Boundary integral solution of the two-dimensional heat equation

Here we consider initial boundary value problems for the heat equation by using the heat potential representation for the solution. Depending on the choice of the representation we are led to a solution of the various boundary integral equations. We discuss the solvability of these equations in anisotropic Sobolev spaces. It turns out that the double-layer heat potential D and its spatial adjoint D′ have smoothing properties similar to the single-layer heat operator. This yields compactness of the operators D and D′. In addition, for any constant c ≠ 0, cI + D′ and cI + D′ are isomorphisms. Based on the coercivity of the single-layer heat operator and the above compactness we establish the coerciveness of the hypersingular heat operator. Moreover, we show an equivalence between the weak solution and the various boundary integral solutions. As a further application we describe a coupling procedure for an exterior initial boundary value problem for the non-homogeneous heat equation.     

Explicit relation between the solutions of the heat and the Hermite heat equation

There are lots of results on the solutions of the heat equation \frac?u?t = \mathop?ⁿ_i=1\frac?²?x²_iu,\frac{\partial u}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\frac{\partial^2}{\partial x^{2}_{i}}u, but much less on those of the Hermite heat equation \frac?U?t = \mathop?ⁿ_i=1(\frac?²?x²_i - x²_i) U\frac{\partial U}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\left(\frac{\partial^2}{\partial x^{2}_{i}} - x^{2}_{i}\right) U due to that its coefficients are not constant and even not bounded. In this paper, we find an explicit relation between the solutions of these two equations, thus all known results on the heat equation can be transferred to results on the Hermite heat equation, which should be a completely new idea to study the Hermite equation. Some examples are given to show that known results on the Hermite equation are obtained easily by this method, even improved. There is also a new uniqueness theorem with a very general condition for the Hermite equation, which answers a question in a paper in Proc. Japan Acad. (2005).     

On the solution of heat conduction problems for thermosensitive bodies heated by convective heat exchange

We propose a method of solving heat conduction problems for thermosensitive bodies, in particular for a hollow cylinder from whose surfaces a convective heat exchange with the external environment occurs.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 83–86.     

On the affine heat equation for non-convex curves

In this paper, we extend to the non-convex case the affine invariant geometric heat equation studied by Sapiro and Tannenbaum for convex plane curves. We prove that a smooth embedded plane curve will converge to a point when evolving according to this flow. This result extends the analogy between the affine heat equation and the well-known Euclidean geometric heat equation.

     

A fast algorithm for the evaluation of heat potentials

Numerical methods for solving the heat equation via potential theory have been hampered by the high cost of evaluating heat potentials. When M points are used in the discretization of the boundary and N time steps are computed, an amount of work of the order O(N²M²) has traditionally been required. In this paper, we present an algorithm which requires an amount of work of the order O(NM), and we observe speedups of five orders of magnitude for large-scale problems. Thus, the method makes it possible to solve the heat equation by potential theory in practical situations.     

Obtaining upper bounds of heat kernels from lower bounds

We show that a near‐diagonal lower bound of the heat kernel of a Dirichlet form on a metric measure space with a regular measure implies an on‐diagonal upper bound. If in addition the Dirichlet form is local and regular, then we obtain a full off‐diagonal upper bound of the heat kernel provided the Dirichlet heat kernel on any ball satisfies a near‐diagonal lower estimate. This reveals a new phenomenon in the relationship between the lower and upper bounds of the heat kernel. © 2007 Wiley Periodicals, Inc.     

On the unsolvability of the boundary inverse problem of heat conduction

It is shown that the boundary inverse nonstationary problem of heat conduction with given heat flux and temperature on one of the boundary surfaces is essentially ill-posed, since it has no solution in the class of continuous functions.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 114–118.     

Asymptotic of the heat kernel in general Benedicks domains

 Using a new inequality relating the heat kernel and the probability of survival, we prove asymptotic ratio limit theorems for the heat kernel (and survival probability) in general Benedicks domains. In particular, the dimension of the cone of positive harmonic measures with Dirichlet boundary conditions can be derived from the rate of convergence to zero of the heat kernel (or the survival probability). Received: 31 March 2002 / Revised version: 12 August 2002 / Published online: 19 December 2002 Mathematics Subject Classification (2000): 60J65, 31B05 Key words or phrases: Positive harmonic functions – Ratio limit theorems – Survival probability     

The spatial critical points not moving along the heat flow

We consider solutions of the heat equation, in domains inR ^N, and their spatial critical points. In particular, we show that a solutionu has a spatial critical point not moving along the heat flow if and only ifu satisfies some balance law. Furthermore, in the case of Dirichlet, Neumann, and Robin homogeneous initial-boundary value problems on bounded domains, we prove that if the origin is a spatial critical point never moving for sufficiently many compactly supported initial data satisfying the balance law with respect to the origin, then the domain must be a ball centered at the origin.     

The stress-strain state of shells under the action of concentrated heat sources

We construct a fundamental solution of a static problem of heat conduction for thin isotropic shells of positive curvature under the action of concentrated heat sources. We study the problems involved in simplifying the heat equations. Four figures. Two tables. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 70–80.     

Approximation of the conductivity coefficient in the heat equation

In this paper, we study the conductivity coefficient determination in the heat equation from observation of the lateral Dirichlet-to-Neumann map. We define a bilinear form function Q_γ associated to the boundary condition and the Dirichlet-to-Neumann map, and prove that the linearized problem d?Q_γ is injective. Based on the idea of complex geometrical optics solutions, we give two approximations to the conductivity coefficient by using the Fourier truncation method and the mollification method. Under the a priori assumption of the conductivity, we estimate the errors between the conductivity coefficient and its approximations by setting a suitable bound of the frequency.     

Mathematical modeling and the study of the heat exchange process in color kinescopes

We propose a mathematical model for computing the heat exchange in a masked color kinescope taking account of the specific geometry of its shell and the actual three-dimensional character of the heat emission in the mask. We study the temperature field in the mask and determine the size of the integral degree of blackness in the cone coating that guarantees a stable operation of the kinescope. Three figures. Bibliography: 7 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 57–63.     

The heat equation under conditions on the moments in higher dimensions

We consider the heat equation on the N‐dimensional cube (0, 1)^N and impose different classes of integral conditions, instead of usual boundary ones. Well‐posedness results for the heat equation under the condition that the moments of order 0 and 1 are conserved had been known so far only in the case of , for which such conditions can be easily interpreted as conservation of mass and barycenter. In this paper we show that in the case of general N the heat equation with such integral conditions is still well‐posed, upon suitably relaxing the notion of solution. Existence of solutions with general initial data in a suitable space of distributions over (0, 1)^N are proved by introducing two appropriate realizations of the Laplacian and checking by form methods that they generate analytic semigroups. The solution thus obtained turns out to solve the heat equation only in a certain distributional sense. However, one of these realizations is tightly related to a well‐known object of operator theory, the Krein–von Neumann extension of the Laplacian. This connection also establishes well‐posedness in a classical sense, as long as the initial data are L²‐functions.     

Volume growth and heat kernel estimates for the continuum random tree

In this article, we prove global and local (point-wise) volume and heat kernel bounds for the continuum random tree. We demonstrate that there are almost–surely logarithmic global fluctuations and log–logarithmic local fluctuations in the volume of balls of radius r about the leading order polynomial term as r → 0. We also show that the on-diagonal part of the heat kernel exhibits corresponding global and local fluctuations as t → 0 almost–surely. Finally, we prove that this quenched (almost–sure) behaviour contrasts with the local annealed (averaged over all realisations of the tree) volume and heat kernel behaviour, which is smooth.      

Nonlinear stochastic wave and heat equations

We study nonlinear wave and heat equations on ℝ ^d driven by a spatially homogeneous Wiener process. For the wave equation we consider the cases of d = 1, 2, 3. The heat equation is considered on an arbitrary ℝ ^d -space. We give necessary and sufficient conditions for the existence of a function-valued solution in terms of the covariance kernel of the noise. Received: 1 April 1998 / Revised version: 23 June 1999 / Published online: 7 February 2000     

Nonlinear heat equation with L 1-data

A nonlinear heat equation with Newton-type boundary conditions with all heat sources being bounded only in L ¹-norms is investigated. An integral solution is then defined and its existence, uniqueness and continuous dependence is proved by using accretivity of the stationary part in L ¹ () L ¹ (). Received September 1, 1997     

A Pizzetti-type formula for the heat operator

We provide an asymptotic expansion of the integral mean of a smooth function over the Heat ball. Namely we generalize to the Heat operator the so-called Pizzetti’s Formula, which expresses the integral mean of a smooth function over an Euclidean ball in terms of a power series with respect to the radius of the ball having the iterated of the ordinary Laplace operator as coefficients. Similarly here, we express the heat integral mean as a power series with respect to the radius of the heat ball, whose coefficients are powers of a distorted heat operator. We also discuss sufficient conditions to have a finite sum. Received: 27 May 2005     

Determination of the characteristics of concentrated heat sources in a half-space under prescribed conditions for the heat exchange and temperature distribution on its surface

A method is given for solving the inverse problem for determining the intensity and location of concentrated heat sources in a half-space under prescribed conditions on the heat exchange with the external medium and a distribution of surface temperature with error. Using methods of the variational calculus, we obtain a system of nonlinear equations for the characteristics of subsurface heat sources, leading to a temperature function on the surface which is closest to the given distribution. Several individual examples are solved.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 46–51, 1989.     

On the heat flow equation of surfaces of constant mean curvature in higher dimensions

In this paper, we consider the heat flow for the Hsystem with constant mean curvature in higher dimensions. We give sufficient conditions on the initial data such that the heat flow develops finite time singularity. We also provide a new set of initial data to guarantee the existence of global regular solution to the heat flow, that converges to zero in W 1,n with the decay rate t 2/(2-n) as time goes to infinity.     

Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation

In the first part of this paper, we get new Li–Yau type gradient estimates for positive solutions of heat equation on Riemannian manifolds with Ricci(M)?−k, k∈R. As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with Ricci curvature bounded from below. In the second part, we establish a Perelman type Li–Yau–Hamilton differential Harnack inequality for heat kernels on manifolds with Ricci(M)?−k, which generalizes a result of L. Ni (2004, 2006) [20] and [21]. As applications, we obtain new Harnack inequalities and heat kernel estimates on general manifolds. We also obtain various entropy monotonicity formulas for all compact Riemannian manifolds.