Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami, Johannes Lankeit
We study the heat equation in the exterior of the unit ball with a linear dynamical boundary condition. Our main aim is to find upper and lower bounds for the rate of convergence to solutions of the Laplace equation with the same dynamical boundary condition as the diffusion coefficient tends to infinity.
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New Hardy type inequalities in sectorial area and as a limit in an exterior of a ball are proved. Sharpness of the inequalities is shown as well.
We are interested in this article in investigating the smoothing effect properties of the solutions of the Schrodinger equation. We deduce global well-posedness results for the cubic Schrodinger equation in the exterior of several convex obstacles of R^{3}.
We consider the defocusing energy-critical NLS in the exterior of the unit ball in three dimensions. For the initial value problem with Dirichlet boundary condition we prove global well-posedness and scattering with large radial initial data in the Sobolev space $\dot H_0^1$. We also point out that the same strategy can be used to treat the energy-supercritical NLS in the exterior of balls with Dirichlet boundary condition and radial $\dot H_0^1$ initial data.
We obtain threshold results for the existence, non-existence and multiplicity of normalized solutions for semi-linear elliptic equations in the exterior of a ball. To the best of our knowledge, it is the first result in the literature addressing this problem. In particular, we show that the prescribed mass can affect the number of normalized solutions and has a stabilizing effect in the mass supercritical case. Furthermore, in the threshold we find a new exponent p = 6 when N = 2, which does not seem to have played a role for this equation in the past. Moreover, our findings are "quite surpris…
This paper is devoted to the study of gradient estimates for the Dirichlet problem of the heat equation in the exterior domain of a compact set. Our results describe the time decay rates of the derivatives of solutions to the Dirichlet problem. Applications of these estimates to bilinear type commutator estimates for Laplace operator with Dirichlet boundary condition in exterior domain are discussed too.
Higher order evolution inequalities with Hardy potential in the exterior of a half-ball
本文研究半空间外部区域上带有Hardy势的半线性高阶时间发展不等式，发现在非齐次Dirichlet边界条件下，其解存在与否的Fujita型临界指数依赖于势参数与维数，但与时间导数阶数无关。
We consider the null controllability problem from the exterior for the one dimensional heat equation on the interval $(0,1)$ associated with the fractional Laplace operator $(-\partial_x^2)^s$, where $0<s<1$. We show that there is a control function which is localized in a non-empty open set $\mathcal{O}\subset \left(\mathbb{R}\setminus(0,1)\right)$, that is, at the exterior of the interval $(0,1)$, such that the system is null controllable at any time $T>0$ if and only if $\frac 12<s<1$.
We consider the mixed problem on the exterior of the unit ball in $\mathbb{R}^{n}$, $n\ge2$, for a defocusing Schrödinger equation with a power nonlinearity $|u|^{p-1}u$, with zero boundary data. Assuming that the initial data are non radial, sufficiently small perturbations of \emph{large} radial initial data, we prove that for all powers $p>n+6$ the solution exists for all times, its Sobolev norms do not inflate, and the solution is unique in the energy class.
This paper is dedicated to the proof of new maximal regularity results involving Besov spaces for the heat equation in the half-space or in bounded or exterior domains of R^n. We strive for time independent a priori estimates in regularity spaces of type L^1(0,T;X) where X stands for some homogeneous Besov space. In the case of bounded domains, the results that we get are similar to those of the whole space or of the half-space. For exterior domains, we need to use mixed Besov norms in order to get a control on the low frequencies. Those estimates are crucial for proving global-in-time results…
In this paper, we show global existence, in spatial dimensions greater than or equal to four, for semilinear wave equations with quadratic nonlinearities exterior to a nontrapping obstacle. This extends the previous work of Shibata-Tsutsumi and Hayashi. Shibata-Tsutsumi showed the result for spatial dimensions greater than or equal to 6, and Hayashi proved the result exterior to a ball.
We show the existence and optimal regularity of the optimal temperature configuration in a problem in heat conduction with minimal temperature constraint, interior heating and exterior insulation. Regularity of the two free boundaries is also studied.
On the loss of mass for the heat equation in an exterior domain with general boundary conditions
本文研究外区域中热方程在不同齐次边界条件下的质量损失，确定其确切数量与完全衰减准则，并给出显式衰减速率。
We consider 1-equivariant wave maps from \R \times (\R^3 \setminus B) to S^3 where B is a ball centered at 0, and the boundary of B gets mapped to a fixed point on S^3. We show that 1-equivariant maps of degree zero scatter to zero irrespective of their energy. For positive degrees, we prove asymptotic stability of the unique harmonic maps in the energy class determined by the degree.
We give necessary and sufficient conditions for the existence of positive radial solutions for a class of fully nonlinear uniformly elliptic equations posed in the complement of a ball in $\mathbb R^N$, and equipped with homogeneous Dirichlet boundary conditions.
We consider the exterior Cauchy-Dirichlet problem for equivariant wave maps from 3+1 dimensional Minkowski spacetime into the three-sphere. Using mixed analytical and numerical methods we show that, for a given topological degree of the map, all solutions starting from smooth finite energy initial data converge to the unique static solution (harmonic map). The asymptotics of this relaxation process is described in detail. We hope that our model will provide an attractive mathematical setting for gaining insight into dissipation-by-dispersion phenomena, in particular the soliton resolution conj…
Asymptotic behaviour of the heat equation in an exterior domain with general boundary conditions I. The case of integrable data
研究外部区域热方程解在不同边界条件下的渐近行为，刻画剩余质量的空间分布并证明最优时间速率。
We examine the Hénon equation $ -Δu =|x|^αu^p$ in $ Ω\subset \mathbb{R}^N$ with $u=0$ on $ \partial Ω$ where $ 0 < α$. We show there exists a sequence $ \{p_k\}_k \subset [ \frac{N+2}{N-2}, p_α(N)]$ with $p_1 < p_2 <p_3 < ...$, $ p_k \nearrow p_α(N)$ such that for any $ \frac{N+2}{N-2} \le p < p_α(N)$, which avoids $ \{p_k\}_k $, there exists a positive classical solution of the Hénon equation, provided $ Ω$ is a sufficiently small perturbation of the unit ball. We also examine the Lane-Emden-Fowler equation in the case of an exterior domain; ie. $ -Δu = u^p$ in $ Ω$, an exterior domain, with …
Double Criticality for a Hardy-Rellich Biharmonic Heat Equation in an Exterior Domain
研究外域上含奇异Hardy-Rellich势、加权非线性和正源项的双调和热方程弱解的存在性与非存在性，识别两种不同临界情形，提出Fujita型临界指数及源项衰减指数的第二临界指数，扩展了Tobakhanov近期工作。
In this paper we study 1-equivariant wave maps of finite energy from 1+3-dimensional Minkowski space exterior to the unit ball at the origin into the 3-sphere. We impose a Dirichlet boundary condition at r=1, meaning that the unit sphere in R^3 gets mapped to the north pole. Finite energy implies that spacial infinity gets mapped to either the north or south pole. In particular, each such equivariant wave map has a well-defined topological degree which is an integer. We establish relaxation of such a map of arbitrary energy and degree to the unique stationary harmonic map in its degree class. …
We study the Hardy inequality when the singularity is placed on the boundary of a bounded domain in $\mathbb{R}^n$ that satisfies both an interior and exterior ball condition at the singularity. We obtain the sharp Hardy constant $n^2/4$ in case the exterior ball is large enough and show the necessity of the large exterior ball condition. We improve Hardy inequality with the best constant by adding a sharp Sobolev term. We next produce criteria that lead to characterizing maximal potentials that improve Hardy inequality. Breaking the criteria one produces successive improvements with sharp con…
The paper is concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet and Neumann problems of harmonic analysis for the three-dimensional unit ball with the corresponding behaviour of the associated ergodic inverse problems for the punched unlimited space. The basis is the theory of semigroups of linear operators mapping a Banach space $X$ into itself. The rates of approximation play a basic role. Another tool is a Drazin-like inverse operator $B$ for the infinitesimal generator $A$ of a semigroup that arises naturally in ergodic theory. This oper…
Large-time estimates for the Dirichlet heat equation in exterior domains
本文研究了外区域中 Dirichlet 热方程的大时间渐近行为，给出了一致范数和 $L^1$ 范数下的显式逼近速率估计，并利用熵方法揭示了热核的渐近轮廓及其与区域形状和质量损失的关系。
In this paper, we consider the asymptotic behavior of solutions to the wave equation with space-dependent damping in an exterior domain. We prove that when the damping is effective, the solution is approximated by that of the corresponding heat equation as time tends to infinity. Our proof is based on semigroup estimates for the corresponding heat equation and weighted energy estimates for the damped wave equation. The optimality of the decay late for solutions is also established.
Asymptotic behaviour of the heat equation in an exterior domain with general boundary conditions II. The case of bounded and of $L^{p}$ data
本文研究了外部区域上热方程在 Dirichlet、Robin 和 Neumann 等齐次边界条件下的渐近行为，证明有界初始数据经修正项后趋近全空间解，并分析了 $L^p$ 初始数据下解衰减至 0 且收敛速度可能任意缓慢的情形。
Consider the equation u_t=Δu-Vu +au^p \text{in} R^n\times (0,T); u(x,0)=ϕ(x)\gneq0, \text{in} R^n, where $p>1$, $n\ge2$, $T\in(0,\infty]$, $V(x)\sim\fracω{|x|^2}$ as $|x|\to\infty$, for some $ω\neq0$, and $a(x)$ is on the order $|x|^m$ as $|x|\to\infty$, for some $m\in (-\infty,\infty)$. A solution to the above equation is called global if $T=\infty$. Under some additional technical conditions, we calculate a critical exponent $p^*$ such that global solutions exist for $p>p^*$, while for $1<p\le p^*$, all solutions blow up in finite time. We also show that when $V\equiv0$, the blow-up/global s…
Heat kernel estimates, fractional Riesz transforms and applications on exterior domains
本文推导了平面外部$C^{1,1}$区域双侧热核精确估计与$\n\boldsymbol{R}^n$外部$C^{1,\boldsymbol{\u214A}ini}$区域热核上界估计，证明了分数阶里斯变换有界性，并给出其在乘积法则、链式法则及非线性薛定谔方程中的应用。
In this paper we consider the " exterior approach " to solve the inverse obstacle problem for the heat equation. This iterated approach is based on a quasi-reversibility method to compute the solution from the Cauchy data while a simple level set method is used to characterize the obstacle. We present several mixed formulations of quasi-reversibility that enable us to use some classical conforming finite elements. Among these, an iterated formulation that takes the noisy Cauchy data into account in a weak way is selected to serve in some numerical experiments and show the feasibility of our st…
We study Schroedinger operators with Robin boundary conditions on exterior domains in $\R^d$. We prove sharp point-wise estimates for the associated semi-groups which show, in particular, how the boundary conditions affect the time decay of the heat kernel in dimensions one and two. Applications to spectral estimates are discussed as well.
Heat Transfer Shape Optimization: Stability and Non-Optimality of the Ball
本文利用流方法和Steklov特征值估计研究Robin边界条件下热传递形状优化中球的稳定性，揭示球在热对流下可能非最优，并给出依赖Robin系数、维数和体积约束的稳定性破坏阈值。
Green functions for the heat and Laplace equations with dynamical boundary conditions in a ball
本文研究球内动态边界条件下热方程与拉普拉斯方程的格林函数，给出拉普拉斯方程格林函数并讨论热核性质，进而构造热方程格林函数与解的两种互补近似，并用特征函数级数隐式表示热方程格林函数。
Two Infinite Families of Solutions for Singular Superlinear Equations on Exterior Domains
本文针对欧氏空间N>2的球外奇异超线性方程，在特定参数条件下证明了两类无穷多径向解的存在性，丰富了此类方程解的相关理论。
On the decay of mass with respect to an invariant measure for semilinear heat equations in exterior domains
本文研究外部区域半线性热方程解关于不变测度的质量衰减，证明当且仅当 $1<p\leq\min\{2,1+2/N\}$ 时质量趋于零，否则解渐近自由，且 $N\geq3$ 时轮廓为高斯修正。
Existence of solutions for nonlinear elliptic PDEs with fractional Laplacians on open balls
本文利用(2s)-稳定分支过程的树状概率表示，在开球上证明了带分数阶拉普拉斯算子的半线性椭圆型偏微分方程粘性解的存在性，并放宽了对多项式非线性次数和项数的限制。
Semilinear Heat Inequalities with a Hardy-Type Potential in an Exterior Geodesic Domain on $\mathbb{S}^N$
研究单位球$\mathbb S^N$（$N\ge3$）上带有Hardy型势$\lambda/\sin^2 r$的半线性热不等式，确定临界指数$p_{\mathrm{crit}}$对解存在性和非存在性的控制，通过构造径向Hardy屏障和精确积分估计证明临界指数的非存在性。
Global well-posedness of the energy-critical complex Ginzburg-Landau equation in exterior domains
本文通过集中紧致性/刚性定理方法，证明了在外围域中能量临界复吉布斯-兰道方程的全球适定性，并推导出非聚焦非线性薛定谔方程和半线性热方程的相应理论。
On the critical behavior for the semilinear biharmonic heat equation with forcing term in an exterior domain
本文研究外区域中带强迫项的半线性双调和热方程在六类齐次边界条件下的临界行为，运用测试函数法与比较原理推导出Fujita意义下的临界指数及Lee-Ni第二临界指数，并给出次临界与临界情况下的有限时间爆破结果和寿命估计。
Stability of purely convective steady-states of fractional Boussinesq equations in an exterior domain
研究了三维无限流体域中球体外的热对流流动，通过分数幂的斯托克斯算子确定粘性力，得出纯导热稳态，并证明了稳态解在 $L^2$ 空间中的全局稳定性。
In this paper we study the initial boundary value problem for two-dimensional semilinear wave equations with small data, in asymptotically Euclidean exterior domains. We prove that if $1<p\le p_c(2)$, the problem admits almost the same upper bound of the lifespan as that of the corresponding Cauchy problem, only with a small loss for $1<p\le 2$. It is interesting to see that the logarithmic increase of the harmonic function in $2$-D has no influence to the estimate of the upper bound of the lifespan for $2<p\le p_c(2)$. One of the novelties is that we can deal with the problem with flat metric…
Strong solutions of fractional Boussinesq equations in an exterior domain
本文研究三维球外区域中由加热引起的热对流问题，通过谱表示建立分数阶Boussinesq方程系统，证明Hilbert空间中小强解的存在性，并由此得到稳态的局部稳定性与渐近吸引性。
We study the boundary regularity for the normalised $\infty$-heat equation $u_t = Δ_{\infty}^Nu$ in arbitrary domains. Perron's Method is used for constructing solutions. We characterize regular boundary points with barrier functions, and prove an Exterior Ball result. A Petrovsky-like criterion is established.
We consider classes of simply connected planar domains which are isophasal, ie, have the same scattering phase $s(ł)$ for all $ł> 0$. This is a scattering-theoretic analogue of isospectral domains. Using the heat invariants and the determinant of the Laplacian, Osgood, Phillips and Sarnak showed that each isospectral class is sequentially compact in a natural $C$-infinity topology. In this paper, we show sequential compactness of each isophasal class of domains. To do this we define the determinant of the exterior Laplacian and use it together with the heat invariants (the heat invariants and …
Global well-posedness of the defocusing, cubic nonlinear wave equation outside of the ball with radial data
本文利用畸变 Fourier 变换建立球外线性波动方程的色散与端点 Strichartz 估计，并结合 Fourier 截断和能量方法证明散焦三次非线性波动方程径向解在 $s>3/4$ 低正则空间中的整体适定性。
Normalized Schrödinger equations with mass-supercritical nonlinearity in exterior domains
本文研究外部区域中质量超临界归一化薛定谔方程，证明了在障碍集为小球或质量较大两种情形下，存在约束 Morse 指标不超过 N+1 的解。
In earlier works, we have shown the uniform decay of the local energy of the damped wave equation in exterior domain when the damper is spatially localized near captive rays. In order to have uniform decay of the total energy, the damper has also to act at space infinity. In this work, we establish uniform decay of both the local and global energies. The rates of decay turns out to be the same as those for the heat equation, which shows that an effective damper at space infinity strengthens the parabolic structure in the equation.
We consider positive solution to the weighted elliptic problem \begin{equation*} \left \{ \begin{array}{ll} -{\rm div} (|x|^θ\nabla u)=|x|^\ell u^p \;\;\; \mbox{in $\mathbb{R}^N \backslash {\overline B}$},\\ u=0 \;\;\; \mbox{on $\partial B$}, \end{array} \right. \end{equation*} where $B$ is the standard unit ball of $\mathbb{R}^N$. We give a complete answer for the existence question when $N':=N+θ>2$. In particular, for $N' > 2$ and $τ:=\ell-θ>-2$, it is shown that the problem admits a unique positive radial solution for $p>p_s:=\frac{N'+2+2τ}{N'-2}$, while for any $ 0<p \leq p_s$, the only no…
Dispersive Estimates for Maxwell's Equations in the Exterior of a Sphere
本文研究了完全导电球外部麦克斯韦方程的高频色散估计，构造了新的广义本征函数，证明了电场传播子的全局衰减率，并指出部分极化以与波动算子相同的速率散射。
The full compressible Navier-Stokes system (FNS) describing the motion of a viscous, compressible, heat-conductive, and Newtonian polytropic fluid in a three-dimensional (3D) exterior domain is studied. For the initial-boundary-value problem with the slip boundary conditions on the velocity and the Neumann one on the temperature, it is shown that there exists a unique global classical solutions with the initial data which are of small energy but possibly large oscillations. In particular, both the density and temperature are allowed to vanish initially. This is the first result about classical…
We study the controllability to trajectories, under positivity constraints on the control or the state, of a one-dimensional heat equation involving the fractional Laplace operator $ (-\partial_x^2)^s$ (with $0<s<1$) on the interval $(-1,1)$. Our control function is localized in an open set $\mathcal O$ in the exterior of $(-1,1)$, that is, $\mathcal O\subset (\mathbb{R} \setminus (-1,1))$. We show that there exists a minimal (strictly positive) time $T_{\rm min}$ such that the fractional heat dynamics can be controlled from any initial datum in $L^2(-1,1)$ to a positive trajectory through the…
Lifespan estimates for semilinear damped wave equation in a two-dimensional exterior domain
本文针对二维外部区域中临界半线性阻尼波动方程，在径向对称条件下，证明了其生命跨度与临界半线性热方程相同的双指数下界和上界估计，并引入了控制L^1范数的新技术及带对数权重的Gagliardo-Nirenberg型估计。
Uniqueness of least energy solutions to the fractional Lane-Emden equation in the ball
本文证明了球域上分数阶Lane-Emden方程在齐次Dirichlet外部条件下最小能量解的唯一性，利用Morse理论和新的Hopf引理型结果，受C. S. Lin启发。
We consider a class of semilinear nonlocal problems with vanishing exterior condition and establish a Ambrosetti-Prodi type phenomenon when the nonlinear term satisfies certain conditions. Our technique makes use of the probabilistic tools and heat kernel estimates.
We study the heat equation on a half-space or on an exterior domain with a linear dynamical boundary condition. Our main aim is to establish the rate of convergence to solutions of the Laplace equation with the same dynamical boundary condition as the diffusion coefficient tends to infinity.
In this article we consider the existence of positive singular solutions on bounded domains and also classical solutions on exterior domains. First we consider positive singular solutions of the following problems: \begin{equation} \label{eq_abst_1}-Δu = (1+g(x)) | \nabla u|^p \qquad \mbox{ in } B_1, \qquad u = 0 \mbox{ on } \;\; \partial B_1, \qquad \mbox{ and} \end{equation} \begin{equation} \label{eq_abst_2} -Δu = | \nabla u|^p \qquad \mbox{ in } Ω, \qquad u = 0 \mbox{ on } \;\; \partial Ω. \end{equation} In the first problem $B_1$ is the unit ball in $ \mathbb{R}^N$ and in the second $Ω$ i…
Asymptotic properties of the Stokes flow in an exterior domain with slowly decaying initial data and its application to the Navier-Stokes equations
本文研究外部区域中慢衰减初始数据下Stokes流的衰减率，并应用于Navier-Stokes方程，得到唯一强解和最优的逐点衰减估计。
Consider the Cauchy problem for the 3-d linear wave equation $\square_{1+3}U=0$ with radial initial data $U(0,x)=Φ(x)=ϕ(|x|)$, $U_t(0,x)=Ψ(x)=ψ(|x|)$. A standard result gives that $U$ belongs to $C([0,T];H^s(\mathbb{R}^3))$ whenever $(Φ,Ψ)\in H^s\times H^{s-1}(\mathbb{R}^3)$. In this note we are interested in the question of how $U$ can be realized as a limit of solutions to initial-boundary value problems on the exterior of vanishing balls $B_\varepsilon$ about the origin. We note that, as the solutions we compare are defined on different domains, the answer is not an immediate consequence of…
We consider refinements of the local smoothing estimates for the Schrödinger equation in domains which are exterior to a strictly convex obstacle in $\RR^n$. By restricting the solution to small, frequency dependent collars of the boundary, it is expected that taking its square integral in space-time should exhibit a larger gain in regularity when compared to the usual gain of half a derivative. By a result of Ivanovici, these refined local smoothing estimates are satisfied by solutions in the exterior of a ball. We show that when such estimates are valid, they can be combined with wave packet…
We design sources for the two-dimensional Helmholtz equation that can cloak an object by cancelling out the incident field in a region, without the sources completely surrounding the object to hide. As in previous work for real positive wavenumbers, the sources are also determined by the Green identities. The novelty is that we prove that the same approach works for complex wavenumbers which makes it applicable to a variety of media, including media with dispersion, loss and gain. Furthermore, by deriving bounds on Graf's addition formulas with complex arguments, we obtain new estimates that a…
Existence and nonexistence of sign-changing solutions for linearly perturbed superlinear equations on exterior domains
研究外区域上含奇异线性扰动项的超线性椭圆方程径向解，通过系数 $K$ 的衰减指数条件，给出无穷多变号解的存在区间及不存在区间。
We study the convergence of the Method of Reflections for the Dirichlet problem of the Poisson and the Stokes equations in perforated domains which consist in the exterior of balls. We prove that the method converges if the balls are contained in a bounded region and the density of the electrostatic capacity of the balls is sufficiently small. If the capacity density is too large or the balls extend to the whole space, the method diverges, but we provide a suitable modification of the method that converges to the solution of the Dirichlet problem also in this case. We give new proofs of classi…
Existence of an Infinite Number of Solutions to a Singular Superlinear p-Laplacian Equation on Exterior Domains
该研究在外区域上考察具有原点奇异性和无穷远超线性增长的 p-Laplacian 方程，通过对系数衰减指数施加区间条件，证明了存在无穷多个在无穷远处趋于零的径向解。
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