On Carleman-type Formulas for Solutions to the Heat Equation

Abstract

We apply the method of integral representations to study the ill-posed Cauchy problem for the heat equa- tion. More precisely we recover a function, satisfying the heat equation in a cylindrical domain, via its values and the values of its normal derivative on a given part of the lateral surface of the cylinder. We prove that the problem is ill-posed in the natural (anisotropic) spaces (Sobolev and H¨older spaces, etc). Finally, we obtain a uniqueness theorem for the problem and a criterion of its solvability and a Carleman-type formula for its solution

On approximation of solutions to the heat equation from Lebesgue class $L^2$ by more regular solutions

“…However, it is not easy to construct an example of a basis with the double orthogonality property, provided by theorem 2.1. A non-complete double orthogonal countable (trigonometric) system was constructed in [19] for cubes ω and Ω in R n if their centres coincide and the ratio of their edges equals to two. Let us indicate one more example; it is related to the case where the cylinder bases of ω T1,T2 , Ω T1,T2 are concentric balls in R n .…”

“…in a cylinder domain R n × R with the Cauchy data on a part of its lateral surface (that naturally arises in the diffusion problems, for instance in the inverse problem of the electrocardiography using models of the charge diffusion in the heart tissues) can be also reduced to the continuation problem for solutions of the heat equations from a lesser cylinder domain to a bigger one, see [18], [19]. Since the solutions to the heat equation are real analytic with respect to the space variables (see, for instance, [20, ch.…”

On the uniqueness theorems for transmissions problems related to models of elasticity, diffusion and electrocardiography

“…Therefore it is identically zero for each t > 0 and then w ≡ 0 in Ω T , cf. [14], [21] for the similar uniqueness theorem related to the heat equation and the parabolic Lamé type systems. Finally, we see that (w i , w e , w b ) = 0 because of (3.21).…”

“…with some data g 0 , g 1 and (possibly, non-linear) term F . Even in the case where F is linear with respect to u e problem (3.28) might be ill-posed in some cases, cf., [14], [21]. Thus, for both linear and the non-linear case, a thorough investigation of (3.28) is necessary.…”

Замечание О Преобразовании Лапласа

“…Явные формулы Карлемана для решения задачи Коши для уравнения теплопроводности в определенных областях X n-мерного евклидова пространства можно найти в [16,18].…”