Arbitrary Open Set Research Articles

Sobolev regularity of Gaussian random fields

In this article, we fully characterize the measurable Gaussian processes (U(x))x∈D whose sample paths lie in the Sobolev space of integer order Wm,p(D),m∈N0,1<p<+∞, where D is an arbitrary open set of Rd. The result is phrased in terms of a form of Sobolev regularity of the covariance function on the diagonal. This is then linked to the existence of suitable Mercer or otherwise nuclear decompositions of the integral operators associated to the covariance function and its cross-derivatives. In the Hilbert case p=2, additional links are made w.r.t. the Mercer decompositions of the said integral operators, their trace and the imbedding of the RKHS in Wm,2(D). We provide simple examples and partially recover recent results pertaining to the Sobolev regularity of Gaussian processes.

Trilinear embedding for divergence-form operators with complex coefficients

We prove a dimension-free Lp(Ω)×Lq(Ω)×Lr(Ω)→L1(Ω×(0,∞)) embedding for triples of elliptic operators in divergence form with complex coefficients and subject to mixed boundary conditions on Ω, and for triples of exponents p,q,r∈(1,∞) mutually related by the identity 1/p+1/q+1/r=1. Here Ω is allowed to be an arbitrary open subset of Rd. Our assumptions involving the exponents and coefficient matrices are expressed in terms of a condition known as p-ellipticity. The proof utilizes the method of Bellman functions and heat flows. As a corollary, we give applications to (i) paraproducts and (ii) square functions associated with the corresponding operator semigroups, moreover, we prove (iii) inequalities of Kato–Ponce type for elliptic operators with complex coefficients. All the above results are the first of their kind for elliptic divergence-form operators with complex coefficients on arbitrary open sets. Furthermore, the approach to (ii),(iii) through trilinear embeddings seems to be new.

The set of mildly regular boundary points has full caloric measure

AbstractWe introduce a new kind of regularity in the Dirichlet problem for the heat equation, called mild regularity. The mildly regular boundary points include all types of regular points previously considered. Just as regular boundary points have barriers associated with them, the mildly regular boundary points have mild barriers, which are just as important. We consider whether the mild regularity of a boundary point for an open set is inherited by an open subset or superset. We detail the connection of mild regularity with the Green function. We consider, in particular, subdomains of an arbitrary open set that satisfy a certain connectivity condition which takes account of the special nature of the temporal variable, called zones of influence. Our main theorem states that, for any zone of influence, the set of boundary points that are not mildly regular is caloric measure null.

The Bishop’s property (β) for class A operators

We say that an operator T on a Hilbert space H has the Bishop?s property (?) if for an arbitrary open set U ? C and analytic functions fn : U ? H with ?(T ? z) fn(z)? converges to 0 uniformly on every compact subset of U as n ? ?then ? fn? converges to 0 uniformly on every compact subset ofU as n ? ?. An operator T on H is called to be hyponormal if T*T ? TT*, and T is called to be class A if T*T ? |T2|. In this papaer, we give an elementary proof of the assertion that every hyponormal operator has the Bishop?s property (?). And we show that every class A operator has the Bishop?s property (?). Moreover, we also show a class A operator T is similar to a hyponormal operator if T is invertible, and hence T has the growth condition (G1).

Reachability results for perturbed heat equations

This work studies the reachable space of infinite dimensional control systems which are null controllable in any positive time, the typical example being the heat equation controlled from the boundary or from an arbitrary open set. The focus is on the robustness of the reachable space with respect to linear or nonlinear perturbations of the generator. More precisely, our first main result asserts that this space is invariant under perturbations which are small (in an appropriate sense). A second main result asserts the invariance of the reachable space with respect to perturbations which are compact (again in an appropriate sense), provided that a Hautus type condition is satisfied. Moreover, our methods give precise information on the behavior of the reachable space when the generator is perturbed by a class of nonlinear operators. When applied to the classical heat equation, our results provide detailed information on the reachable space when the generator is perturbed by a small potential or by a class of non local operators, and in particular in one space dimension, we deduce from our analysis that the reachable space for perturbations of the 1-d heat equation is a space of holomorphic functions. We also show how our approach leads to reachability results for a class of semi-linear parabolic equations.

Exterior controllability properties for a fractional Moore–Gibson–Thompson equation

The three concepts of exact, null and approximate controllabilities are analyzed from the exterior of the Moore–Gibson–Thompson equation associated with the fractional Laplace operator subject to the nonhomogeneous Dirichlet type exterior condition. Assuming that \(b>0\) and \(\alpha -\frac{\tau c^2}{b}>0\), we show that if \(0<s<1\) and \(\varOmega \subset {\mathbb {R}}^N\) (\(N\ge 1\)) is a bounded domain with a Lipschitz continuous boundary \(\partial \varOmega \), then there is no control function g such that the following system $$\begin{aligned} {\left\{ \begin{array}{ll} \tau u_{ttt} + \alpha u_{tt}+c^2(-\varDelta )^{s} u + b(-\varDelta )^{s} u_{t}=0 &{} \text{ in } \; \varOmega \times (0,T),\\ u=g\chi _{{\mathcal {O}}} &{} \text{ in } \; ({\mathbb {R}}^N\setminus \varOmega )\times (0,T) ,\\ u(\cdot ,0) = u_0, u_t(\cdot ,0) = u_1, u_{tt}(\cdot ,0)=u_2 &{} \text{ in } \; \varOmega , \end{array}\right. } \end{aligned}$$is exactly or null controllable in time \(T>0\). However, we prove that for \(0<s<1\), the system is approximately controllable for every \(g\in H^1((0,T);L^{2}({\mathcal {O}}))\), where \(\mathcal O\subset {\mathbb {R}}^N\setminus {\overline{\varOmega }}\) is an arbitrary non-empty open set.

Generic power series on subsets of the unit disk

We examine the boundary behaviour of the generic power series f with coefficients chosen from a fixed bounded set Λ in the sense of Baire category. Notably, we prove that for any open subset U of the unit disk D with a nonreal boundary point on the unit circle, f(U) is a dense set of ℂ. As it is demonstrated, this conclusion does not necessarily hold for arbitrary open sets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property is given.

A quantitative approach on the solvability of evolution problems in open sets of certain geometries

We study the engagement of the boundary data in solving certain evolution problems – that incorporate the standard second order, symmetric, uniformly elliptic operators – in arbitrary open sets. We extend the range of the open sets considered, by proposing a unified approach for both bounded and unbounded sets. The core behind this approach is a revision of (i) the extension operator theory and (ii) the elliptic regularity theory. We illustrate the abstract results for the case of the non vanishing, non linear Schrödinger equation (NLSE) with the defocusing pure power non linearity.

Caloric Measure Null Sets

We give a systematic treatment of caloric measure null sets on the essential boundary $\partial_eE$ of an arbitrary open set $E$ in ${\bf R}$. We discuss two characterisations of such sets and present some basic properties. We investigate the dependence of caloric measure null sets on the open set $E$. Thus, if $D$ is an open subset of $E$ and $Z\subseteq\partial_eE\cap\partial_eD$, we show that $Z$ is caloric measure null for $D$ if it is caloric measure null for $E$. We also give conditions on $E$ and $Z$ which imply that the reverse implication is true. We know from \cite{watson2011} that any polar subset of $\partial_eD$ is caloric measure null for $D$, but the reverse implication is not generally true. In our final result we show that, for subsets of a certain component of $\partial_eD$, caloric measure null sets are necessarily polar.

On density of smooth functions in weighted fractional Sobolev spaces

We prove that smooth C∞ functions are dense in weighted fractional Sobolev spaces on an arbitrary open set, under some mild conditions on the weight. We also obtain a similar result in non-weighted spaces defined by some kernel similar to x↦|x|−d−sp. One may consider the results to be a version of the Meyers–Serrin theorem.

Approximation in measure: the Dirichlet problem, universality and the Riemann hypothesis

We use approximation in measure to solve an asymptotic Dirichlet problem on arbitrary open sets and to show that many functions, including the Riemann zeta-function, are universal in measure. Connections with the Riemann hypothesis are suggested.

On grand Sobolev spaces and pointwise description of Banach function spaces

We obtain a pointwise description of functions belonging to function spaces with the lattice property. In particular, it is valid for Banach function spaces provided that the Hardy–Littlewood maximal operator is bounded. We also study weighted grand Sobolev spaces, defined on arbitrary open sets Ω (of finite or infinite measure) in Rn, and provide pointwise description of them.

An asymptotic holomorphic boundary problem on arbitrary open sets in Riemann surfaces

We show that if U is an arbitrary open subset of a Riemann surface and φ an arbitrary continuous function on the boundary ∂U, then there exists a holomorphic function φ˜ on U such that, for every p∈∂U, φ˜(x)→φ(p), as x→p outside a set of density 0 at p relative to U. These “solutions to a boundary problem” are not unique. In fact they can be required to have interpolating properties and also to assume all complex values near every boundary point. Our result is new even for the unit disc.

On the exit time from open sets of some semi-Markov processes

In this paper we characterize the distribution of the first exit time from an arbitrary open set for a class of semi-Markov processes obtained as time-changed Markov processes. We estimate the asymptotic behaviour of the survival function (for large $t$) and of the distribution function (for small $t$) and we provide some conditions for absolute continuity. We have been inspired by a problem of neurophyshiology and our results are particularly usefull in this field, precisely for the so-called Leaky Integrate-and-Fire (LIF) models: the use of semi-Markov processes in these models appear to be realistic under several aspects, for example, it makes the intertimes between spikes a r.v. with infinite expectation, which is a desiderable property. Hence, after the theoretical part, we provide a LIF model based on semi-Markov processes.

Fractional Gaussian estimates and holomorphy of semigroups

Let $$\Omega \subset {\mathbb {R}}^N$$ be an arbitrary open set, $$0<s<1$$ and denote by $$(e^{-t(-\Delta )_{{{\mathbb {R}}}^N}^s})_{t\ge 0}$$ the semigroup on $$L^2({{\mathbb {R}}}^N)$$ generated by the fractional Laplace operator. In the first part of the paper, we show that if T is a self-adjoint semigroup on $$L^2(\Omega )$$ satisfying a fractional Gaussian estimate in the sense that $$|T(t)f|\le Me^{-bt(-\Delta )_{{{\mathbb {R}}}^N}^s}|f|$$, $$0\le t \le 1$$, $$f\in L^2(\Omega )$$, for some constants $$M\ge 1$$ and $$b\ge 0$$, then T defines a bounded holomorphic semigroup of angle $$\frac{\pi }{2}$$ that interpolates on $$L^p(\Omega )$$, $$1\le p<\infty $$. Using a duality argument, we prove that the same result also holds on the space of continuous functions. In the second part, we apply the above results to the realization of fractional order operators with the exterior Dirichlet conditions.

A Constructive Proof of Helmholtz’s Theorem

Summary It is a known result that any vector field ${\boldsymbol{u}}$ that is locally Hölder continuous on an arbitrary open set $\Omega\subset \mathbb{R}^3$ can be written on $\Omega$ as the sum of a gradient and a curl. Should $\Omega$ be unbounded, no conditions are required on the behaviour of ${\boldsymbol{u}}$ at infinity. We present a direct, self-contained proof of this theorem that only uses elementary techniques and has a constructive character. It consists in patching together local solutions given by the Newtonian potential that are then modified by harmonic approximations—based on solid spherical harmonics—to assure convergence near infinity for the resulting series.

On average controllability of random heat equations with arbitrarily distributed diffusivity

In this paper, we study the average controllability of a random heat equation, with the diffusivity serving as the random variable drawn from a general probability distribution. We show that the solutions of such random heat equations are both null and approximately controllable in average from an arbitrary open set of the domain and in an arbitrarily small time, recovering the known result when the random diffusivity is uniformly or exponentially distributed.

Traces for homogeneous Sobolev spaces in infinite strip-like domains

In this paper we construct a trace operator for homogeneous Sobolev spaces defined on infinite strip-like domains. We identify an intrinsic seminorm on the resulting trace space that makes the trace operator bounded and allows us to construct a bounded right inverse. The intrinsic seminorm involves two features not encountered in the trace theory of bounded Lipschitz domains or half-spaces. First, due to the strip-like structure of the domain, the boundary splits into two infinite disconnected components. The traces onto each component are not completely independent, and the intrinsic seminorm contains a term that measures the difference between the two traces. Second, as in the usual trace theory, there is a term in the seminorm measuring the fractional Sobolev regularity of the trace functions with a difference quotient integral. However, the finite width of the strip-like domain gives rise to a screening effect that bounds the range of the difference quotient perturbation. The screened homogeneous fractional Sobolev spaces defined by this screened seminorm on arbitrary open sets are of independent interest, and we study their basic properties. We conclude the paper with applications of the trace theory to partial differential equations.

Besov spaces on open sets

This paper is devoted to giving definitions of Besov spaces on an arbitrary open set Ω of Rn via the spectral theorem for the Schrödinger operator with the Dirichlet boundary condition. The crucial point is to introduce some test function spaces on Ω. The fundamental properties of Besov spaces are also shown, such as embedding relations and duality, etc. Furthermore, the isomorphism relations are established among the Besov spaces in which regularity of functions is measured by the Dirichlet Laplacian and the Schrödinger operators.

Approximate Controllability from the Exterior of Space-Time Fractional Diffusive Equations

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain with a Lipschitz continuous boundary. We study the controllability of the space-time fractional diffusive equation $\{\mathbb D_t^\alpha u+(-\Delta)^su=0 \mbox{ in}\;(0,T)\times\Omega, u=g\chi_{(0,T)\times\mathcal O} %& \mbox{ in}\;(0,T)\times(\mathbb{R}^N\setminus\Omega), u(0,\cdot)=u_0%& \mbox{ in}\;\Omega\}$, where $u=u(t,x)$ is the state to be controlled and $g=g(t,x)$ is the control function which is localized in a nonempty open subset $\mathcal O$ of $\mathbb{R}^N\setminus\Omega$. Here, $0<\alpha\le 1$, $0<s<1,$ and $T>0$ are real numbers. After giving an explicit representation of solutions, we show that the system is always approximately controllable for every $T>0$, $u_0\in L^2(\Omega)$, and $g\in \mathcal D((0,T)\times\mathcal O),$ where $\mathcal O\subseteq\mathbb{R}^N\setminus\Omega$ is an arbitrary nonempty open set. The results obtained are sharp in the sense that such a system is never null controllable if $0<\alpha<1$. The proof of our result is based on a new unique continuation principle for the eigenvalues problem associated with the fractional Laplace operator subject to the zero Dirichlet exterior condition that we have established.