M. van den Berg, P. Gilkey, R. Seeley
We study the heat content asymptotics with either Dirichlet or Robin boundary conditions where the initial temperature exhibits radial blowup near the boundary. We show that there is a complete small-time asymptotic expansion and give explicit geometrical formulas for the first few terms in the expansion.
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The Robin heat kernel and its expansion via Robin eigenfunctions
该研究证明了带光滑边界的紧黎曼流形上任意实Robin参数下Robin热核的存在唯一性，给出非负参数的直接证法并发展新技术处理负参数情形，将经典结果拓展到负参数区间。
This paper is devoted to the study of a nonlinear heat equation associated with Dirichlet-Robin conditions. At first, we use the Faedo -- Galerkin and the compactness method to prove existence and uniqueness results. Next, we consider the properties of solutions. We obtain that if the initial condition is bounded then so is the solution and we also get asymptotic behavior of solutions as. Finally, we give numerical results
The principal aim of this short note is to extend a recent result on Gaussian heat kernel bounds for self-adjoint $L^2(\Om; d^n x)$-realizations, $n\in\bbN$, $n\geq 2$, of divergence form elliptic partial differential expressions $L$ with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains $\Om \subset \bbR^n$, where $$ Lu = - \sum_{j,k=1}^n\partial_j a_{j,k}\partial_k u. $$ The (nonlocal) Robin-type boundary conditions are then of the form $$ ν\cdot A\nabla u + Θ\big[u\big|_{\partial\Om}\big]=0 \, \text{on} \, \partialΩ, $$ where $Θ$ represents an appropriate operator actin…
Well-Posedness and Regularity of the Heat Equation with Robin Boundary Conditions in the Two-Dimensional Wedge
本文研究了二维楔形中带有Robin边界条件的热方程的适定性与正则性，通过构建数学框架实现了任意高正则性，无需假设楔形开口角的大小。
We investigate the heat kernel with Robin boundary condition and prove comparison theorems for heat kernel on geodesic balls and on minimal submanifolds. We also prove an eigenvalue comparison theorem for the first Robin eigenvalues on minimal submanifolds. This generalizes corresponding results for the Dirichlet and Neumann heat kernels.
Some remarks on optimal insulation with Robin boundary conditions
本文研究由对流热传递驱动的Robin边值问题最优隔热模型，证明当对流换热系数足够大且隔热总量足够小时，以椭圆算子第一特征值为目标的优化问题会发生对称性破缺。
We study the asymptotic behavior of the heat content on a compact Riemannian manifold with boundary and with singular specific heat and singular initial temperature distributions imposing Robin boundary conditions. Assuming the existence of a complete asymptotic series we determine the first three terms in that series. In addition to the general setting, the interval is studied in detail as are recursion relations among the coefficients and the relationship between the Dirichlet and Robin settings.
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.
For a heat equation with Robin's boundary conditions which depends on a parameter $α>0$, we prove that its unique weak solution $ρ^α$ converges, when $α$ goes to zero or to infinity, to the unique weak solution of the heat equation with Neumann's boundary conditions or the heat equation with periodic boundary conditions, respectively. To this end, we use uniform bounds on a Sobolev norm of $ρ^α$ obtained from the hydrodynamic limit of the symmetric slowed exclusion process, plus a careful analysis of boundary terms.
Non-Positivity of the heat equation with non-local Robin boundary conditions
本文研究非局部罗宾边界条件下热方程的非正性问题，证明在特定算子B下半群具有超收缩性并最终正。
In this paper, a one-phase Stefan-type problem for a semi-infinite material which has as its main feature a variable latent heat that depends on the power of the position and the velocity of the moving boundary is studied. Exact solutions of similarity type are obtained for the cases when Neumann or Robin boundary conditions are imposed at the fixed face. Required relationships between data are presented in order that these problems become equivalent to the problem where a Dirichlet condition at the fixed face is considered. Moreover, in the case where a Robin condition is prescribed, the limi…
Time-dependent Robin heat equation via Markovian switching
本文研究了带有罗宾边界条件的热方程，其中反应参数通过连续时间马尔可夫链建模。通过泛函分析方法，分析了两种随机框架下的系统，推导了费曼-科茨公式并证明了快速切换极限下的平均原理。
We consider in this paper a reaction-diffusion system under a KPP hypothesis in a cylindrical domain in the presence of a shear flow. Such systems arise in predator-prey models as well as in combustion models with heat losses. Similarly to the single equation case, the existence of a minimal speed c* and of traveling front solutions for every speed c > c* has been shown both in the cases of heat losses distributed inside the domain or on the boundary. Here, we deal with the accordance between the two models by choosing heat losses inside the domain which tend to a Dirac mass located on the bou…
We consider a family of initial boundary value problems governed by a fractional diffusion equation with Caputo derivative in time, where the parameter is the Newton heat transfer coefficient linked to the Robin condition on the boundary. For each problem we prove existence and uniqueness of solution by a Fourier approach. This will enable us to also prove the convergence of the family of solutions to the solution of the limit problem, which is obtained by replacing the Robin boundary condition with a Dirichlet boundary condition.
Active thermography is a non-destructive testing technique to detect the internal structure of a heat conductor, which is widely applied in industrial engineering. In this paper, we consider the problem of identifying an unknown cavity with Robin boundary condition inside a heat conductor from boundary measurements. To set up the inverse problem mathematically, we first state the corresponding forward problem and show its well-posedness in an anisotropic Sobolev space by the integral equation method. Then, taking the Neumann-to-Dirichlet map as mathematically idealized measured data for the ac…
We study thermal insulating of a bounded body $Ω\subset \mathbb{R}^n$. Under a prescribed heat source $f\geq 0$, we consider a model of heat transfer between $Ω$ and the environment determined by convection; this corresponds, before insulation, to Robin boundary conditions. The body is then surrounded by a layer of insulating material of thickness of size $\varepsilon>0$, and whose conductivity is also proportional to $\varepsilon$. This corresponds to the case of a small amount of insulating material, with excellent insulating properties. We then compute the $Γ$-limit of the energy functional…
The classical half line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $bq(0,t)+q_x(0,t)=0$ is replaced with a dynamic Robin condition; $b=b(t)$ is allowed to vary in time. We present a solution representation, and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this…
One of the principal topics of this paper concerns the realization of self-adjoint operators $L_{Θ, \Om}$ in $L^2(\Om; d^n x)^m$, $m, n \in \bbN$, associated with divergence form elliptic partial differential expressions $L$ with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains $\Om \subset \bbR^n$. In particular, we develop the theory in the vector-valued case and hence focus on matrix-valued differential expressions $L$ which act as $$ Lu = - \biggl(\sum_{j,k=1}^n\partial_j\bigg(\sum_{β= 1}^m a^{α,β}_{j,k}\partial_k u_β\bigg) \bigg)_{1\leqα\leq m}, \quad u=(u_1,...,u_m)…
Ultracontractivity of Heat semigroups in $\mathrm{L}^{2}\left( Ω\right)$ with non-local Robin boundary conditions using Nash's inequality
研究有界 Lipschitz 域上带一般 Robin 边界条件的热方程，通过 Nash 不等式证明解半群的超压缩性。
Ultracontractivity of heat semigroups with non-local Robin boundary conditions via two-sided bounds for a positive eigenfunction
该研究在非局部Robin边界条件下，通过构造控制椭圆算子的正特征函数并得到其双侧界，结合Nash不等式与对偶性，证明了高维有界Lipschitz区域上热半群的超收缩性并给出短时间范数阶。
This is a note on a recent paper of De Simoi-Kaloshin-Wei \cite{DKW}. We show that using their results combined with wave trace invariants of Guillemin-Melrose and the heat trace invariants of Zayed for the Laplacian with Robin boundary conditions, one can extend the Dirichlet/Neumann spectral rigidity results of \cite{DKW} to the case of Robin boundary conditions. We will consider the same generic subset as in \cite{DKW} of smooth strictly convex $\mathbb Z_2$-symmetric planar domains sufficiently close to a circle, however we pair them with arbitrary $\mathbb Z_2$-symmetric smooth Robin func…
We consider the inhomogeneous heat equation on the half-space $\mathbb R_{+}^{d}$ with Neumann boundary conditions. We prove a space-time Gevrey regularity of the solution, with a radius of analyticity uniform up to the boundary of the half-space. We also address the case of homogeneous Robin boundary conditions. Our results generalize the case of homogeneous Dirichlet boundary conditions established by Kukavica and Vicol in [10].
We study the thermal insulation of a bounded body $Ω\subset\mathbb{R}^n$, under a prescribed heat source $f>0$, via a bulk layer of insulating material. We consider a model of heat transfer between the insulated body and the environment determined by convection; this corresponds to Robin boundary conditions on the free boundary of the layer. We show that a minimal configuration exists and that it satisfies uniform density estimates.
We study the heat content function, the heat trace function, and questions of isospectrality for the Laplacian with Dirichlet boundary conditions on a compact manifold with smooth boundary in the context of finite coverings and warped products.
The vector-valued Allen-Cahn equation with potentials of high-dimensional double-wells under Robin boundary conditions
本文研究了Robin边界条件下高维双井势的向量值Allen-Cahn方程，证明了解向具有固定接触角的平均曲率流的局部时间收敛性，并推导出极限锐界面系统，推广了无边界效应向量值与带边界能标量值的已有分析。
We show in the smooth category that the heat trace asymptotics and the heat content asymptotics can be made to grow arbitrarily rapidly. In the real analytic context, however, this is not true and we establish universal bounds on their growth.
Let (M,g) be a compact Riemannian manifold without boundary. Let D be a compact subdomain of M with smooth boundary. We examine the heat content asymptotics for the heat flow from D into M where both the initial temperature and the specific heat are permitted to have controlled singularities on the boundary of D. The operator driving the heat process is assumed to be an operator of Laplace type
Qualitative properties of the heat content
本文研究黎曼流形与欧几里得空间中热含量的单调性与凸性，引入严格递减温度集作为单调性的充分条件，并构造反例说明非单调及单调非凸情形。
We study the asymptotic behaviour of the heat content on a compact Riemannian manifold with boundary and with singular specific heat and singular initial temperature distributions. Assuming the existence of a complete asymptotic series we determine the first three terms in that series. In addition to the general setting, the interval is studied in detail.
Let $D$ be a bounded, connected, open set in Euclidean space $\mathbb{R}^{2}$ with polygonal boundary. Suppose $D$ has initial temperature $1$ and the complement of $D$ has initial temperature $0$. We obtain the asymptotic behaviour of the heat content of $D$ as time $t \downarrow 0$. We then apply this result to compute the heat content of a particular fractal polyhedron as $t \downarrow 0$.
We obtain (i) lower and upper bounds for the heat content of an open set in $\mathbb{R}^m$ with $R$-smooth boundary and finite Lebesgue measure, (ii) a necessary and sufficient geometric condition for finiteness of the heat content in $\mathbb{R}^m$, and corresponding lower and upper bounds, (iii) lower and upper bounds for the heat loss of an open set in $\mathbb{R}^m$ with finite Lebesgue measure.
Let $Ω$ be an open set in a complete, smooth, non-compact, $m$-dimensional Riemannian manifold $M$ without boundary, where $M$ satisfies a two-sided Li-Yau gaussian heat kernel bound. It is shown that if $Ω$ has infinite measure, and if $Ω$ has finite heat content $H_Ω(T)$ for some $T>0$, then $H_Ω(t)<\infty$ for all $t>0$. Comparable two-sided bounds for $H_Ω(t)$ are obtained for such $Ω$.
The heat kernel on curvilinear polygonal domains in surfaces
本文构造了曲面上曲线多边形区域在多种边界条件下的热核，通过热迹的短时渐近展开证明角点是谱不变量。
In this short note, we study the local well-posedness of a 3D model for incompressible Navier-Stokes equations with partial viscosity. This model was originally proposed by Hou-Lei in \cite{HouLei09a}. In a recent paper, we prove that this 3D model with partial viscosity will develop a finite time singularity for a class of initial condition using a mixed Dirichlet Robin boundary condition. The local well-posedness analysis of this initial boundary value problem is more subtle than the corresponding well-posedness analysis using a standard boundary condition because the Robin boundary conditio…
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。
Optimal insulation and concentration breaking for nonlinear Robin boundary value problems
针对p-Laplace算子驱动的有界域，通过Γ-收敛分析薄层绝缘材料的能量泛函，研究固定总质量下热含量的优化，并揭示边界连通性不足或温度分布恒定导致绝缘层无法完全覆盖边界的浓度破碎现象。
The hybrid spectral problem where the field satisfies Dirichlet conditions (D) on part of the boundary of the relevant domain and Neumann (N) on the remainder is discussed in simple terms. A conjecture for the C_1 coefficient is presented and the conformal determinant on a 2-disc, where the D and N regions are semi-circles, is derived. Comments on higher coefficients are made. A hemisphere hybrid problem is introduced that involves Robin boundary conditions and leads to logarithmic terms in the heat--kernel expansion which are evaluated explicitly.
Convexity of inverse spectral Green kernels and nondegeneracy of Robin centers in convex domains
本文证明凸域上逆谱格林核的凸性，从而对满足N>2s的每个s>0建立唯一Robin中心，并进一步证明Navier多调和情形下该中心的非退化性。
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。
Heat content asymptotics for sets with positive reach
本文研究欧氏空间中正可达有界集的热含量短时渐近性，拓展了光滑边界集合的研究范畴，采用与已有研究完全不同的技术，得到了略有差异的结果。
In this short paper, we derive an alternative proof for some known [van den Berg & Gilkey 2015] short-time asymptotics of the heat content in compact full-dimensional submanifolds $S$ with smooth boundary. This includes formulae like \begin{equation*} \int_{S} \exp(tΔ)\left( f \mathbb 1_S\right)\, \mathrm{d}x = \int_S f \,\mathrm{d}x - \sqrt{\frac{t}π} \int_{\partial S} f \,\mathrm{d}A + o(\sqrt t),\quad t \rightarrow 0\,. \end{equation*} and (partially new) explicit expressions for similar expansions involving other powers of $\sqrt t$. By the same method, we also obtain short-time asymptotic…
We study a shape optimization problem involving a solid $K\subset\mathbb{R}^n$ that is maintained at constant temperature and is enveloped by a layer of insulating material $Ω$ which obeys a generalized boundary heat transfer law. We minimize the energy of such configurations among all $(K,Ω)$ with prescribed measure for $K$ and $Ω$, and no topological or geometrical constraints. In the convection case (corresponding to Robin boundary conditions on $\partialΩ$) we obtain a full description of minimizers, while for general heat transfer conditions, we prove the existence and regularity of solut…
We discuss the heat content asymptotics associated with the heat flow out of a smooth compact manifold in a larger compact Riemannian manifold. Although there are no boundary conditions, the corresponding heat content asymptotics involve terms localized on the boundary. The classical pseudo-differential calculus is used to establish the existence of the complete asymptotic series and methods of invariance theory are used to determine the first few terms in the asymptotic series in terms of geometric data. The operator driving the heat process is assumed to be an operator of Laplace type.
Existence result for a nonlinear mixed boundary value problem for the heat equation
本文借助 Leray Schauder 不动点定理，证明了穿孔域中热方程在 Neumann 与非线性 Robin 混合边界条件下，于抛物 Schauder 空间中至少存在一个解。
A recursive method for computing singular solutions in corners with homogeneous Dirichlet-Robin boundary condition with power-law coefficient variation
提出一种递归方法计算角点域中具有幂律系数变化的齐次 Dirichlet-Robin 边界条件 Laplace 方程渐近解，证明递归过程收敛性并给出临界情况闭式解，可应用于断裂力学及热传导等工程问题。
Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions
本文提出了一种基于热半群形式的积分公式，用于在有限域上离散化谱分数拉普拉斯算子，结合适合的积分四则公式和有限元方法，无需计算拉普拉斯算子的本征对，能处理不同边界条件，通过数值模拟验证了方法的有效性。
Heat content and spectrum for subordinated sub-Laplacians
该研究针对任意 Carnot 群上次拉普拉斯从属算子，结合多类方法推导其特征值估计、热含量渐近行为等谱性质，完善了相关算子的分析理论。
Observability inequality, log-type Hausdorff content and heat equations
本文研究有界域与全空间热方程的能观测性不等式，以对数型 Hausdorff 容度刻画观测集，并借助谱不等式、Remez 不等式及容度工具证明其成立条件与一维最优尺度。
Heat content and horizontal mean curvature on the Heisenberg group
本文研究第一Heisenberg群中光滑有界区域的次Riemann热含量短时间渐近，利用Brown运动给出概率证明，并以水平周长和总水平平均曲率刻画其前几个系数。
Let $Ω$ be an open set in a geodesically complete, non-compact, $m$-dimen-sional Riemannian manifold $M$ with non-negative Ricci curvature, and without boundary. We study the heat flow from $Ω$ into $M-Ω$ if the initial temperature distribution is the characteristic function of $Ω$. We obtain a necessary and sufficient condition which ensures that an open set $Ω$ with infinite measure has finite heat content for all $t>0$. We also obtain upper and lower bounds for the heat content of $Ω$ in $M$. Two-sided bounds are obtained for the heat loss of $Ω$ in $M$ if the measure of $Ω$ is finite.
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。
An inverse problem for a stationary heat transfer process is studied for a totally isolated bar on its lateral surface, of negligible diameter, made up of two consecutive sections of different, isotropic and homogeneous materials. At the left boundary, a Dirichlet type condition is imposed that represents a constant temperature source while a Robin type condition that models the heat dissipation by convection is considered at the right one. Many articles in the literature focus on thermal and stress analysis at the interface but no one is dedicated to the estimation of the contact point locati…
Shape perturbation of a nonlinear mixed problem for the heat equation
该研究针对带孔洞区域的热方程非线性混合问题，分析孔洞形状扰动对解的影响，证明近恒等映射下解的存在性、映射的$C^\infty$光滑性及局部唯一性，所用工具为位势论与隐函数定理。
We study and classify smooth bounded domains in an analytic Riemannian manifold which are critical for the heat content at all times t>0. We do that by first computing the first variation of the heat content, and then showing that a domain is critical if and only if it has the so-called constant flow property, so that we can use a previous classification result established by the author. The outcome is that a domain is critical for the heat content at all times if and only if it admits an isoparametric foliation, that is, a foliation whose leaves are all parallel to the boundary and have const…
We consider sub-Laplacians in open bounded sets in a homogeneous Carnot group and study their spectral properties. We prove that these operators have a pure point spectrum, and prove the existence of the spectral gap. In addition, we give applications to the small ball problem for a hypoelliptic Brownian motion and the large time behavior of the heat content in a regular domain.
Beyond Three Terms: Exponential Bounds in the Neumann Heat Trace of Polygons
本文通过建立角点、边和内部的热核局部性原理，用模型热核逼近多边形上的热核，给出了诺伊曼热迹渐近展开中指数小余项的显式估计，并讨论了该指数的锐度及与已知结果的关系，最后探讨了推广到Robin边界条件的问题。
We obtain upper bounds on the heat content and on the torsional rigidity of a complete Riemannian manifold M, assuming a generalized Hardy inequality for the Dirichlet Laplacian on M.
In this paper, we prove first-order asymptotics on a bounded open set of the heat content when the ambient space is an ${\sf RCD}(K,N)$ space, under a regularity condition for the boundary that we call measured interior geodesic condition of size $ε$. We carefully study such a condition, relating it to the properties of the disintegration of the signed distance function from $\partial Ω$ studied by Cavalletti and Mondino.
Let $\mathbf{X}=\{X_t\}_{t\geq 0}$ be a Lévy process in $\mathbb{R}^d$ and $Ω$ be an open subset of $\mathbb{R}^d$ with finite Lebesgue measure. In this article we consider the quantity $H(t)=\int_Ω \mathbb{P}^x (X_t\inΩ^c) \, \mathrm{d}x$ which is called the heat content. We study its asymptotic expansion for isotropic $α$-stable Lévy processes and more general Lévy processes, under mild assumptions on the characteristic exponent.
On the heat content of compact quantum graphs
本文研究紧有限度量图上带边界Dirichlet条件拉普拉斯算子的热含量，证明了组合性质的闭合公式，给出小时间渐近展开并得到新手术原理与比较原理，公式对所有t>0成立且短时间衰减率更强。
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