Showing papers in "Potential Analysis in 2020"

Beurling–Ahlfors Commutators on Weighted Morrey Spaces and Applications to Beltrami Equations

Abstract: Let $p\in (1, \infty )$ , ? ? (0,1) and $w\in A_{p}({\mathbb C}).$ In this article, the authors obtain a boundedness (resp., compactness) characterization of the Beurling–Ahlfors commutator $[\mathcal B, b]$ on the weighted Morrey space $L_{w}^{p, \kappa }(\mathbb C)$ via $\text {BMO}({\mathbb C})$ [resp., $\text {CMO}({\mathbb C})$ ], where $\mathcal B$ denotes the Beurling–Ahlfors transform and $b\in \text {BMO}({\mathbb C})$ [resp., $\text {CMO}({\mathbb C})$ ]. Moreover, an application to the Beltrami equation is also given.

Heat Kernel Estimates of Fractional Schrödinger Operators with Negative Hardy Potential

Abstract: We obtain two-sided estimates for the heat kernel (or the fundamental function) associated with the following fractional Schrodinger operator with negative Hardy potential Δα/2 − λ|x|−α $$ {\Delta}^{\alpha/2} -\lambda |x|^{-\alpha} $$ on , where α ∈ (0, d ∧ 2) and λ > 0. The proof is purely analytical and elementary. In particular, for upper bounds of heat kernel we use the Chapman-Kolmogorov equation and adopt self-improving argument.

On the Essential Self-Adjointness of Singular Sub-Laplacians

Abstract: We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. We also show that, in the compact case, this criterion implies discreteness of the sub-Laplacian spectrum even though the total volume of the manifold is infinite. As a consequence of our result, the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp’s measure) is essentially self-adjoint on the equiregular connected components of a sub-Riemannian manifold. This settles a conjecture formulated by Boscain and Laurent (Ann. Inst. Fourier (Grenoble) 63(5), 1739–1770, 2013), under mild regularity assumptions on the singular region, and when the latter does not contain characteristic points.

On the Boundary Theory of Subordinate Killed Lévy Processes

Abstract: Let Z be a subordinate Brownian motion in $\mathbb {R}^{d}$ , d ≥ 2, via a subordinator with Laplace exponent ϕ. We kill the process Z upon exiting a bounded open set $D\subset \mathbb {R}^{d}$ to obtain the killed process ZD, and then we subordinate the process ZD by a subordinator with Laplace exponent ψ. The resulting process is denoted by YD. Both ϕ and ψ are assumed to satisfy certain weak scaling conditions at infinity. We study the potential theory of YD, in particular the boundary theory. First, in case that D is a κ-fat bounded open set, we show that the Harnack inequality holds. If, in addition, D satisfies the local exterior volume condition, then we prove the Carleson estimate. In case D is a smooth open set and the lower weak scaling index of ψ is strictly larger than 1/2, we establish the boundary Harnack principle with explicit decay rate near the boundary of D. On the other hand, when ψ(λ) = λγ with γ ∈ (0, 1/2], we show that the boundary Harnack principle near the boundary of D fails for any bounded C1,1 open set D. Our results give the first example where the Carleson estimate holds true, but the boundary Harnack principle does not. One of the main ingredients in the proofs is the sharp two-sided estimates of the Green function of YD. Under an additional condition on ψ, we establish sharp two-sided estimates of the jumping kernel of YD which exhibit some unexpected boundary behavior. We also prove a boundary Harnack principle for non-negative functions harmonic in a smooth open set E strictly contained in D, showing that the behavior of YD in the interior of D is determined by the composition ψ ∘ ϕ.

Optimisation of the Lowest Robin Eigenvalue in the Exterior of a Compact Set, II: Non-Convex Domains and Higher Dimensions

Abstract: We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact set in any dimension, subject to attractive Robin boundary conditions. As an improvement upon our previous work (Krejciřik and Lotoreichik J. Convex Anal. 25, 319–337, 2018), we show that under either a constraint of fixed perimeter or area, the maximiser within the class of exteriors of simply connected planar sets is always the exterior of a disk, without the need of convexity assumption. Moreover, we generalise the result to disconnected compact planar sets. Namely, we prove that under a constraint of fixed average value of the perimeter over all the connected components, the maximiser within the class of disconnected compact planar sets, consisting of finitely many simply connected components, is again a disk. In higher dimensions, we prove a completely new result that the lowest point in the spectrum is maximised by the exterior of a ball among all sets exterior to bounded convex sets satisfying a constraint on the integral of a dimensional power of the mean curvature of their boundaries. Furthermore, it follows that the critical coupling at which the lowest point in the spectrum becomes a discrete eigenvalue emerging from the essential spectrum is minimised under the same constraint by the critical coupling for the exterior of a ball.

Stability of Stochastic Functional Differential Equations with Regime-Switching: Analysis Using Dupire’s Functional Itô Formula

Abstract: This work focuses on almost sure and Lp stability of stochastic functional differential equations by using Lyapunov functionals with the help of the recently developed Dupire’s functional Ito formula. Novel conditions for stability, which are different from those in the existing literature, are given in terms of Lyapunov functionals. It is demonstrated that the conditions are useful for stochastic stabilization. It is also shown that adding a diffusion term can stabilize an unstable system of deterministic differential equations with Markov switching. Furthermore, a robustness result is obtained, which states that the stability of stochastic differential equations with regime-switching is preserved under delayed perturbations when the delay is small enough.

Harnack Inequality for Quasilinear Elliptic Equations in Generalized Orlicz-Sobolev Spaces

Abstract: In this paper we prove, by a new method, the Harnack inequality for positive solutions of quasilinear elliptic equations in the generalized Orlicz-Sobolev space setting. Our approach is based on the usage of the Φ-functions associated to generalized Φ-functions and the Moser’s iteration technique. As a consequence, we obtain the Holder continuity of bounded solutions of such equations.

Green’s Function for Second Order Elliptic Equations in Non-divergence Form

Abstract: We construct the Green function for second order elliptic equations in non-divergence form when the mean oscillations of the coefficients satisfy the Dini condition and the domain has C1,1 boundary. We also obtain pointwise bounds for the Green functions and its derivatives.

Boundedness of Operators on Certain Weighted Morrey Spaces Beyond the Muckenhoupt Range

Abstract: We prove that for operators satistying weighted inequalities with Ap weights the boundedness on a certain class of Morrey spaces holds with weights of the form |x|αw(x) for w ∈ Ap. In the case of power weights the shift with respect to the range of Muckenhoupt weights was observed by N. Samko for the Hilbert transform, by H. Tanaka for the Hardy-Littlewood maximal operator, and by S. Nakamura and Y. Sawano for Calderon-Zygmund operators and others. We extend the class of weights and establish the results in a very general setting, with applications to many operators. For weak type Morrey spaces, we obtain new estimates even for the Hardy-Littlewood maximal operator. Moreover, we prove the necessity of certain Aq condition.

Irreducibility and Asymptotics of Stochastic Burgers Equation Driven by α -stable Processes

Abstract: The irreducibility, moderate deviation principle and ψ-uniformly exponential ergodicity with ψ(x) := 1 + ∥x∥0 are proved for stochastic Burgers equation driven by the α-stable processes for α ∈ (1, 2), where the first two are new for the present model, and the last strengthens the exponential ergodicity under total variational norm derived in Dong et al. (J. Stat. Phys. 154:929–949, 2014).

Dirichlet Heat Kernel for the Laplacian in a Ball

Abstract: We provide sharp two-sided estimates on the Dirichlet heat kernel k1(t, x, y) for the Laplacian in a ball. The result accurately describes the exponential behaviour of the kernel for small times and significantly improves the qualitatively sharp results known so far. As a consequence we obtain the full description of the kernel k1(t, x, y) in terms of its global two-sided sharp estimates. Such precise estimates were possible to obtain due to the enrichment of analytical methods with probabilistic tools.

Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature

Abstract: Let Pt be the (Neumann) diffusion semigroup Pt generated by a weighted Laplacian on a complete connected Riemannian manifold M without boundary or with a convex boundary. It is well known that the Bakry-Emery curvature is bounded below by a positive constant ≪> 0 if and only if $$W_{p}(\mu_{1}P_{t}, \mu_{2}P_{t})\leq e^{-\ll t} W_{p} (\mu_{1},\mu_{2}),\ \ t\geq 0, p\geq 1 $$ holds for all probability measures μ1 and μ2 on M, where Wp is the Lp Wasserstein distance induced by the Riemannian distance. In this paper, we prove the exponential contraction $$W_{p}(\mu_{1}P_{t}, \mu_{2}P_{t})\leq ce^{-\ll t} W_{p} (\mu_{1},\mu_{2}),\ \ p \geq 1, t\geq 0$$ for some constants c,≪> 0 for a class of diffusion semigroups with negative curvature where the constant c is essentially larger than 1. Similar results are derived for SDEs with multiplicative noise by using explicit conditions on the coefficients, which are new even for SDEs with additive noise.

Decoupled Mild Solutions of Path-Dependent PDEs and Integro PDEs Represented by BSDEs Driven by Cadlag Martingales

Abstract: We focus on a class of path-dependent problems which include path-dependent PDEs and Integro PDEs (in short IPDEs), and their representation via BSDEs driven by a cadlag martingale. For those equations we introduce the notion of decoupled mild solution for which, under general assumptions, we study existence and uniqueness and its representation via the aforementioned BSDEs. This concept generalizes a similar notion introduced by the authors in recent papers in the framework of classical PDEs and IPDEs. For every initial condition (s, η), where s is an initial time and η an initial path, the solution of such BSDE produces a couple of processes (Ys, η, Zs, η). In the classical (Markovian or not) literature the function $u(s,\eta ):= Y^{s,\eta }_{s}$ constitutes a viscosity type solution of an associated PDE (resp. IPDE); our approach allows not only to identify u as the unique decoupled mild solution, but also to solve quite generally the so called identification problem, i.e. to also characterize the (Zs, η)s, η processes in term of a deterministic function v associated to the (above decoupled mild) solution u.

Uniform Shapiro-Lopatinski Conditions and Boundary Value Problems on Manifolds with Bounded Geometry

Abstract: We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem (P,C) is equivalent to the uniform regularity of the natural family (Px,Cx) of associated boundary value problems in local coordinates. We verify that this property is satisfied for the Dirichlet boundary conditions and strongly elliptic operators via a compactness argument. We then introduce a uniform Shapiro-Lopatinski regularity condition, which is a modification of the classical one, and we prove that it characterizes the boundary value problems that satisfy the usual regularity property. We also show that the natural Robin boundary conditions always satisfy the uniform Shapiro-Lopatinski regularity condition, provided that our operator satisfies the strong Legendre condition. This is achieved by proving that “well-posedness implies regularity” via a modification of the classical “Nirenberg trick”. When combining our regularity results with the Poincare inequality of (Ammann-Grose-Nistor, preprint 2015), one obtains the usual well-posedness results for the classical boundary value problems in the usual scale of Sobolev spaces, thus extending these important, well-known theorems from smooth, bounded domains, to manifolds with boundary and bounded geometry. As we show in several examples, these results do not hold true anymore if one drops the bounded geometry assumption. We also introduce a uniform Agmon condition and show that it is equivalent to the coerciveness. Consequently, we prove a well-posedness result for parabolic equations whose elliptic generator satisfies the uniform Agmon condition.

Local Continuity and Harnack’s Inequality for Double-Phase Parabolic Equations

Abstract: We consider parabolic equations of the form $$ u_{t}-\text{div} \left( | abla u|^{p-2} abla u+ a(x,t)| abla u|^{q-2} abla u\right)= 0, a(x,t)\geq 0. $$ In the range $\frac {2n}{n+1}

Asymptotic Lipschitz regularity for tug-of-war games with varying probabilities

Abstract: We prove an asymptotic Lipschitz estimate for value functions of tug-of-war games with varying probabilities defined in Ω ⊂ ℝn. The method of the proof is based on a game-theoretic idea to estimate the value of a related game defined in Ω ×Ω via couplings.

Multiplicity and Concentration Results for Fractional Schrödinger-Poisson Equations with Magnetic Fields and Critical Growth

Abstract: We deal with the following fractional Schrodinger-Poisson equation with magnetic field $$\varepsilon^{2s}(-{\Delta})_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u+|u|^{{2}_{s}^{*}-2}u \quad \text{ in } \mathbb{R}^{3}, $$ where e > 0 is a small parameter, $s\in (\frac {3}{4}, 1)$, t ∈ (0, 1), ${2}_{s}^{*}=\frac {6}{3-2s}$ is the fractional critical exponent, $(-{\Delta })^{s}_{A}$ is the fractional magnetic Laplacian, $V:\mathbb {R}^{3}\rightarrow \mathbb {R}$ is a positive continuous potential, $A:\mathbb {R}^{3}\rightarrow \mathbb {R}^{3}$ is a smooth magnetic potential and $f:\mathbb {R}\rightarrow \mathbb {R}$ is a subcritical nonlinearity. Under a local condition on the potential V, we study the multiplicity and concentration of nontrivial solutions as $\varepsilon \rightarrow 0$. In particular, we relate the number of nontrivial solutions with the topology of the set where the potential V attains its minimum.

Unbounded Weighted Composition Operators on Fock space

Abstract: In this paper, we consider unbounded weighted composition operators acting on Fock space, and investigate some important properties of these operators, such as $\mathcal {C}$ -selfadjoint (with respect to weighted composition conjugations), Hermitian, normal, and cohyponormal. In addition, the paper shows that unbounded normal weighted composition operators are contained properly in the class of $\mathcal {C}$ -selfadjoint operators with respect to weighted composition conjugations.

The torsion function of convex domains of high eccentricity

Abstract: The torsion function of a convex planar domain Ω has convex level sets, but explicit formulae are known only for rectangles and ellipses. Here we study the torsion function when the eccentricity of the domain is large. We obtain an approximation for the torsion function on any convex planar domain by viewing the domain as a perturbation of a rectangle in order to define an approximate Green’s function for the Laplacian. For a class of convex domains we use this approximation to establish sharp bounds on the Hessian and the infinitesimal shape of the level sets around its maximum. We also use these results to construct examples demonstrating contrasting behaviour of the torsion function and the first eigenfunction of the Dirichlet Laplacian around their respective maxima.

Discrete Harmonic Analysis Associated with Ultraspherical Expansions

Abstract: In this paper we study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted lp-boundedness properties of maximal operators and Littlewood-Paley g-functions defined by Poisson and heat semigroups generated by the difference operator $$ {\Delta}_{\lambda} f(n):=a_{n}^{\lambda} f(n+1)-2f(n)+a_{n-1}^{\lambda} f(n-1),\quad n\in \mathbb{N}, \lambda >0, $$ where $a_{n}^{\lambda } :=\{(2\lambda +n)(n+1)/[(n+\lambda )(n+1+\lambda )]\}^{1/2}$ , $n\in \mathbb {N}$ , and $a_{-1}^{\lambda }:=0$ . We also prove weighted lp-boundedness properties of transplantation operators associated with the system $\{\varphi _{n}^{\lambda } \}_{n\in \mathbb {N}}$ of ultraspherical functions, a family of eigenfunctions of Δλ. In order to show our results we previously establish a vector-valued local Calderon-Zygmund theorem in our discrete setting.

Pathwise Uniqueness for a Class of SPDEs Driven by Cylindrical α-Stable Processes

Abstract: By studying the infinite dimensional Kolmogorov equation with non-local operator, we show the pathwise uniqueness for stochastic partial differential equation driven by cylindrical α-stable process with Holder continuous drift.

Quasiopen Sets, Bounded Variation and Lower Semicontinuity in Metric Spaces

Abstract: In the setting of a complete metric space that is equipped with a doubling measure and supports a Poincare inequality, we show that the total variation of functions of bounded variation is lower semicontinuous with respect to L1-convergence in every 1-quasiopen set. To achieve this, we first prove a new characterization of the total variation in 1-quasiopen sets. Then we utilize the lower semicontinuity to show that the variation measures of a sequence of functions of bounded variation converging in the strict sense are uniformly absolutely continuous with respect to the 1-capacity.

Desingularization of a Steady Vortex Pair in the Lake Equation

Abstract: We construct a family of steady solutions of the lake model perturbed by some small Coriolis force, that converge to a singular vortex pair. The desingularized solutions are obtained by maximization of the kinetic energy over a class of rearrangements of sign changing functions. The precise localization of the asymptotic singular vortex pair is proved to depend on the depth function and the Coriolis parameter, and it is independent on the geometry of the lake domain. We apply our result to construct a singular rotating vortex pair in a rotation invariant lake.

Reilly-Type Inequalities for Paneitz and Steklov Eigenvalues

Abstract: We prove Reilly-type upper bounds for different types of eigenvalue problems on submanifolds of Euclidean spaces with density. This includes the eigenvalues of Paneitz-like operators as well as three types of generalized Steklov problems. In the case without density, the equality cases are discussed and we prove some stability results for hypersurfaces which derive from a general pinching result about the moment of inertia.

Ergodicity and Kolmogorov Equations for Dissipative SPDEs with Singular Drift: a Variational Approach

Abstract: We prove existence of invariant measures for the Markovian semigroup generated by the solution to a parabolic semilinear stochastic PDE whose nonlinear drift term satisfies only a kind of symmetry condition on its behavior at infinity, but no restriction on its growth rate is imposed. Thanks to strong integrability properties of invariant measures μ, solvability of the associated Kolmogorov equation in L1(μ) is then established, and the infinitesimal generator of the transition semigroup is identified as the closure of the Kolmogorov operator. A key role is played by a generalized variational setting.

Various Concepts of Riesz Energy of Measures and Application to Condensers with Touching Plates

Abstract: We develop further the concept of weak α-Riesz energy with α ∈ (0,2] of Radon measures μ on $\mathbb R^{n}$ , $n\geqslant 3$ , introduced in our preceding study and defined by $\int \limits (\kappa _{\alpha /2}\mu )^{2} dm$ , m denoting the Lebesgue measure on $\mathbb R^{n}$ . Here κα/2μ is the potential of μ relative to the α/2-Riesz kernel |x − y|α/2−n. This concept extends that of standard α-Riesz energy, and for μ with κα/2μ ∈ L2(m) it coincides with that of Deny-Schwartz energy defined with the aid of the Fourier transform. We investigate minimum weak α-Riesz energy problems with external fields in both the unconstrained and constrained settings for generalized condensers (A1,A2) such that the closures of A1 and A2 in $\mathbb R^{n}$ are allowed to intersect one another. (Such problems with the standard α-Riesz energy in place of the weak one would be unsolvable, which justifies the need for the concept of weak energy when dealing with condenser problems.) We obtain sufficient and/or necessary conditions for the existence of minimizers, provide descriptions of their supports and potentials, and single out their characteristic properties. To this end we have discovered an intimate relation between minimum weak α-Riesz energy problems over signed measures associated with (A1,A2) and minimum α-Green energy problems over positive measures carried by A1. Crucial for our analysis of the latter problems is the perfectness of the α-Green kernel, established in our recent paper. As an application of the results obtained, we describe the support of the α-Green equilibrium measure.

Dimension drop for harmonic measure on Ahlfors regular boundaries

Abstract: We show that given a domain ${\Omega }\subseteq \mathbb {R}^{d+1}$ with uniformly non-flat Ahlfors s-regular boundary with s ≥ d, the dimension of its harmonic measure is strictly less than s.

Two-Dimensional Brownian Random Interlacements

Abstract: We introduce the model of two-dimensional continuous random interlacements, which is constructed using the Brownian trajectories conditioned on not hitting a fixed set (usually, a disk). This model yields the local picture of Wiener sausage on the torus around a late point. As such, it can be seen as a continuous analogue of discrete two-dimensional random interlacements (Comets et al. Commun. Math. Phys. 343, 129–164, 2016). At the same time, one can view it as (restricted) Brownian loops through infinity. We establish a number of results analogous to these of Comets and Popov (Ann. Probab. 45, 4752–4785, 2017), Comets et al. (Commun. Math. Phys. 343, 129–164, 2016), as well as the results specific to the continuous case.

Weighted Calderon-Zygmund Estimates for Weak Solutions of Quasi-Linear Degenerate Elliptic Equations

Abstract: This paper studies regularity estimates in Sobolev spaces for weak solutions of a class of degenerate and singular quasi-linear elliptic problems of the form div[A(x,u,∇u)] = div[F] with non-homogeneous Dirichlet boundary conditions over bounded non-smooth domains. The coefficients A could be be singular, degenerate or both in x in the sense that they behave like some weight function μ, which is in the A2 class of Muckenhoupt weights. Global and interior weighted W1,p(Ω,ω)-regularity estimates are established for weak solutions of these equations with some other weight function ω. The results obtained are even new for the case μ = 1 because of the dependence on the solution u of A. In case of linear equations, our W1,p-regularity estimates can be viewed as the Sobolev’s counterpart of the Holder’s regularity estimates established by B. Fabes, C. E. Kenig, and R. P. Serapioni.

Blow-Up Results for Space-Time Fractional Stochastic Partial Differential Equations

Abstract: Consider non-linear time-fractional stochastic reaction-diffusion equations of the following type, $$ \partial^{\beta}_{t}u_{t}(x)=- u(-{\Delta})^{\alpha/2} u_{t}(x)+I^{1-\beta}[b(u)+ \sigma(u)\stackrel{\cdot}{F}(t,x)] $$ in (d + 1) dimensions, where ν > 0,β ∈ (0, 1), α ∈ (0, 2]. The operator $\partial ^{\beta }_{t}$ is the Caputo fractional derivative while − (−Δ)α/2 is the generator of an isotropic α-stable Levy process and I1−β is the Riesz fractional integral operator. The forcing noise denoted by $\stackrel {\cdot }{F}(t,x)$ is a Gaussian noise. These equations might be used as a model for materials with random thermal memory. We derive non-existence (blow-up) of global random field solutions under some additional conditions, most notably on b, σ and the initial condition. Our results complement those of P. Chow in (Commun. Stoch. Anal. 3(2):211–222, 2009), Chow (J. Differential Equations 250(5):2567–2580, 2011), and Foondun et al. in (2016), Foondun and Parshad (Proc. Amer. Math. Soc. 143(9):4085–4094, 2015) among others.