triangle is contained at the singular point and it does not propagate.     

Regularity of the Monge–Ampère equation in Besov’s spaces

Let $\mu = e^{-V} \ dx$ be a probability measure and $T = \nabla \Phi $ be the optimal transportation mapping pushing forward $\mu $ onto a log-concave compactly supported measure $\nu = e^{-W} \ dx$ . In this paper, we introduce a new approach to the regularity problem for the corresponding Monge–Ampère equation $e^{-V} = \det D^2 \Phi \cdot e^{-W(\nabla \Phi )}$ in the Besov spaces $W^{\gamma ,1}_{loc}$ . We prove that $D^2 \Phi \in W^{\gamma ,1}_{loc}$ provided $e^{-V}$ belongs to a proper Besov class and $W$ is convex. In particular, $D^2 \Phi \in L^p_{loc}$ for some $p>1$ . Our proof does not rely on the previously known regularity results.     

Inverse problem of finding n coefficients of lower derivatives in a parabolic equation

We consider the problem of reconstructing the vector function $\vec b(x) = (b_1 ,...,b_n )$ in the term $(\vec b,\nabla u)$ in a linear parabolic equation. This coefficient inverse problem is considered in a bounded domain Ω ? R ⁿ . To find the above-mentioned function $\vec b(x)$ , in addition to initial and boundary conditions we pose an integral observation of the form $\int_0^T {u(x,t)\vec \omega (t)dt = \vec \chi (x)} $ , where $\vec \omega (t) = (\omega _1 (t),...,\omega _n (t))$ is a given weight vector function. We derive sufficient existence and uniqueness conditions for the generalized solution of the inverse problem. We present an example of input data for which the assumptions of the theorems proved in the paper are necessarily satisfied.     

Construction of {\mathcal L}^{{p}}-strong Feller Processes via Dirichlet Forms and Applications to Elliptic Diffusions

We provide a general construction scheme for $\mathcal L^p$ -strong Feller processes on locally compact separable metric spaces. Starting from a regular Dirichlet form and specified regularity assumptions, we construct an associated semigroup and resolvent of kernels having the $\mathcal L^p$ -strong Feller property. They allow us to construct a process which solves the corresponding martingale problem for all starting points from a known set, namely the set where the regularity assumptions hold. We apply this result to construct elliptic diffusions having locally Lipschitz matrix coefficients and singular drifts on general open sets with absorption at the boundary. In this application elliptic regularity results imply the desired regularity assumptions.     

Laplace Equation on a Domain With a Cuspidal Point in Little Hölder Spaces

In this paper, we give new results about existence, uniqueness and regularity properties for solutions of Laplace equation $$\Delta u = h \quad {\rm in} \, \Omega$$ where Ω is a cusp domain. We impose nonhomogeneous Dirichlet conditions on some part of ?Ω. The second member h will be taken in the little Hölder space ${h^{2 \sigma}(\bar{\Omega})}$ with ${\sigma \, \in \, ]0, \, 1/2[}$ . Our approach is based essentially on the study of an abstract elliptic differential equation set in an unbounded domain. We will use the continuous interpolation spaces and the generalized analytic semigroup theory.     

Shape derivatives for minima of integral functionals

For \(\Omega \) varying among open bounded sets in \(\mathbb R ^n\) , we consider shape functionals \(J (\Omega )\) defined as the infimum over a Sobolev space of an integral energy of the kind \(\int _\Omega [ f (\nabla u) + g (u) ]\) , under Dirichlet or Neumann conditions on \(\partial \Omega \) . Under fairly weak assumptions on the integrands \(f\) and \(g\) , we prove that, when a given domain \(\Omega \) is deformed into a one-parameter family of domains \(\Omega _\varepsilon \) through an initial velocity field \(V\in W ^ {1, \infty } (\mathbb R ^n, \mathbb R ^n)\) , the corresponding shape derivative of \(J\) at \(\Omega \) in the direction of \(V\) exists. Under some further regularity assumptions, we show that the shape derivative can be represented as a boundary integral depending linearly on the normal component of \(V\) on \(\partial \Omega \) . Our approach to obtain the shape derivative is new, and it is based on the joint use of Convex Analysis and Gamma-convergence techniques. It allows to deduce, as a companion result, optimality conditions in the form of conservation laws.     

Approximation of analytical functions by k-positive linear operators in the closed domain

This work treats the problem of convergence for the sequences of linear \(k\) -positive operators on a space of functions that are analytic in a closed domain. By convergence in this space, we mean a uniform convergence in a closed domain that contains the original domain strictly inside itself, while the linear \(k\) -positive operators are naturally associated with Faber polynomials related to the considered domain. Until now, this problem has been solved in the space of functions analytic in an open bounded domain with the topology of compact convergence.