We consider a second order differential operator $\mathscr{A}$ on an (typically unbounded) open and Dirichlet regular set $Ω\subset \mathbb{R}^d$ and subject to nonlocal Dirichlet boundary conditions of the form \[ u(z) = \int_Ωu(x)μ(z, dx) \quad \mbox{ for } z\in \partial Ω. \] Here, $μ: \partialΩ\to \mathscr{M}(Ω)$ is a $σ(\mathscr{M}(Ω), C_b(Ω))$-continuous map taking values in the probability measures on $Ω$. Under suitable assumptions on the coefficients in $\mathscr{A}$, which may be unbounded, we prove that a realization $A_μ$ of $\mathscr{A}$ subject to the nonlocal boundary condition, generates a (not strongly continuous) semigroup on $L^\infty(Ω)$. We also establish a sufficient condition for this semigroup to be Markovian and prove that in this case, it enjoys the strong Feller property. We also study the asymptotic behavior of the semigroup.
We consider the initial value problem for the semilinear plate equation with nonlocal nonlinearity. We prove the existence of global attractor and then establish the regularity and finite dimensionality of this attractor.
We consider the Cauchy problem posed in the whole space for the following nonlocal heat equation: u_t = J * u - u, where J is a symmetric continuous probability density. Depending on the tail of J, we give a rather complete picture of the problem in optimal classes of data by: (i) estimating the initial trace of (possibly unbounded) solutions; (ii) showing existence and uniqueness results in a suitable class; (iii) giving explicit unbounded polynomial solutions.
In this paper we analyze the asymptotic behavior of the Dirichlet fractional Laplacian $(-Δ_{\mathbb R^{n+k}})^{s}$, with $s\in (0, 1)$, on bounded domains in $\mathbb R^{n+k}$ that become unbounded in the last $k$-directions. A dimension reduction phenomenon is observed and described via $Γ$-convergence.
We study function spaces and extension results in relation with Dirichlet problems involving integrodifferential operators. For such problems, data are prescribed on the complement of a given domain in the Euclidean space. We introduce a function space that serves as a trace space for nonlocal Dirichlet problems and study related extension results.
We prove the Harnack inequality for general nonlocal elliptic equations with zero order terms. As an application we prove the existence of the principal eigenvalue in general domains. Furthermore, we study the eigenvalue problem associated to the existence of self-similar solutions to the parabolic problem and provide estimates on the decay rate.
The existence of Feller semigroups arising in the theory of multidimensional diffusion processes is studied. Unbounded perturbations of elliptic operators (in particular, integro-differential operators) are considered in plane bounded regions. Their domain of definition is given by a nonlocal boundary condition involving the integral over the closure of the region with respect to a nonnegative Borel measure. The support of the measure may intersect with the boundary, and the measure need not be small. We formulate sufficient conditions on unbounded perturbations of elliptic operator and on the…
In this paper we analyse the spectrum of nonlocal Dirichlet problems with non-singular kernels in bounded open sets. The novelty is the continuity of eigenvalues with respect to domain perturbation via Lebesgue measure. Also, under additional smooth condition on the kernel and domain, we prove differentiability of simple eigenvalues computing their first derivative.
Unbounded operators corresponding to nonlocal elliptic problems on a bounded region $G\subset\mathbb R^2$ are considered. The domain of these operators consists of functions from the Sobolev space $W_2^m(G)$ being generalized solutions of the corresponding $2m$-order elliptic equation with right-hand side from $L_2(G)$ and satisfying homogeneous nonlocal boundary conditions. It is known that such unbounded operators have the Fredholm property. It is proved in the paper that low-order terms in the differential equation do not affect the index of the operator. Conditions under which nonlocal per…
We study translation-invariant integrodifferential operators that generate Lévy processes. First, we investigate different notions of what a solution to a nonlocal Dirichlet problem is and we provide the classical representation formula for distributional solutions. Second, we study the question under which assumptions distributional solutions are twice differentiable in the classical sense. Sufficient conditions and counterexamples are provided.
In this note we set up the elliptic and the parabolic Dirichlet problem for linear nonlocal operators. As opposed to the classical case of second order differential operators, here the "boundary data" are prescribed on the complement of a given bounded set. We formulate the problem in the classical framework of Hilbert spaces and prove unique solvability using standard techniques like the Fredholm alternative.
The aim of this work is to study comparability of nonlocal Dirichlet forms. We provide sufficient conditions on the kernel for local and global comparability. As an application we prove a-priori estimates in Hölder spaces for solutions to integrodifferential equations. These solutions are defined with the help of symmetric nonlocal Dirichlet forms.
We study a Dirichlet problem in the entire space for some nonlocal degenerate elliptic operators with internal nonlinearities. With very mild assumptions on the boundary datum, we prove existence and uniqueness of the solution in the viscosity sense. If we further assume uniform ellipticity then the solution is shown to be classical, and even smooth if both the operator and the boundary datum are smooth.
We study the obstacle problem for unbounded sets in a proper metric measure space supporting a (p,p)-Poincare inequality. We prove that there exists a unique solution. We also prove that if the measure is doubling and the obstacle is continuous, then the solution is continuous, and moreover p-harmonic in the set where it does not touch the obstacle. This includes, as a special case, the solution of the Dirichlet problem for p-harmonic functions with Sobolev type boundary data.
This paper considers the initial-boundary value problem for the nonlocal Cahn--Hilliard equation $$ \partial_tφ+ (-Δ+1)(a(\cdot)φ-J\astφ+ G'(φ)) = 0 \quad \mbox{in}\ Ω\times(0, T) $$ in an unbounded domain $Ω\subset \mathbb{R}^N$ with smooth bounded boundary, where $N\in\mathbb{N}$, $T>0$, and $a(\cdot), J, G$ are given functions. In the case that $Ω$ is a bounded domain and $-Δ+1$ is replaced with $-Δ$, this problem has been studied by using a Faedo--Galerkin approximation scheme considering the compactness of the Neumann operator $-Δ+1$ (cf. Colli--Frigeri--Grasselli (2012), Gal--Grasselli (…
In this article, we consider the spectral problem for the mixed local and nonlocal $p$-Laplace operator. We discuss the existence and regularity of eigenfunction of the associated Dirichlet $(p,q)$-eigenvalue problem in a bounded domain $Ω\subset\mathbb{R}^N$ under the assumption that $1<p<\infty$ and $1<q<p^{*}$ where $p^{*}=\frac{N p}{N-p}$ if $1<p<N$ and $p^{*}=\infty$ if $p\geq N$.
We present the theory of the Dirichlet problem for nonlocal operators which are the generators of general pure-jump symmetric Lévy processes whose Lévy measures need not be absolutely continuous. We establish basic facts about the Sobolev spaces for such operators, in particular we prove the existence and uniqueness of weak solutions. We present strong and weak variants of maximum principle, and L^\infty bounds for solutions. We also discuss the related extension problem in C^{1,1} domains.
We present an approach to handle Dirichlet type nonlocal boundary conditions for nonlocal diffusion models with a finite range of nonlocal interactions. Our approach utilizes a linear extrapolation of prescribed boundary data. A novelty is, instead of using local gradients of the boundary data that are not available a priori, we incorporate nonlocal gradient operators into the formulation to generalize the finite differences-based methods which are pervasive in literature; our particular choice of the nonlocal gradient operators is based on the interplay between a constant kernel function and …
We prove a stochastic representation formula for the viscosity solution of Dirichlet terminal-boundary value problem for a degenerate Hamilton-Jacobi-Bellman integro-partial differential equation in a bounded domain. We show that the unique viscosity solution is the value function of the associated stochastic optimal control problem. We also obtain the dynamic programming principle for the associated stochastic optimal control problem in a bounded domain.
In this note we study the boundary regularity of solutions to nonlocal Dirichlet problems of the form $Lu=0$ in $Ω$, $u=g$ in $\mathbb R^N\setminusΩ$, in non-smooth domains $Ω$. When $g$ is smooth enough, then it is easy to transform this problem into an homogeneous Dirichlet problem with a bounded right hand side, for which the boundary regularity is well understood. Here, we study the case in which $g\in C^{0,α}$, and establish the optimal Hölder regularity of $u$ up to the boundary. Our results extend previous results of Grubb for $C^\infty$ domains $Ω$.
The aim of this paper is to study a nonlocal problem with a mixed Dirichlet-Neumann exterior condition. We prove existence, nonexistence and multiplicity of positive energy solutions and describe the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.
In this paper we study an equation driven by a nonlocal anisotropic operator with homogeneous Dirichlet boundary conditions. We find at least three non trivial solutions: one positive, one negative and one of unknown sign, using variational methods and Morse theory. We present some results about regularity of solutions as $L^{\infty}$-bound and Hopf's lemma, for the latter we first consider a non negative nonlinearity and then a strictly negative one. Moreover, we prove that, for the corresponding functional, local minimizers with respect to a $C^0$-topology weighted with a suitable power of t…
A system of segregated boundary-domain integral equations (BDIEs) is obtained from the Dirichlet problem for the diffusion equation in non-homogeneous media defined on an exterior two-dimensional domain. We use a parametrix different from the one employed by in (Dufera, 2019). The system of BDIEs is formulated in terms of parametrix-based surface and volume potentials whose mapping properties are analysed in weighted Sobolev spaces. The system of BDIEs is shown to be equivalent to the original boundary value problem and uniquely solvable in appropriate weighted Sobolev spaces suitable for infi…
We analyze the asymptotic behavior of the eigenvalues of nonlinear elliptic problems under Dirichlet boundary conditions and mixed (Dirichlet, Neumann) boundary conditions on domains becoming unbounded. We make intensive use of Picone identity to overcome nonlinearity complications. Altogether the use of Picone identity makes the proof easier with respect to the known proof in the linear case. Surprisingly the asymptotic behavior under mixed boundary conditions critically differs from the case of pure Dirichlet boundary conditions for some class of problems.
Age-structured models with nonlocal diffusion arise naturally in describing the population dynamics of biological species and the transmission dynamics of infectious diseases in which individuals disperse nonlocally and interact each other and the age structure of individuals matters. In the first part of our series papers, we study the principal spectral theory of age-structured models with nonlocal diffusion of Dirichlet type. First, we provide two criteria on the existence of principal eigenvalues by using the theory of resolvent positive operators with their perturbations. Then we define t…
Recently, we constructed a class of nonlocal Poisson model on manifold under Dirichlet boundary with global $\mathcal{O}(δ^2)$ truncation error to its local counterpart, where $δ$ denotes the nonlocal horizon parameter. In this paper, the well-posedness of such manifold model is studied. We utilize Poincare inequality to control the lower order terms along the $2δ$-boundary layer in the weak formulation of model. The second order localization rate of model is attained by combining the well-posedness argument and the truncation error analysis. Such rate is currently optimal among all nonlocal m…
For a smooth, bounded domain $Ω$, $s\in(0,1)$, $p\in \left(1,\frac{n+2s}{n-2s}\right)$ we consider the nonlocal equation $$ ε^{2s} (-Δ)^s u+u=u^p \quad {\mbox{in}}Ω$$ with zero Dirichlet datum and a small parameter $ε>0$. We construct a family of solutions that concentrate as $ε\to 0$ at an interior point of the domain in the form of a scaling of the ground state in entire space. Unlike the classical case $s=1$, the leading order of the associated reduced energy functional in a variational reduction procedure is of polynomial instead of exponential order on the distance from the boundary, due …
We consider a Dirichlet to Neumann operator $\mathcal{L}_a$ arising in a model for water waves, with a nonlocal parameter $a\in(-1,1)$. We deduce the expression of the operator in terms of the Fourier transform, highlighting a local behavior for small frequencies and a nonlocal behavior for large frequencies. We further investigate the $ Γ$-convergence of the energy associated to the equation $ \mathcal{L}_a(u)=W'(u) $, where $W$ is a double-well potential. When $a\in(-1,0]$ the energy $Γ$-converges to the classical perimeter, while for $a\in(0,1)$ the $Γ$-limit is a new nonlocal operator, tha…
Nonlocal gradient operators are prototypical nonlocal differential operators thatare very important in the studies of nonlocal models. One of the simplest variational settings for such studies is the nonlocal Dirichlet energies wherein the energy densities are quadratic in the nonlocal gradients. There have been earlier studies to illuminate the link between the coercivity of the Dirichlet energies and the interaction strengths of radially symmetric kernels that constitute nonlocal gradient operators in the form of integral operators. In this work we adopt a different perspective and focus on …
We introduce a generalized index for certain meromorphic, unbounded, operator-valued functions. The class of functions is chosen such that energy parameter dependent Dirichlet-to-Neumann maps associated to uniformly elliptic partial differential operators, particularly, non-self-adjoint Schrödinger operators, on bounded Lipschitz domains, and abstract operator-valued Weyl-Titchmarsh $M$-functions and Donoghue-type $M$-functions corresponding to closed extensions of symmetric operators belong to it. The principal purpose of this paper is to prove index formulas that relate the difference of the…
The Fredholm index of unbounded operators defined on generalized solutions of nonlocal elliptic problems in plane bounded domains is investigated. It is known that nonlocal terms with smooth coefficients having zero of a certain order at the conjugation points do not affect the index of the unbounded operator. In this paper, we construct examples showing that the index may change under nonlocal perturbations with coefficients not vanishing at the points of conjugation of boundary-value conditions.
Given a Schrödinger differential expression on an exterior Lipschitz domain we prove strict inequalities between the eigenvalues of the corresponding selfadjoint operators subject to Dirichlet and Neumann or Dirichlet and mixed boundary conditions, respectively. Moreover, we prove a strict inequality between the eigenvalues of two different elliptic differential operators on the same domain with Dirichlet boundary conditions.
Global weak solutions to the isentropic compressible Navier-Stokes equations with vacuum and unbounded density in a half-plane under Dirichlet boundary conditions
School of Mathematics and Statistics, Southwest University（西南大学数学与统计学院）
Let $D$ be a bounded $C^2$-domain. Consider the following Dirichlet initial-boundary problem of nonlocal operators with a drift: $$ \partial_t u={\mathscr L}^{(α)}_κu+b\cdot \nabla u+f\ \mathrm{in}\ \mathbb R_+\times D,\ \ u|_{\mathbb R_+\times D^c}=0,\ u(0,\cdot)|_{D}=φ, $$ where $α\in(0,2)$ and $\mathscr L^{(α)}_κ$ is an $α$-stable-like nonlocal operator with kernel function $κ(x,z)$ bounded from above and below by positive constants, and $b:\mathbb R^d\to\mathbb R^d$ is a bounded $C^β$-function with $α+β>1$, $f: \mathbb R_+\times D\to\mathbb R$ is a $C^γ$-function in $D$ uniformly in $t$ wi…
For a second order formally symmetric elliptic differential expression we show that the knowledge of the Dirichlet-to-Neumann map or Robin-to-Dirichlet map for suitably many energies on an arbitrarily small open subset of the boundary determines the self-adjoint operator with a Dirichlet boundary condition or with a (possibly non-self-adjoint) Robin boundary condition uniquely up to unitary equivalence. These results hold for general Lipschitz domains, which can be unbounded and may have a non-compact boundary, and under weak regularity assumptions on the coefficients of the differential expre…
We prove continuity for bounded weak solutions of a nonlinear nonlocal parabolic type equation associated to a Dirichlet form with a rough kernel. The equation is allowed to be singular at the level zero, and solutions may change sign. If the nonlinearity in the equation does not oscillate too much at the origin, the solution is proved to be moreover Hölder continuous. The results are new even when the Dirichlet form is the one corresponding to the fractional Laplacian.
We investigate the qualitative properties of solution to the Zaremba type problem in unbounded domain for the non-divergence elliptic equation with possible degeneration at infinity. The main result is Phragmén-Lindelöf type principle on growth/decay of a solution at infinity depending on both the structure of the Neumann portion of the boundary and the "thickness" of its Dirichlet portion. The result is formulated in terms of so-called $s$-capacity of the Dirichlet portion of the boundary, while the Neumann boundary should satisfy certain "admissibility" condition in the sequence of layers co…
We consider the Cauchy problem for the Helmholtz equation with a domain in R^d, d>2 with N cylindrical outlets to infinity with bounded inclusions in R^{d-1}. Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Maz'ya proposed an alternating iterative method for solving Cauchy problems associated with elliptic,self-adjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We cons…
In this article, we prove the existence of at least three positive solutions for the following nonlocal singular problem \begin{equation*} (P_\la)\left\{ \begin{split} (-\De)^su &= \la\frac{f(u)}{u^q}, \; \; u>0 \;\; \text{in}\;\; \Om,\\ u &= 0\;\; \text{in}\;\; \mb R^n \setminus \Om \end{split} \right. \end{equation*} where $(-\De)^s$ denotes the fractional Laplace operator for $s\in (0,1)$, $n>2s$, $q \in (0,1)$, $\la>0$ and $\Om$ is smooth bounded domain in $\mb R^n$. Here $f :[0,\infty) \to [0,\infty)$ is a continuous nondecreasing map satisfying $\lim\limits_{u\to \infty}\frac{f(u)}{u^{q+…
