Local Boundedness Research Articles

Regularity results for a class of obstacle problems with p, q−growth conditions

In this paper we prove the the local Lipschitz continuity for solutions to a class of obstacle problems of the type min{ ∫ΩF(x, Dz) : z ∈ 𝛫ψ(Ω)}. Here 𝛫ψ(Ω) is the set of admissible functions z ∈ u0 + W1,p(Ω) for a given u0 ∈ W1,p(Ω) such that z ≥ ψ a.e. in Ω, ψ being the obstacle and Ω being an open bounded set of ℝn, n ≥ 2. The main novelty here is that we are assuming that the integrand F(x, Dz) satisfies (p, q)-growth conditions and as a function of the x-variable belongs to a suitable Sobolev class. We remark that the Lipschitz continuity result is obtained under a sharp closeness condition between the growth and the ellipticity exponents. Moreover, we impose less restrictive assumptions on the obstacle with respect to the previous regularity results. Furthermore, assuming the obstacle ψ is locally bounded, we prove the local boundedness of the solutions to a quite large class of variational inequalities whose principal part satisfies non standard growth conditions.

On the local regularity theory for the magnetohydrodynamic equations

Local regularity results are obtained for the MHD equations using as global framework the setting of parabolic Morrey spaces. Indeed, by assuming some local boundedness assumptions (in the sense of parabolic Morrey spaces) for weak solutions of the MHD equations it is possible to obtain a gain of regularity for such solutions in the general setting of the Serrin regularity theory. This is the first step of a wider program that aims to study both local and partial regularity theories for the MHD equations.

Cofinal completion vis-á-vis Cauchy continuity and total boundedness

A function f from a metric space (X,d) to another metric space (Y,ρ) is said to be Cauchy-continuous if for every Cauchy sequence (xn) in (X,d), (f(xn)) is Cauchy in (Y,ρ). It is well known that a metric space is complete if and only if every real-valued continuous function defined on it is Cauchy-continuous. Here we consider a well-studied intermediate class of metric spaces which lies between the class of compact metric spaces and that of complete metric spaces called cofinally complete metric spaces. In this paper, we use Cauchy-continuous functions and some Lipschitz-type functions to study a more general case, that is, to study the metric spaces which have a cofinal completion. We also study a geometric functional that measures the local total boundedness of a metric space at its each point and characterize the aforesaid metric spaces using it.

Meromorphic Extensions of $$(\cdot , W)$$-Meromorphic Functions

In this paper we establish some classes of subspaces W of the dual $$F'$$ of a locally convex space F such that every F-valued (F, W)-meromorphic function (with/without local boundedness) on a domain D in $$\mathbb {C}^n,$$ in the sense $$u \circ f$$ is meromorphic for all $$u \in W,$$ is meromorphic. Further, combining those results with studing on (BB)-Zorn property we give conditions for Frechet spaces E, F and subspaces W of $$F'$$ under which (F, W)-meromorphic functions can be meromorphically extended to a domain D of E from a subset $$D \cap E_B$$ where $$E_B$$ is the linear hull of some balanced convex compact subset B of E. Using these results we get the answers of the following questions: (1) When does the domain of meromorphy of a $$(\cdot , W)$$ -meromorphic function on a Riemann domain D over a Frechet space coincide with the envelope of holomorphy of D? (2) When will $$(\cdot , W)$$ -meromorphic functions be able to extend meromorphically through an analytic subset of codimension $$\ge 2$$ of a domain in a Frechet space?

PD Tracking for a Class of Underactuated Robotic Systems With Kinetic Symmetry

In this letter, we study stability properties of Proportional-Derivative (PD) controlled underactuated robotic systems for trajectory tracking applications. Stability of PD control laws for fully actuated systems is an established result, and we extend it for the class of underactuated robotic systems. We will first show some well known examples where PD tracking control laws do not yield tracking; some of which can even lead to instability. We will then show that for a subclass of robotic systems, PD tracking control laws, indeed, yield desirable tracking guarantees. We will show that for a specified time interval, and for sufficiently large enough PD gains (input saturations permitting), local boundedness of the tracking error can be guaranteed. In addition, for a class of systems with the kinetic symmetry property, stronger conditions like convergence to desirable bounds can be guaranteed. This class is not restrictive and includes robots like the acrobot, the cart-pole, and the inertia-wheel pendulums. Towards the end, we will provide necessary simulation results in support of the theoretical guarantees presented.

Growth conditions and regularity, an optimal local boundedness result

We prove local boundedness of local minimizers of scalar integral functionals [Formula: see text], [Formula: see text] where the integrand satisfies [Formula: see text]-growth of the form [Formula: see text] under the optimal relation [Formula: see text].

Off-singularity bounds and Hardy spaces for Fourier integral operators

We define a scale of Hardy spaces H F I O p ( R n ) \mathcal {H}^{p}_{FIO}(\mathbb {R}^n) , p ∈ [ 1 , ∞ ] p\in [1,\infty ] , that are invariant under suitable Fourier integral operators of order zero. This builds on work by Smith for p = 1 p=1 [J. Geom. Anal. 8 (1998), pp. 629–653]. We also introduce a notion of off-singularity decay for kernels on the cosphere bundle of R n \mathbb {R}^n , and we combine this with wave packet transforms and tent spaces over the cosphere bundle to develop a full Hardy space theory for oscillatory integral operators. In the process we extend the known results about L p L^{p} -boundedness of Fourier integral operators, from local boundedness to global boundedness for a larger class of symbols.

Pointwise estimates of solutions to the weighted porous medium equation and the fast diffusion one via weighted Riesz potentials

For the weighted parabolic equation v(x)ut − div (ω(x)um − 1 ∇ u) = f(x, t), u ≥ 0, m ≠ 1, we prove the local boundedness of weak solutions in terms of the weighted Riesz potential on the right-hand side of the equation.

Global solution for a general cross-diffusion two-competitive-predator and one-prey system with predator-taxis

This paper is concerned with a general cross-diffusion system modeling the population dynamics of two-competitive-predator and one-prey with predator-taxis. Firstly, through the use of contraction mapping principle, the Schauder estimates and Lp-estimates of parabolic equations, we investigate the local existence and boundedness of the unique classical solution for this system. Furthermore, the global existence and uniqueness of classical solution for this system are obtained base on some priori estimates. Finally, several numerical simulations are presented to demonstrate the effects of the predator-taxis to the population dynamics and support the theoretical results. The novel point of this work is that, in addition to the self-diffusion of one predator species, the cross-diffusion not only between predator and prey (predator-taxis) but also between two predator species (compete for available resources) are considered.

Local boundedness for minimizers of convex integral functionals in metric measure spaces

In this paper we consider the convex integral functional $ I := \int _\Omega {\Phi (g_u)\,d\mu } $ in the metric measure space $(X,d,\mu )$, where $X$ is a set, $d$ is a metric, µ is a Borel regular measure satisfying the doubling condition, Ω is a bounded open subset of $X$, $u$ belongs to the Orlicz-Sobolev space $N^{1,\Phi }(\Omega )$, Φ is an N-function satisfying the $\Delta _2$-condition, $g_u$ is the minimal Φ-weak upper gradient of $u$. By improving the corresponding method in the Euclidean space to the metric setting, we establish the local boundedness for minimizers of the convex integral functional under the assumption that $(X,d,\mu )$ satisfies the $(1,1)$-Poincaré inequality. The result of this paper can be applied to the Carnot-Carathéodory space spanned by vector fields satisfying Hörmander's condition.

Sobolev regularity of polar fractional maximal functions

We study the Sobolev regularity on the sphere Sd of the uncentered fractional Hardy–Littlewood maximal operator M˜β at the endpoint p=1, when acting on polar data. We first prove that if q=dd−β, 0<β<d and f is a polar W1,1(Sd) function, we have ‖∇M˜βf‖q≲d,β‖∇f‖1. We then prove that the map f↦|∇M˜βf| is continuous from W1,1(Sd) to Lq(Sd) when restricted to polar data. Our methods allow us to give a new proof of the continuity of the map f↦|∇M˜βf| from Wrad1,1(Rd) to Lq(Rd). Moreover, we prove that a conjectural local boundedness for the centered fractional Hardy–Littlewood maximal operator Mβ implies the continuity of the map f↦|∇Mβf| from W1,1 to Lq, in the context of polar functions on Sd and radial functions on Rd.

Solution maps of polynomial variational inequalities

In this paper, we investigate several properties of the solution maps of variational inequalities with polynomial data. First, we prove some facts on the $$R_0$$ -property, the local boundedness, and the upper semicontinuity of the solution maps. Second, we establish results on the solution existence and the local upper-Holder stability under the copositivity condition. Third, when the constraint set is semi-algebraic, we discuss the genericity of the $$R_0$$ -property and the finite-valuedness of the solution maps.

Regularity of quasi-minimizers for non-uniformly elliptic integrals

In this paper we consider a class of non-uniformly elliptic integral functionals and we prove the local boundedness of the quasi-minimizers. Our approach is based on a suitable adaptation of the celebrated De Giorgi proof, and it relies on an appropriate Caccioppoli-type inequality.

Quenched invariance principle for random walks among random degenerate conductances

We consider the random conductance model in a stationary and ergodic environment. Under suitable moment conditions on the conductances and their inverse, we prove a quenched invariance principle for the random walk among the random conductances. The moment conditions improve earlier results of Andres, Deuschel and Slowik (Ann. Probab. 43 (2015) 1866–1891) and are the minimal requirement to ensure that the corrector is sublinear everywhere. The key ingredient is an essentially optimal deterministic local boundedness result for finite difference equations in divergence form.

Hyperbolic Maxwell variational inequalities of the second kind

We analyze a class of hyperbolic Maxwell variational inequalities of the second kind. By means of a local boundedness assumption on the subdifferential of the underlying nonlinearity, we prove a well-posedness result, where the main tools for the proof are the semigroup theory for Maxwell’s equations, the Yosida regularization and the subdifferential calculus. The second part of the paper focuses on a more general case omitting the local boundedness assumption. In this case, taking into account more regular initial data and test functions, we are able to prove a weaker existence result through the use of the minimal section operator associated with the Nemytskii operator of the governing subdifferential. Eventually, we transfer the developed well-posedness results to the case involving Faraday’s law, which in particular allows us to improve the regularity property of the electric field in the weak existence result.

Exceptional sets for the definition of quasiregularity

We establish some quasiregularity conditions in terms of local boundedness of the inferior linear dilatation outside some exceptional sets E.

On the generalized [formula omitted] classes of De Giorgi-Ladyzhenskaya-Ural'tseva and pointwise estimates of solutions to high-order elliptic equations via Wolff potentials

We consider quasilinear elliptic 2m-order (m⩾2) partial differential equations which prototype is∑|α|=m(−1)|α|Dα(|Dmu|p−2Dαu)=f(x),x∈Ω, where Ω is a bounded open set in Rn, n=mp and f∈L1(Ω). Using an analogue of the Kilpeläinen-Malý method, we obtain the local boundedness and continuity of arbitrary weak solution u∈Wm,p(Ω) via the Wolff potential Wm,pf.

Local Boundedness and Harnack Inequality for Solutions of Linear Nonuniformly Elliptic Equations

AbstractWe study local regularity properties for solutions of linear, nonuniformly elliptic equations. Assuming certain integrability conditions on the coefficient field, we prove local boundedness and Harnack inequality. The assumed integrability assumptions are essentially sharp and improve upon classical results by Trudinger. We then apply the deterministic regularity results to the corrector equation in stochastic homogenization and establish sublinearity of the corrector. © 2019 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, Inc.

Local properties for weak solutions of nonlocal heat equations

In this paper, using the De Giorgi–Nash–Moser theory, we obtain an interior parabolic Hölder regularity for weak solutions of nonlocal heat equations given by an integro-differential operator LK as follows; LKu+∂tu=0in Ω×(−T,0]u=gin ((Rn∖Ω)×(−T,0])∪(Ω×{t=−T})where g∈C(Rn×[−T,0])∩L∞(Rn×(−T,0])∩HTs(Rn) and Ω⊂Rn is a bounded domain with Lipschitz boundary. In addition, we get the local boundedness of such weak solutions.

Limit Theorems for Cylindrical Martingale Problems Associated with Lévy Generators

We prove limit theorems for cylindrical martingale problems associated with Levy generators. Furthermore, we give sufficient and necessary conditions for the Feller property of well-posed problems with continuous coefficients. We discuss two applications. First, we derive continuity and linear growth conditions for the existence of weak solutions to infinite-dimensional stochastic differential equations driven by Levy noise. Second, we derive continuity, local boundedness and linear growth conditions for limit theorems and the Feller property of weak solutions to stochastic partial differential equations driven by Wiener noise.